Apparatus for operating double vector and encrypting system including the same

Abstract

There is provided an apparatus for summing bivectors, e.g. double vectors (alternatively referred to as bivectors) each having a plurality of pairs of elements selected from a predetermined finite field, the apparatus including (a) a device for inputting bivectors X1 and X2, and parameters A (e.g., a3 to a13) for defining a curve therethrough, (b) a first memory for storing the bivector X1 therein, (c) a second memory for storing the bivector X2 therein, (d) a third memory for storing the parameters A therein, and (e) a device for reading the bivectors X1 and X2, and the parameters A out of the first, second and third memories, respectively, and, when the bivectors X1 and X2 are supposed to be coordinate value rows of points in point-sets Q1 and Q2 on the curve defined with the parameters A, operating a bivector X3 comprised of coordinate value row of points in a point-set Q3 equal to a sum of the point-sets Q1 and Q2 in Jacobian group of the curve defined with the parameters A. The above-mentioned apparatus may be used for a public key encryption system in a smaller size.

Claims

What is claimed is:

1. An encryption device that encrypts messages by summing double vectors, the encryption device including an apparatus for summing double vectors each comprising a plurality of pairs of elements selected from a predetermined finite field, said apparatus comprising:

(a) means for inputting double vectors X1 and X2, and parameters A defining a curve therethrough, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(b) first storage means for storing said double vector X1 therein;

(c) second storage means for storing said double vector X2 therein;

(d) third storage means for storing said parameters A therein; and

(e) means for reading said double vectors X1 and X2, and said parameters A out of said first, second and third storage means, respectively, wherein said double vectors X1 and X2 represent coordinate value rows of points in respective point-sets Q1 and Q2 on said curve defined with said parameters A, operating a double vector X3 comprised of coordinate value row of points in a point-set Q3 equal to a sum of said point-sets Q1 and Q2 in Jacobian group of said curve defined with said parameters A.

2. An encryption device that encrypts messages by summing double vectors, the encryption device including an apparatus for summing double vectors each comprising a plurality of pairs of elements selected from a predetermined finite field, said apparatus comprising:

(a) means for inputting double vectors X1 and X2, and parameters A defining a curve therethrough, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(b) first storage means for storing said double vector X1 therein;

(c) second storage means for storing said double vector X2 therein;

(d) third storage means for storing said parameters A therein;

(e) a union-set operating device for reading said double vectors X1 and X2, and said parameters A out of said first, second and third storage means, respectively, wherein said double vectors X1 and X2 represent respective coordinate value rows of points in point-sets on said curve defined with said parameters A, operating a double vector T1 comprised of coordinate value row of points in a union-set of said point-sets indicated by said double vectors X1 and X2;

(f) fourth storage means for storing said double vector T1 operated by said union-set operating device;

(g) a first point-set operating device for reading said double vector T1 out of said fourth storage means, and said parameters A out of said third storage means, wherein said double vector T1 represents coordinate value row of points on said curve defined with said parameters A, operating a double vector T2 comprised of coordinate value row of points in a point-set indicative of the inverse of said point-set indicated by said double vector T1 in Jacobian group of said curve defined with said parameters A;

(h) fifth storage means for storing said double vector T2 operated by said first point-set operating device;

(i) a second point-set operating device for reading said double vector T2 out of said fifth storage means, and said parameters A out of said third storage means, wherein said double vector T2 represents coordinate value row of points on said curve defined with said parameters A, operating a double vector X3 comprised of coordinate value row of points in a point-set indicative of the inverse of said point-set indicated by said double vector T2 in Jacobian group of said curve defined with said parameters A; and

(j) means for outputting said double vector X3 operated by said second point-set operating device.

3. The encryption device as set forth in claim 2, wherein said apparatus further comprises:

(k) sixth storage means for storing parameters B therein; and

(l) seventh storage means for storing a double vector S1 therein, and wherein said first point-set operating device comprising:

(g-1) a common curve operating device for reading said double vector T1 out of said fourth storage means, and said parameters A out of said third storage means, wherein said double vector T1 represents coordinate value row of points on said curve defined with said parameters A, operating parameters B of a curve passing through all points constituting said point-set expressed by said double vector T1;

(g-2) an intersection-set operating device for reading said parameters B out of said sixth storage means, and said parameters A out of said third storage means, and operating a double vector S1 comprised of coordinate value row of points in an intersection between a curve defined with said parameters A and a curve defined with said parameters B;

(g-3) a difference-set operating device for reading said double vector T1 out of said fourth storage means, and said double vector S1 out of said seventh storage means, wherein said double vectors T1 and S1 represent respective coordinate value rows of points on said curve defined with said parameters A, operating said double vector T2 comprised of coordinate value row of points in a point-set obtained by subtracting a point-set indicated by said double vector T1 from a point-set indicated by said double vector S1; and

(g-4) means for outputting said double vector T2 operated by said difference-set operating device.

4. The encryption device as set forth in claim 2, wherein said apparatus further comprises:

(k) eighth storage means for storing parameters C therein; and

(l) ninth storage means for storing a double vector S2 therein, and wherein said second point-set operating device comprising:

(i-1) a common curve operating device for reading said double vector T2 out of said fifth storage means, and said parameters A out of said third storage means, wherein said double vector T2 represents coordinate value row of points on said curve defined with said parameters A, operating parameters C of a curve passing through all points indicated by said double vector T2;

(i-2) an intersection-set operating device for reading said parameters C out of said eighth storage means, and said parameters A out of said third storage means, and operating a double vector S2 comprised of coordinate value row of points in an intersection between a curve defined with said parameters A and a curve defined with said parameters C;

(i-3) a difference-set operating device for reading said double vector T2 out of said 5 fifth storage means, and said double vector S2 out of said ninth storage means, wherein said double vectors T2 and S2 respective coordinate value rows of points on said curve defined with said parameters A, operating said double vector X3 comprised of coordinate value row of points in a point-set obtained by subtracting a point-set indicated by said double vector T2 from a point-set indicated by said double vector S2; and

(i-4) means for outputting said double vector X3 operated by said difference-set operating device.

5. The encryption device as set forth in claim 3, wherein the apparatus further comprises:

(k) eighth storage means for storing parameters C therein; and

(l) ninth storage means for storing a double vector S2 therein, and wherein said second point-set operating device comprising:

(i-1) a common curve operating device for reading said double vector T2 out of said fifth storage means, and said parameters A out of said third storage means, wherein said double vector T2 represents coordinate value row of points on said curve defined with said parameters A, operating parameters C of a curve passing through all points indicated by said double vector T2;

(i-2) an intersection-set operating device for reading said parameters C out of said eighth storage means, and said parameters A out of said third storage means, and operating a double vector S2 comprised of coordinate value row of points in an intersection between a curve defined with said parameters A and a curve defined with said parameters C;

(i-3) a difference-set operating device for reading said double vector T2 out of said fifth storage means, and said double vector S2 out of said ninth storage means, wherein said double vectors T2 and S2 represent coordinate respective value rows of points on said curve defined with said parameters A, operating said double vector X3 comprised of coordinate value row of points in a point-set obtained by subtracting a point-set indicated by said double vector T2 from a point-set indicated by said double vector S2; and

(i-4) means for outputting said double vector X3 operated by said difference-set operating device.

6. A recording medium readable by a computer, storing a program therein for causing a computer to act as an encryption device that encrypts messages by summing double vectors, the encryption device including an apparatus for summing double vectors each comprising a plurality of pairs of elements selected from a predetermined finite field, said apparatus comprising:

(a) means for inputting double vectors X1 and X2, and parameters A defining a curve therethrough, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(b) first storage means for storing said double vector X1 therein;

(c) second storage means for storing said double vector X2 therein;

(d) third storage means for storing said parameters A therein; and

(e) means for reading said double vectors X1 and X2, and said parameters A out of said first, second and third storage means, respectively, wherein said double vectors X1 and X2 represent respective coordinate value rows of points in point-sets Q1 and Q2 on said curve defined with said parameters A, operating a double vector X3 comprised of coordinate value row of points in a point-set Q3 equal to a sum of said point-sets Q1 and Q2 in Jacobian group of said curve defined with said parameters A.

7. A recording medium readable by a computer, storing a program therein for causing a computer to act as an encryption device that encrypts messages by summing double vectors, the encryption device including an apparatus for summing double vectors each comprising a plurality of pairs of elements selected from a predetermined finite field, said apparatus comprising:

(a) means for inputting double vectors X1 and X2, and parameters A defining a curve therethrough, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(b) first storage means for storing said double vector X1 therein;

(c) second storage means for storing said double vector X2 therein;

(d) third storage means for storing said parameters A therein; and

(e) a union-set operating device for reading said double vectors X1 and X2, and said parameters A out of said first, second and third storage means, respectively, wherein said double vectors X1 and X2 represent coordinate value rows of points on said curve defined with said parameters A, operating a double vector T1 comprised of coordinate value row of points in a union-set of said point-set indicated by said double vectors X1 and X2;

(f) fourth storage means for storing said double vector T1 operated by said union-set operating device;

(g) a first point-set operating device for reading said double vector T1 out of said fourth storage means, and said parameters A out of said third storage means, wherein said double vector T1 represents coordinate value row of points on said curve defined with said parameters A, operating a double vector T2 comprised of coordinate value row of points in a point-set indicative of the inverse of said point-set indicated by said double vector T1 in Jacobian group of said curve defined with said parameters A;

(h) fifth storage means for storing said double vector T2 operated by said first point-set operating device;

(i) a second point-set operating device for reading said double vector T2 out of said fifth storage means, and said parameters A out of said third storage means, wherein said double vector T2 represents coordinate value row of points on said curve defined with said parameters A, operating a double vector X3 comprised of coordinate value row of points in a point-set indicative of the inverse of said point-set indicated by said double vector T2 in Jacobian group of said curve defined with said parameters A; and

(j) means for outputting said double vector X3 operated by said second point-set operating device.

8. The recording medium as set forth in claim 7, wherein said apparatus further comprises:

(k) sixth storage means for storing parameters B therein; and

(l) seventh storage means for storing a double vector S1 therein, and wherein said first point-set operating device comprising:

(g-1) a common curve operating device for reading said double vector T1 out of said fourth storage means, and said parameters A out of said third storage means, wherein said double vector T1 represents coordinate value row of points on said curve defined with said parameters A, operating parameters B of a curve passing through all points constituting said point-set indicated by said double vector T1;

(g-2) an intersection-set operating device for reading said parameters B out of said sixth storage means, and said parameters A out of said third storage means, and operating a double vector S1 comprised of coordinate value row of points in an intersection between a curve defined with said parameters A and a curve defined with said parameters B;

(g-3) a difference-set operating device for reading said double vector T1 out of said fourth storage means, and said double vector S1 out of said seventh storage means, wherein said double vectors T1 and S1 represent respective coordinate value rows of points on said curve defined with said parameters A, operating said double vector T2 comprised of coordinate value row of points in a point-set obtained by subtracting a point-set indicated by said double vector T1 from a point-set indicated by said double vector S1; and

(g-4) means for outputting said double vector T2 operated by said difference-set operating device.

9. The recording medium as set forth in claim 7, wherein said apparatus further comprises:

(k) eighth storage means for storing parameters C therein; and

(l) ninth storage means for storing a double vector S2 therein, and wherein said second point-set operating device comprising:

(i-1) a common curve operating device for reading said double vector T2 out of said fifth storage means, and said parameters A out of said third storage means, wherein said double vector T2 represents coordinate value row of points on said curve defined with said parameters A, operating parameters C of a curve passing through all points constituting said point-set indicated by said double vector T2;

(i-2) an intersection-set operating device for reading said parameters C out of said eighth storage means, and said parameters A out of said third storage means, and operating a double vector S2 comprised of coordinate value row of points in an intersection between a curve defined with said parameters A and a curve defined with said parameters C;

(i-3) a difference-set operating device for reading said double vector T2 out of said fifth storage means, and said double vector S2 out of said ninth storage means, wherein said double vectors T2 and S2 represent respective coordinate value rows of points on said curve defined with said parameters A, operating said double vector X3 comprised of coordinate value row of points in a point-set obtained by subtracting a point-set indicated by said double vector T2 from a point-set indicated by said double vector S2; and

(i-4) means for outputting said double vector X3 operated by said difference-set operating device.

10. An encryption device that encrypts messages by doubling a double vector, the encryption device including an apparatus for doubling a double vector comprising a plurality of pairs of elements selected from a predetermined finite field, said apparatus comprising:

(a) means for inputting a double vector X, and parameters A for defining a curve therethrough, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(b) first storage means for storing said double vector X therein;

(c) second storage means for storing said parameters A therein; and

(d) means for reading said double vector X out of said first storage means, and said parameters A out of said second storage means, wherein said double vector X represents coordinate value row of points in a point-set Q on said curve defined with said parameters A, operating a double vector Y comprised of coordinate value row of points in a point-set R equal to a doubled Q in Jacobian group of said curve defined with said parameters A.

11. An encryption device that encrypts messages by doubling a double vector, the encryption device including an apparatus for doubling a double vector comprising a plurality of pairs of elements selected from a predetermined finite field, said apparatus comprising:

(a) means for inputting a double vector X, and parameters A for defining a curve therethrough, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(b) first storage means for storing said double vector X therein;

(c) second storage means for storing a double vector Xa which is a copy of said double vector X;

(d) third storage means for storing said parameters A therein;

(e) a double vector adding device for reading said double vector X out of said first storage means, said double vector Xa out of said second storage means, and said parameters A out of said third storage means, and adding said double vector X to said double vector Xa to thereby have a sum of 2X; and

(f) means for outputting said double vector 2X operated by said double vector adding device.

12. A recording medium readable by a computer, storing a program therein for causing a computer to act as an encryption device that encrypts messages by doubling a double vector, the encryption device including an apparatus for doubling a double vector comprising a plurality of pairs of elements selected from a predetermined finite field, said apparatus comprising:

(a) means for inputting a double vector X, and parameters A for defining a curve therethrough, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(b) first storage means for storing said double vector X therein;

(c) second storage means for storing said parameters A therein; and

(d) means for reading said double vector X out of said first storage means, and said parameters A out of said second storage means, wherein said double vector X represents coordinate value row of points in a point-set Q on said curve defined with said parameters A, operating a double vector Y comprised of coordinate value row of points in a point-set R equal to a doubled Q in Jacobian group of said curve defined with said parameters A.

13. A recording medium readable by a computer, storing a program therein for causing a computer to act as an encryption device that encrypts messages by doubling a double vector, the encryption device including an apparatus for doubling a double vector comprising a plurality of pairs of elements selected from a predetermined finite field, said apparatus comprising:

(a) means for inputting a double vector X, and parameters A for defining a curve therethrough, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(b) first storage means for storing said double vector X therein;

(c) second storage means for storing a double vector Xa which is a copy of said double vector X;

(d) third storage means for storing said parameters A therein;

(e) a double vector adding device for reading said double vector X out of said first storage means, said double vector Xa out of said second storage means, and said parameters A out of said third storage means, and adding said double vector X to said double vector Xa to thereby have a sum of 2X; and

(f) means for outputting said double vector 2X operated by said double vector adding device.

14. An encryption device that encrypts messages by multiplying a double vector by an integer, the encryption device including an apparatus for multiplying a double vector by an integer, said double vector comprising a plurality of pairs of elements selected from a predetermined finite field, said apparatus comprising:

(a) means for inputting therethrough an integer N, a double vector X, and parameters A for defining a curve, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(b) first storage means for storing said integer N therein;

(c) second storage means for storing said double vector X therein;

(d) third storage means for storing said parameters A therein; and

(e) means for reading said integer N out of said first storage means, said double vector X out of said second storage means, and said parameters A out of said third storage means, wherein said double vector X represents coordinate value row of points in a point-set Q on said curve defined with said parameters A, operating a double vector Z comprised of coordinate value row of points in a point-set R equal to said point-set Q multiplied by said integer N in Jacobian group of said curve defined with said parameters A.

15. An encryption device that encrypts messages by multiplying a double vector by an integer, the encryption device including an apparatus for multiplying a double vector by an integer, said double vector comprising a plurality of pairs of elements selected from a predetermined finite field, said apparatus comprising:

(a) means for inputting therethrough an integer N, a double vector X, and parameters A for defining a curve, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(b) first storage means for storing said integer N therein;

(c) second storage means for storing said double vector X therein;

(d) third storage means for storing a double vector Y which is a copy of said double vector X;

(e) fourth storage means for storing a double vector Z therein;

(f) fifth storage means for storing said parameters A therein;

(g) sixth storage means for storing an integer R therein;

(h) a double vector adding device for summing double vectors;

(i) a double vector doubling device for doubling a double vector;

(j) means for reading said integer N out of said first storage means, calculating a remainder R obtained when said integer N is divided by 2, and storing the obtained R in said sixth storage means;

(k) means for reading said integer N out of said first storage means, calculating a quotient by dividing said integer N by 2, and storing the obtained quotient in said first storage means as a renewed integer N;

(l) means for reading said integer R out of said sixth storage means, if said integer R is equal to 1, reading said double vector Y out of said third storage means, said double vector Z out of said fourth storage means, and said parameters A out of said fifth storage means, inputting said double vectors Y and Z and said parameters A into said double vector adding device, calculating a sum of said double vectors Y and Z, and storing the calculated sum in said fourth storage means, and

(m) means for reading said integer N out of said first storage means, if the read-out integer N is greater than 0, reading said double vector Y out of said third storage means and said parameters A out of said fifth storage means, inputting said double vector Y and said parameters A into said double vector doubling device, doubling said double vector Y, and storing the doubled double vector Y in said third storage means, and if said integer N is equal to 0, reading said double vector Z out of said fourth storage means.

16. A recording medium readable by a computer, storing a program therein for causing a computer to act as an encryption device that encrypts messages by multiplying a double vector by an integer, the encryption device including an apparatus for multiplying a double vector by an integer, said double vector comprising a plurality of pairs of elements selected from a predetermined finite field, said apparatus comprising:

(a) means for inputting therethrough an integer N, a double vector X, and parameters A for defining a curve, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(b) first storage means for storing said integer N therein;

(c) second storage means for storing said double vector X therein;

(d) third storage means for storing said parameters A therein; and

(e) means for reading said integer N out of said first storage means, said double vector X out of said second storage means, and said parameters A out of said third storage means, wherein said double vector X represents coordinate value row of points in a point-set Q on said curve defined with said parameters A, operating a double vector Z comprised of coordinate value row of points in a point-set R equal to said point-set Q multiplied by said integer N in Jacobian group of said curve defined with said parameters A.

17. A recording medium readable by a computer, storing a program therein for causing a computer to act as an encryption device that encrypts messages by multiplying a double vector by an integer, the encryption device including an apparatus for multiplying a double vector by an integer, said double vector comprising a plurality of pairs of elements selected from a predetermined finite field, said apparatus comprising:

(a) means for inputting therethrough an integer N, a double vector X, and parameters A for defining a curve, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(b) first storage means for storing said integer N therein;

(c) second storage means for storing said double vector X therein;

(d) third storage means for storing a double vector Y which is a copy of said double vector X;

(e) fourth storage means for storing a double vector Z therein;

(f) fifth storage means for storing said parameters A therein;

(g) sixth storage means for storing an integer R therein;

(h) a double vector adding device for summing double vectors;

(i) a double vector doubling device for doubling a double vector;

(j) means for reading said integer N out of said first storage means, calculating a remainder R obtained when said integer N is divided by 2, and storing the thus obtained R in said sixth storage means;

(k) means for reading said integer N out of said first storage means, calculating a quotient by dividing said integer N by 2, and storing the obtained quotient in said first storage means as a renewed integer N;

(l) means for reading said integer R out of said sixth storage means, if said integer R is equal to 1, reading said double vector Y out of said third storage means, said double vector Z out of said fourth storage means, and said parameters A out of said fifth storage means, inputting said double vectors Y and Z and said parameters A into said double vector adding device, calculating a sum of said double vectors Y and Z, and storing the calculated sum in said fourth storage means, and

(m) means for reading said integer N out of said first storage means, if the read-out integer N is greater than 0, reading said double vector Y out of said third storage means and said parameters A out of said fifth storage means, inputting said double vector Y and said parameters A into said double vector doubling device, doubling said double vector Y, and storing the doubled double vector Y in said third storage means, and if said integer N is equal to 0, reading said double vector Z out of said fourth storage means.

18. A system for distributing a public key comprising:

parameters A defining a curve, and a double vector Q comprising a plurality of pairs of elements selected from a predetermined finite field are informed in advance to all users, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

a user terminal U randomly selects an integer Nu and keeps the selected integer Nu secret,

a user terminal V randomly selects an integer Nv and keeps the selected integer Nv secret, said user terminal U transmits a double vector Qu (Qu=Nu.times.Q) to said user terminal V,

said double vector Qu being obtained by multiplying said double vector Q by said integer Nu through the use of said integer Nu, said double vector Q, and said parameters A,

said user terminal V transmits a double vector Qv (Qv=Nv.times.Q) to said user terminal U, said double vector Qv being obtained by multiplying said double vector Q by said integer Nv through the use of said integer Nv, said double vector Q, and said parameters A,

said user terminal U multiplies said double vector Qv by said integer Nu through the use of said double vector Qv having been transmitted from said user terminal V, said integer Nu, and said parameters A, to thereby obtain a double vector K (K=Nu.times.Qv=Nu.times.Nv.times.Q) as a common key K, and

said user terminal V multiplies said double vector Qu by said integer Nu through the use of said double vector Qu having been transmitted from said user terminal U, said integer Nv, and said parameters A, to thereby obtain a double vector K (K=Nv X Qu=Nv.times.Nu.times.Q) as a common key K.

19. A system for distributing a public key, comprising a center and a plurality of user terminals,

said center comprising:

(a) means for receiving a request for parameters A defining a curve, and a double vector Q comprising a plurality of pairs of elements selected from a predetermined finite field, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A; and

(b) means for disclosing said double vector Q and said parameters A to a user terminal making a request,

said user terminal comprising:

(a) first means for requesting said center to transmit said double vector Q and parameters A both made open;

(b) second means for receiving and retaining said double vector Q and parameters A, and transmitting them to a later mentioned device for multiplying a double vector by an integer;

(c) third means for randomly selecting an integer Nu, keeping the thus selected integer Nu secret, and transmitting said integer Nu to a device for multiplying a double vector by an integer;

(d) said device for multiplying a double vector by an integer, for receiving said double vector Q and said parameters A from said second means, receiving said integer Nu from third means, and calculating a double vector Qu by multiplying said double vector Q by said integer Nu;

(e) fourth means for transmitting said double vector Qu to other user terminals;

(f) fifth means for receiving a double vector Qv transmitted from other user terminals, and transmitting said double vector Qv to said device,

said device receiving said double vector Qv transmitted from said other user terminals, said integer Nu stored in said third means, and said parameters A retained in said second means, and multiplying said double vector Qv by said integer Nu to thereby obtain a double vector K; and

(g) sixth means for storing said double vector K as a secret key.

20. A recording medium readable by a computer, storing a program therein for accomplishing a system for distributing a public key, said system comprising a center and a plurality of user terminals, said program causing a computer to act as said center comprising:

(a) means for receiving a request for parameters A defining a curve, and a double vector Q comprising a plurality of pairs of elements selected from a predetermined finite field, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A; and

(b) means for disclosing said double vector Q and said parameters A to a user terminal making a request,

said program causing a computer to act as said user terminal comprising:

(a) first means for requesting said center to transmit said double vector Q and parameters A both made open;

(b) second means for receiving and retaining said double vector Q and parameters A, and transmitting them to a device for multiplying a double vector by an integer;

(c) third means for randomly selecting an integer Nu, keeping the thus selected integer Nu secret, and transmitting said integer Nu to the device for multiplying a double vector by an integer;

(d) said device for multiplying a double vector by an integer, for receiving said double vector Q and said parameters A from said second means, receiving said integer Nu from third means, and calculating a double vector Qu by multiplying said double vector Q by said integer Nu;

(e) fourth means for transmitting said double vector Qu to other user terminals;

(f) fifth means for receiving a double vector Qv transmitted from other user terminals, and transmitting said double vector Qv to said device,

said device receiving said double vector Qv transmitted from said other user terminals, said integer Nu stored in said third means, and said parameters A

retained in said second means, and multiplying said double vector Qv by said integer Nu to thereby obtain a double vector K; and

(g) sixth means for storing said double vector K as a secret key.

21. An El-Gamal type encryption system comprising:

parameters A defining a curve, and a double vector Q comprising a plurality of pairs of elements selected from a predetermined finite field are in advance informed of to all users, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

a user terminal U randomly selects an integer Nu, and keeps the thus selected integer Nu secret,

said user terminal U transmits a double vector Qu (Qu=Nu.times.Q) to other users as a public key, said double vector Qu being obtained by multiplying said double vector Q by said integer Nu through the use of said integer Nu, said double vector Q, and said parameters A,

said user terminal U encrypts a text through the use of said integer Nu and a public key Qv of a user terminal V to which said user terminal U intends to transmit said text, and

said user terminal V having received the thus encrypted text decrypts said encrypted text through the use of an integer Nv which said user terminal V retains in secret.

22. An El-Gamal type encryption system comprising a center and a plurality of user terminals,

said center comprising:

(a) first means for receiving public keys disclosed by said user terminals;

(b) second means for receiving a request to transmit parameters A defining a curve, a double vector Q comprising a plurality of pairs of elements selected from a predetermined finite field, a public key Qu to a user terminal, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(c) third means for disclosing said double vector Q, said parameters A, and said public key Qu to said user terminal making said request, when said second means receives said request,

said user terminal as a transmitter, comprising:

(a) fourth means for requesting said center to transmit said double vector Q, said parameters A, and said public keys Qv of other user terminals;

(b) fifth means for receiving and retaining said double vector Q, said parameters A, and said public key Qv which have been disclosed by said center in accordance with a request from said fourth means, and transmitting them to a first device for multiplying a double vector by an integer;

(c) sixth means for randomly selecting an integer Nu, keeping the thus selected integer Nu secret, and transmitting said integer Nu to first device for multiplying a double vector by an integer;

(d) said first device for multiplying a double vector by an integer, for receiving said double vector Q and said parameters A from said second means, receiving said integer Nu from said sixth means, and calculating a double vector Qu by multiplying said double vector Q by said integer Nu;

(e) seventh means for receiving said double vector Qu from said first device, and transmitting said double vector Qu to said center for disclosing as a public key;

(f) a second device for selecting a random number Ru and keeping the selected random number Ru secret, and transmitting said random number Ru to said first device;

said first device receiving said double vector Q transmitted from said second means, said parameters A, and said random number Ru stored in said second device, and multiplying said double vector Q by said random number Ru to thereby obtain a double vector C1 as a cipher, and storing the cipher in first storage means,

said first device receiving said public key Qv of other user terminals stored in said second means, said parameters A, and said random number Ru, stored in said second device, and multiplying said double vector Qv by said random number Ru to thereby have a double vector T1, and transmitting the double vector T1 to eighth means,

(g) eighth means for calculating a sum t1 of first elements in each of groups included in said double vector T1, and making a cipher C2 to which a correspondence M is added; and

(h) ninth means for cooperating with said eighth means to transmit said ciphers C1 and C2 to other user terminals,

said user terminal as a receiver, comprising:

(a) tenth means for receiving and retaining said ciphers C1 and C2 transmitted from said user terminals as a transmitter;

said first device receiving said cipher C1, an integer Nv retained in said sixth means, and said parameters A, and calculating a double vector T2 by multiplying said double vector C1 by said integer Nv; and

(b) eleventh means for receiving said cipher C2 and said double vector T2, calculating a sum t2 of first elements in each of groups included in said double vector T2, and decrypting said correspondence M by subtracting said sum t2 from said cipher C2.

23. A recording medium readable by a computer, storing a program therein for accomplishing an El-Gamal type encryption system comprising a center and a plurality of user terminals, said program causing a computer to act as said center comprising:

(a) first means for receiving public keys disclosed by said user terminals;

(b) second means for receiving a request to transmit parameters A defining a curve, a double vector Q comprising a plurality of pairs of elements selected from a predetermined finite field, a public key Qu to a user terminal, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

(c) third means for disclosing said double vector Q, said parameters A, and said public key Qu to said user terminal making said request, when said second means receives said request,

said program causing a computer to act as said user terminal as a transmitter, said user terminal comprising:

(a) fourth means for requesting said center to transmit said double vector Q, said parameters A, and said public keys Qv of other user terminals;

(b) fifth means for receiving and retaining said double vector Q, said parameters A, and said public key Qv which have been disclosed by said center in accordance with a request from said fourth means, and transmitting them to a first device for multiplying a double vector by an integer;

(c) sixth means for randomly selecting an integer Nu, keeping the thus selected integer Nu secret, and transmitting said integer Nu to the first device for multiplying a double vector by an integer;

(d) said first device for multiplying a double vector by an integer, for receiving said double vector Q and said parameters A from said third means, receiving said integer Nu from said sixth means, and calculating a double vector Qu by multiplying said double vector Q by said integer Nu;

(e) seventh means for receiving said double vector Qu from said first device, and transmitting said double vector Qu to said center for disclosing as a public key;

(f) a second device for selecting a random number Ru and keeping the selected random number Ru secret, and transmitting said random number Ru to said first device;

said first device receiving said double vector Q transmitted from said second means, said parameters A, and said random number Ru stored in said second device, and multiplying said double vector Q by said random number Ru to thereby obtain a double vector C1 as a cipher, and storing the cipher in first storage means,

said first device receiving said public key Qv of other user terminals stored in said second means, said parameters A, and said random number Ru, stored in said second device, and multiplying said double vector Qv by said random number Ru to thereby have a double vector T1, and transmitting the double vector T1 to eighth means,

(g) eighth means for calculating a sum t1 of first elements in each of groups included in said double vector T1, and making a cipher C2 to which a correspondence M is added; and

(h) ninth means for cooperating with said eighth means to transmit said ciphers C1 and C2 to other user terminals,

said program causing a computer to act as said user terminal as a receiver, said user terminal comprising:

(a) tenth means for receiving and retaining said ciphers C1 and C2 transmitted from said user terminals as a transmitter,

said first device receiving said cipher C1, an integer Nv retained in said sixth means, and said parameters A, and calculating a double vector T2 by multiplying said double vector C1 by said integer Nv; and

(b) eleventh means for receiving said cipher C2 and said double vector T2, calculating a sum t2 of first elements in each of groups included in said double vector T2, and decrypting said correspondence M by subtracting said sum t2 from said cipher C2.

24. An El-Gamal type signature system wherein parameters A defining a curve, and a double vector Q comprising a plurality of pairs of elements selected from a predetermined finite field are informed in advance to all users, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A;

a certifier terminal U randomly selects an integer Nu as a signature key, and keeps the selected integer Nu secret,

said certifier terminal U discloses a double vector Qu (Qu=Nu.times.Q) as a verification key, said double vector Qu being obtained by multiplying said double vector Q by said integer Nu,

said certifier terminal U makes a signature text for a correspondence M through the use of any integer and said signature key Nu, and transmits the signature text to a verification terminal V together with said correspondence M, and

said verification terminal V verifies said correspondence M through the use of said signature text and said verification key Qu of said certifier terminal U.

25. An El-Gamal type signature system comprising a center and a plurality of certifier terminals and verifier terminals,

said center comprising:

(a) first means for receiving verification keys disclosed by said certifier terminals;

(b) second means for receiving a request from one of said certifier and verifier terminals to transmit a double vector Q comprising a plurality of pairs of elements selected from a predetermined finite field, parameters A defining a curve, and a verification key Qu, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A; and

(c) third means for disclosing said double vector Q, said parameters A, and said verification key Qu to said one of said certifier and verifier terminals making said request, when said second means receives said request,

said certifier terminal comprising:

(a) fourth means for requesting said center to transmit said double vector Q and said parameters A;

(b) fifth means for receiving and retaining said double vector Q and said parameters A which have been disclosed by said center in accordance with a request from said fourth means, and transmitting them to a first device for multiplying a double vector by an integer;

(c) sixth means for randomly selecting an integer Nu, keeping the selected integer Nu secret as a signature key, and transmitting said signature key Nu to the first device for multiplying a double vector by an integer;

(d) said first device for multiplying a double vector by an integer, for receiving said double vector Q and said parameters A from said second means, receiving said signature key Nu from said sixth means, and calculating a double vector Qu by multiplying said double vector Q by said integer Nu;

(e) seventh means for receiving said double vector Qu from said first device, and transmitting said double vector Qu to said center for disclosing as a verification key;

(f) a second device for selecting a random number K, keeping the thus selected random number K secret, and transmitting said random number K to said first device;

said first device receiving said double vector Q and said parameters A transmitted from said second means, and said random number K stored in said second device, multiplying said double vector Q by said random number K to thereby obtain a double vector R as a signed text, and storing said signed text R in eighth means; and

(g) ninth means for receiving a correspondence M, said signed text R (R=K.times.Q) from said first device, said random number K from said second device, and said signature key Nu from said sixth means, and calculating S (S=(M-Nu.times.x(R))K.sup.-1 mod O(Q), wherein x(R) indicates a sum of first elements in each of groups included in a double vector R, and O(Q) indicates an order of said double vector Q as a signed text;

said signed text R, said signed text S, and said correspondence M being transmitted to said verifier terminal from said eighth, tenth and ninth means,

said verifier terminal comprising:

(a) eleventh means for requesting said center to transmit said double vector Q, said parameters A, and said verification key all of which have been disclosed;

(b) twelfth means for receiving and retaining said double vector Q, said parameters A, and said verification key Qu, and transmitting them to a third device for multiplying a double vector by an integer;

(c) said third device for multiplying a double vector by an integer, said third device receiving said double vector Q, said parameters A, and said correspondence M, calculating M.times.Q by multiplying said double vector Q by said M to thereby obtain a double vector T1 as a result, and storing the calculated double vector T1 in first storage means,

said third device receiving a sum x(R) of first elements in each of groups included said double vector R having been received from said eighth means, receiving said verification key Qu and said parameters A from said twelfth means, calculating x(R).times.Qu to thereby obtain a double vector T2 as a result, and storing the thus calculated double vector T2 in second storage means,

said third device receiving said double vector R, said signed text S, and said parameters A, calculating S.times.R by multiplying said double vector R by said S to thereby obtain a double vector R3 as a result, and storing said double vector T3 in third storage means;

(d) a fourth device for summing double vectors, said fourth device receiving said double vectors T2 and T3, and said parameters A, calculating (T2+T3) to thereby obtain a double vector T4 as a result, and storing the calculated double vector T4 in fourth storage means; and

(e) verification means for confirming whether said double vector T1 stored in said first storage means is identical with said double vector T4 stored in said fourth storage means, to thereby verify whether said correspondence M is made by said certifier terminal U.

26. A recording medium readable by a computer, storing a program therein for accomplishing an El-Gamal type signature system comprising a center and a plurality of certifier terminals and verifier terminals,

said program causing a computer to act as said center comprising:

(a) first means for receiving verification keys disclosed by said certifier terminals;

(b) second means for receiving a request from one of said certifier and verifier terminals to transmit a double vector Q comprising a plurality of pairs of elements selected from a predetermined finite field, parameters A defining a curve, and a verification key Qu, said curve having the equation:

a13Xx.sup.4 +a12Xy.sup.3 +a11Xxy.sup.2 +a10Xx.sup.2 y+a9Xx.sup.3 +a8Xy.sup.2 +a7Xxy+a6Xx.sup.2 +a5Xy+a4Xx+a3=0 where a3 to a13 indicate the parameters A; and

(c) third means for disclosing said double vector Q, said parameters A, and said verification key Qu to said one of said certifier and verifier terminals making said request, when said second means receives said request,

said program causing a computer to act as said certifier terminal comprising:

(a) fourth means for requesting said center to transmit said double vector Q and said parameters A;

(b) fifth means for receiving and retaining said double vector Q and said parameters A which have been disclosed by said center in accordance with a request from said fourth means, and transmitting them to a first device for multiplying a double vector by an integer;

(c) sixth means for randomly selecting an integer Nu, keeping the selected integer Nu secret as a signature key, and transmitting said signature key Nu to the first device for multiplying a double vector by an integer;

(d) said first device for multiplying a double vector by an integer, for receiving said double vector Q and said parameters A from said second means, receiving said signature key Nu from said sixth means, and calculating a double vector Qu by multiplying said double vector Q by said integer Nu;

(e) seventh means for receiving said double vector Qu from said first device, and transmitting said double vector Qu to said center for disclosing as a verification key;

(f) a second device for selecting a random number K, keeping the selected random number K secret, and transmitting said random number K to said first device;

said first device receiving said double vector Q and said parameters A transmitted from said second means, and said random number K stored in said second device, multiplying said double vector Q by said random number K to thereby obtain a double vector R as a signed text, and storing said signed text R in eighth means;

(g) ninth means for retaining a correspondence M therein; and

(h) tenth means for receiving said correspondence M from said tenth means, said signed text R (R=K.times.Q) from said first device, said random number K from said second device, said signature key Nu from said sixth means, and said correspondence M from said tenth means, and calculating S (S=(M-Nu.times.x(R))K.sup.-1 mod O(Q), wherein x(R) indicates a sum of first elements in each of groups included in a double vector R, and O(Q) indicates an order of said double vector Q) as a signed text;

said signed text R, said signed text S, and said correspondence M being transmitted to said verifier terminal from said eighth, tenth and ninth means,

said program causing a computer to act as said verifier terminal comprising:

(a) eleventh means for requesting said center to transmit said double vector Q, said parameters A, and said verification key all of which have been disclosed;

(b) twelfth means for receiving and retaining said double vector Q, said parameters A, and said verification key Qu, and transmitting them to a third device for multiplying a double vector by an integer;

(c) a third device for multiplying a double vector by an integer, said third device receiving said double vector Q, said parameters A, and said correspondence M, calculating M.times.Q by multiplying said double vector Q by said M to thereby obtain a double vector T1 as a result, and storing the calculated double vector T1 in first storage means,

said third device receiving a sum x(R) of first elements in each of groups included said double vector R having been received from said eighth means, receiving said verification key Qu and said parameters A from said twelfth means, calculating x(R).times.Qu to thereby have a double vector T2 as a result, and storing the calculated double vector T2 in second storage means,

said third device receiving said double vector R, said signed text, and said parameters A, calculating S.times.R by multiplying said double vector R by said S to thereby obtain a double vector R3 as a result, and storing said double vector T3 in third storage means;

(d) a fourth device for summing double vectors, said fourth device receiving said double vectors T2 and T3, and said parameters A, calculating (T2+T3) to thereby have a double vector T4 as a result, and storing the calculated double vector T4 in fourth storage means; and

(e) verification means for confirming whether said double vector T1 stored in said first storage means is identical with said double vector T4 stored in said fourth storage means, to thereby verify whether said correspondence M is made by said certifier terminal U.

Description

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to encryption techniques for data security, and more particularly to a system for distributing a public key for network users to share a secret key through the use of a public key, a public key encryption system such as an El-Gamal type encryption system for network users to make a mutual secret communication through the use of a public key, and an El-Gamal type verification system, which is one of electronic signature systems, for network users to verify a correspondence and/or a transmitter, and apparatuses for operating a bivector, to be used for those systems, such as an apparatus for multiplying a bivector by an integer.

2. Description of the Related Art

Various techniques belonging to a public key encryption system wherein secret communication is made in open network base security thereof on difficulty in solving an issue of a discrete logarithm in a finite field GF(p).

For instance, a system of distributing DH type public key having been suggested by W. Diffie and M. Hellman, New directions in cryptography, IEEE, Trans. Inf. Theory, IT-22, 6, pp. 644-654, and El-Gamal cryptography and signature systems having been suggested by T. E. El-Gamal, A public key cryptosystem and a signature scheme based on discrete logarithm, Proc. Crypto 84, 1984, base security thereof on that an issue of a discrete logarithm in a finite field GF(p) is quite difficult to solve.

Hereinbelow is explained the issue of a discrete logarithm in a finite field GF(p). It is now supposed that p indicates a prime number, and that GF(p) operates an integer N equal to or greater than 0, but smaller than p (N=0, 1, 2, - - - , p-1), with the prime number being used as a modulo. It is also supposed that the following equation is established.

Y=.alpha..sup.X mod p (1.ltoreq.X.ltoreq.p-1)

In the equation, a indicates .alpha. certain fixed primitive root of GF(p). That is, elements of GF(p), 1, 2 - - - , p-1, other than 0 can be represented in the form of .alpha..sup.K where K indicates a certain number. Under those suppositions, X is called a logarithm of Y in GF(p) with the prime number p acting as a base.

It is easy to calculate Y on the basis of X. Specifically, what is required to do so is to merely conduct multiplication by the number of 2.times.log.sub.2 X. To the contrary, it is quite difficult to calculate X on the basis of Y, even if there would be employed an algorithm which is best among presently known algorithms. An amount of calculation for obtaining X on the basis of Y is almost the same as an amount of calculation for prime factor factorization of a composite number having almost the same magnitude as that of the prime number p. A difficulty in calculating X on the basis of Y is called a discrete logarithm problem.

In accordance with the above-mentioned DH type public key distribution system, a first user A and a second user B can share a common key K, which is secret data, with the common key K being kept secret to others, even though open network is utilized. This is based on that the above-mentioned discrete logarithm problem is quite difficult to solve.

A prime number p and a primitive root .alpha. are in advance informed to others as open data. The first user A randomly selects an integer X.sub.A in the range of 0 and (p-1), and the thus selected integer X.sub.A is kept secret. Similarly, the second user B randomly selects an integer X.sub.B in the range of 0 and (p-1), and the thus selected integer X.sub.B is kept secret. The first user A calculates the following equation.

Y.sub.A =.alpha..sup.XA mod p (1.ltoreq.Y.sub.A.ltoreq.p-1)("XA" means "X.sub.A ". The same applies to "XB", "XU" etc., hereinbelow.)

Then, the first user A transmits a calculation result Y.sub.A to the second user B. Similarly, the second user B calculates the following equation.

Y.sub.B =.alpha..sup.XB mod p (1.ltoreq.Y.sub.B.ltoreq.p-1)

Then, the second user B transmits a calculation result Y.sub.B to the first user A.

After the calculation results Y.sub.A and Y.sub.B have been exchanged, the first user A calculates the common key K, as follows.

K=Y.sub.B.sup.XA mod p=(.alpha..sup.XB mod p).sup.XA mod p=.alpha..sup.XAXB mod p (1.ltoreq.K.ltoreq.p-1)

Similarly, the second user B calculates the common key K, as follows.

K=Y.sub.A.sup.XB mod p=(.alpha..sup.XA mod p).sup.XB mod p=.alpha..sup.XAXB mod p (1.ltoreq.K.ltoreq.p-1)

Thus, the first and second users A and B can share the common key K (K=.alpha..sup.XAXB mod p) in secret.

Thereafter, the first and second users A and B can make secret communication therebetween through the use of the common key K. In the above-mentioned procedure, only the calculation results Y.sub.A and Y.sub.B are on open network. Since it would be necessary to solve the discrete logarithm problem in order to obtain the integers X.sub.A and X.sub.B both of which are secret data, a third party cannot know the common key K on the premise that the discrete logarithm problem is quite difficult to solve.

In accordance with the above-mentioned El-Gamal encryption system, it is possible to make a secret communication on open network as follows, based on the fact that the discrete logarithm problem is difficult to solve.

A prime number p and a primitive root .alpha. are in advance informed to others as open data. Each of users U randomly selects an integer X.sub.U, and the thus selected integer X.sub.U is kept secret. In addition, each of users U calculates the following equation.

Y.sub.U =.alpha..sup.XU mod p (1.ltoreq.Y.sub.U.ltoreq.p-1)

Then, each of users U transmits the calculation result Y.sub.U to other users as a public key.

Herein, it is supposed that a first user A transmits a correspondence M to a second user B in secret. First, the first user A makes the following ciphers C1 and C2 through the use of a random number K which only the first user A knows, and a public key Y.sub.B of the second user B.

C1=.alpha..sup.K mod p

C2=M.times.Y.sub.B.sup.K mod p

Then, the first user A transmits the ciphers C1 and C2 to the second user B. The second user B having received the ciphers can obtain the correspondence M by calculating the following equation through the use of an integer X.sub.B which only the second user B knows.

M=C1.sup.-XB.times.C2 mod p

In the above-mentioned El-Gamal encryption system, only the ciphers C1 and C2 are on open network. Since it would be necessary to solve the discrete logarithm problem in order to obtain the random number K and the correspondence M both of which are secret data, a secret communication can be made on the premise that the discrete logarithm problem is quite difficult to solve.

In accordance with the above-mentioned El-Gamal signature system, electronic signature can be accomplished as follows, based on the fact that it is quite difficult to solve the discrete logarithm problem.

A prime number p and a primitive root .alpha. are in advance informed to others as open data. A certifier U randomly selects an integer X.sub.U as a signature key, and the thus selected integer X.sub.U is kept secret. In addition, the certifier U calculates the following equation.

Y.sub.U =.alpha..sup.XU mod p (1.ltoreq.Y.sub.U.ltoreq.p-1)

Then, the certifier U discloses the calculation result Y.sub.U to others as a verification key.

Herein, it is supposed that a verifier V verifies a signature made to a correspondence M of the certifier U. First, the certifier U makes the following signatures R and S through the use of a random number K which only the certifier knows, and a signature key X.sub.U of the certifier U itself.

R=.alpha..sup.K mod p

S=(M=X.sub.U.times.R).times.K.sup.-1 mod p

Then, the certifier U transmits a correspondence M together with the signatures R and S to the verifier V. The verifier V having received the signatures R and S verifies whether the following equation is established through the use of a verification key Yu of the certifier U.

.alpha..sup.M =Yu.sup.R.times.R.sup.S mod p

In the above-mentioned El-Gamal signature system, only the correspondence M and the signatures R and S are on open network. Since it would be necessary to solve the discrete logarithm problem in order to obtain the signature key X.sub.U which is secret data, it would be quite difficult or almost impossible for a person other than the certifier U to impersonate the certifier U, and hence, electronic signature can be accomplished on the premise that the discrete logarithm problem is quite difficult to solve.

As having been explained so far, most of the public key encryption systems base its security on the fact that the discrete logarithm problem in a finite field GF(p) is difficult to solve. However, recent development in a super computer and various arithmetic algorithms is making it possible to solve the discrete logarithm problem in a finite field GF(p) with a relatively small amount of calculation.

As a countermeasure thereto, it is recommended to employ a prime number p having 1024 bit, namely, having an order of about 300 or greater in decimalism. However, it would be necessary to prepare a large-scale circuit for finite field operation in order to make an operation on a finite field GF(p), using a prime number p having about 300 or more orders. This prevents various techniques in a public key encryption system from being put to practical use.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide an apparatus for making an operation, to be used for an encryption system, and other various techniques for accomplishing a public key encryption system in a smaller scale.

As having been explained, the conventional public key encryption systems are based on the fact that it is quite difficult or almost impossible to solve the discrete logarithm problem of a finite field, more accurately, of multiplicative group of a finite field. The principle of the present invention is that Jacobian group of algebraic curves in a finite field is employed in place of multiplicative group a finite field.

Hereinbelow is explained Jacobian group of algebraic curves. Any algebraic curve has a characteristic comprised of a positive integer called a genus. It is now supposed that a curve C has a genus G. It is now possible to define an addition among a set of any G number of points on the curve C, as follows. There are defined following two sets X1 and X2 each composed of the G number of points on the curve C.

X1={P.sub.11, P.sub.12, - - - , P.sub.1G }

X2={P.sub.21, P.sub.22, - - - , P.sub.2G }

A curve B is defined as a curve having the smallest degree among curves passing all points belonging to the sets X1 and X2. The thus defined curve B intersects with the curve C at another G number of points as well as points belonging to the sets X1 or X2. Another number of points are defined as Q.sub.1, Q.sub.2, - - - , Q.sub.G. Herein, a curve A is defined as a curve having the smallest degree among curves passing through all the G number of points, Q1, Q2, - - - , Q.sub.G. Thus defined curve A intersects with the curve C at another G number of points, R.sub.1, R.sub.2, - - - , R.sub.G as well as the G number of points, Q.sub.1, Q.sub.2, - - - , Q.sub.G. An addition of the sets X1 and X2 makes Y={R.sub.1, R.sub.2, - - - , R.sub.G }.

A set of any G number of points on the curve C wherein an addition is defined as mentioned above is called Jacobian group of the curve C in a finite field GF(p). A number of elements of Jacobian group, namely, a number of sets each composed of any G number of points on the curve C is equal to about p.sup.G. An arithmetically detailed explanation is made, for instance, by J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1986.

In order to accomplish various techniques belonging to a public key encryption system, which have sufficient strength with respect to cryptography, it is necessary to use a group having a sufficient number of element, even if multiplicative group in a finite field or Jacobian group of algebraic curves were employed. Generally, a number of elements of multiplicative group in a finite field GF(p) is equal to (p-1), whereas a number of elements of Jacobian group of algebraic curves having a genus G in a finite field GF(p) is equal to about p.sup.G. Accordingly, if Jacobian group of algebraic curves having a genus G in a finite field is employed in place of multiplicative group of a finite field, it would be possible to make an order of p in a finite field GF(p) about 1/G smaller than an order of p obtained when multiplicative group of a finite field is employed, on the assumption that a strength with respect to cryptography is kept at the same level, namely, a number of elements in groups to be used is kept almost the same.

Thus, various techniques of a public key encryption system in accordance with the present invention make it possible to employ a smaller-sized finite field without reduction in a strength with respect to cryptography, which ensures that a sufficient strength with respect to cryptography can be accomplished by means of a smaller-sized apparatus at less costs.

Specifically, the present invention provides the following apparatuses and recording mediums.

In one aspect of the present invention, there is provided an apparatus for summing double vectors (alternatively referred to as bivectors) each having a plurality of pairs of elements selected from a predetermined finite field, the apparatus including (a) a device for inputting double vectors X1 and X2, and a parameter A for defining a curve therethrough, (b) a first storage memory for storing the double vectors X1 therein, (c) a second storage memory for storing the double vectors X2 therein, (d) a third storage memory for storing the parameter A therein, and (e) a device for reading the double vectors X1 and X2, and the parameter A out of the first, second and third storage memories, respectively, and, when the double vectors X1 and X2 are coordinate value rows of points in point-sets Q1 and Q2 on the curve defined with the parameter A, respectively, operating a double vectors X3 comprised of coordinate value row of points in a point-set Q3 equal to a sum of the point-sets Q1 and Q2 in Jacobian group of the curve defined with the parameter A.

There is further provided an apparatus for summing bivectors each having a plurality of pairs of elements selected from a predetermined finite field, the apparatus including (a) a device for inputting bivectors X1 and X2, and a parameter A for defining a curve therethrough, (b) a first storage memory for storing the bivector X1 therein, (c) a second storage memory for storing the bivector X2 therein, (d) a third storage memory for storing the parameter A therein, and (e) a union-set operating device for reading the bivectors X1 and X2, and the parameter A out of the first, second and third storage memories, respectively, and, when the bivectors X1 and X2 are supposed to be coordinate value rows of points in point-sets on the curve defined with the parameter A, respectively, operating a bivector T1 comprised of coordinate value row of points in a union-set of the point-sets indicated by X1 and X2, (f) a fourth storage memories for storing the bivector T1 operated by the union-set operating device, (g) a first point-set operating device for reading the bivector T1 out of the fourth storage memory, and the parameter A out of the third storage memory, and, when the bivector T1 is supposed to be coordinate value row of points on the curve defined with the parameter A, operating a bivector T2 comprised of coordinate value row of points in a point-set indicative of inverse of the point-set expressed by T1 in Jacobian group of the curve defined with the parameter A, (h) a fifth storage memory for storing the bivector T2 operated by the first point-set operating device, (i) a second point-set operating device for reading the bivector T2 out of the fifth storage memory, and the parameter A of the third storage memory, and, when the bivector T2 is supposed to be coordinate value row of pints on the curve defined with the parameter A, operating a bivector X3 comprised of coordinate value row of points in a point-set indicative of inverse of the point-set expressed by T2 in Jacobian group of the curve defined with the parameter A, and (j) a device for outputting the bivector X3 operated by the second point-set operating device.

The above-mentioned apparatus may further include (k) a sixth storage memory for storing a parameter B therein, and (l) a seventh storage memory for storing a bivector S1 therein, and wherein the first point-set operating device includes (g-1) a common curve operating device for reading the bivector T1 out of the fourth storage memory, and the parameter A out of the third storage memory, and, when the bivector T1 is supposed to be coordinate value row of points on the curve defined with the parameter A, operating a parameter B of a curve passing through all points constituting the point-sets expressed by T1, (g-2) an intersection-set operating device for reading the parameter B out of the sixth storage memory, and the parameter A out of the third storage memory, and operating a bivector S1 comprised of coordinate value row of points in an intersection between a curve defined with the parameter A and a curve defined with the parameter B, (g-3) a difference-set operating device for reading the bivector T1 out of the fourth storage memory, and the bivector S1 out of the seventh storage memory, and, when the bivectors T1 and S1 are supposed to be coordinate value rows of points on the curve defined with the parameter A, respectively, operating the bivector T2 comprised of coordinate value row of points in a point-set obtained by subtracting a point-set indicated by the bivector T1 from a point-set indicated by the bivector S1, and (g-4) a device for outputting the bivector T2 operated by the difference-set operating device.

It is preferable that the above-mentioned apparatus may further include (k) an eighth storage memory for storing a parameter C therein, and (l) a ninth storage memory for storing a bivector S2 therein, and wherein the second point-set operating device includes (i-1) a common curve operating device for reading the bivector T2 out of the fifth storage memory, and the parameter A out of the third storage memory, and, when the bivector T2 is supposed to be coordinate value row of points on the curve defined with the parameter A, operating a parameter C of a curve passing through all points indicated by T2, (i-2) an intersection-set operating device for reading the parameter C out of the eighth storage memory, and the parameter A out of the third storage memory, and operating a bivector S2 comprised of coordinate value row of points in an intersection between a curve defined with the parameter A and a curve defined with the parameter C, (i-3) a difference-set operating device for reading the bivector T2 out of the fifth storage memory, and the bivector S2 out of the ninth storage memory, and, when the bivectors T2 and S2 are supposed to be coordinate value rows of points on the curve defined with the parameter A, respectively, operating the bivector X3 comprised of coordinate value row of points in a point-set obtained by subtracting a point-set indicated by the bivector T2 from a point-set indicated by the bivector S2, and (i-4) a device for outputting the bivector X3 operated by the difference-set operating device.

In another aspect of the present invention, there is provided a recording medium readable by a computer, storing a program therein for causing a computer to act as an apparatus for summing bivectors each having a plurality of pairs of elements selected from a predetermined finite field, the apparatus including (a) a device for inputting bivectors X1 and X2, and a parameter A for defining a curve therethrough, (b) a first storage memory for storing the bivector X1 therein, (c) a second storage memory for storing the bivector X2 therein, (d) a third storage memory for storing the parameter A therein, and (e) a device for reading the bivectors X1 and X2, and the parameter A out of the first, second and third storage memories, respectively, and, when the bivectors X1 and X2 are supposed to be coordinate value rows of points in point-sets Q1 and Q2 on the curve defined with the parameter A, respectivley, operating a bivector X3 comprised of coordinate value row of points in a point-set Q3 equal to a sum of the point-sets Q1 and Q2 in Jacobian group of the curve defined with the parameter A.

There is further provided a recording medium readable by a computer, storing a program therein for causing a computer to act as an apparatus for summing bivectors each comprising a plurality of pairs of elements selected from a predetermined finite field, the apparatus including (a) a device for inputting bivectors X1 and X2, and a parameter A for defining a curve therethrough, (b) a first storage memory for storing the bivector X1 therein, (c) a second storage memory for storing the bivector X2 therein, (d) a third storage memory for storing the parameter A therein, and (e) a union-set operating device for reading the bivectors X1 and X2, and the parameter A out of the first, second and third storage memories, respectively, and, when the bivectors X1 and X2 are supposed to be coordinate value rows of points on the curve defined with the parameter A, operating a bivector T1 comprised of coordinate value row of points in a union-set of the point-sets indicated by X1 and X2, (f) a fourth storage memory for storing the bivector T1 operated by the union-set operating device, (g) a first point-set operating device for reading the bivector T1 out of the fourth storage memory, and the parameter A out of the third storage memory, and, when the bivector T1 is supposed to be coordinate value row of points on the curve defined with the parameter A, operating a bivector T2 comprised of coordinate value row of points in a point-set indicative of inverse of the point-set indicated by T1 in Jacobian group of the curve defined with the parameter A, (h) a fifth storage memory for storing the bivector T2 operated by the first point-set operating device, (i) a second point-set operating device for reading the bivector T2 out of the fifth storage memory, and the parameter A out of the third storage memory, and, when the bivector T2 is supposed to be coordinate value row of points on the curve defined with the parameter A, operating a bivector X3 comprised of coordinate value row of points in a point-set indicative of inverse of the point-set indicated by T2 in Jacobian group of the curve defined with the parameter A, and (j) a device for outputting the bivector X3 operated by the second point-set operating device.

There is still further provided an apparatus for doubling a bivector having a plurality of pairs of elements selected from a predetermined finite field, the apparatus including (a) a device for inputting a bivector X, and a parameter A for defining a curve therethrough, (b) a first storage memory for storing the bivector X therein, (c) a second storage memory for storing the parameter A therein, and (d) a device for reading the bivector X out of the first storage memory, and the parameter A out of the second storage memory, and, when the bivector X is supposed to be coordinate value row of points in a point-set Q on the curve defined with the parameter A, operating a bivector Y comprised of coordinate value row of points in a point-set R equal to a doubled Q in Jacobian group of the curve defined with the parameter A.

There is yet further provided an apparatus for doubling a bivector having a plurality of pairs of elements selected from a predetermined finite field, the apparatus including (a) a device for inputting a bivector X, and a parameter A for defining a curve therethrough, (b) a first storage memory for storing the bivector X therein, (c) a second storage memory for storing a bivector Xa which is a copy of the bivector X, (d) a third storage memory for storing the parameter A therein, (e) a bivector adding device for reading the bivector X out of the first storage memory, the bivector Xa out of the second storage memory, and the parameter A out of the third storage memory, and adding the bivector X to the bivector Xa to thereby have a sum of 2X, and (f) a device for outputting the bivector 2X operated by the bivector adding device.

There is still further provided a recording medium readable by a computer, storing a program therein for causing a computer to act as an apparatus for doubling a bivector having a plurality of pairs of elements selected from a predetermined finite field, the apparatus including (a) a device for inputting a bivector X, and a parameter A for defining a curve therethrough, (b) a first storage memory for storing the bivector X therein, (c) a second storage memory for storing the parameter A therein, and (d) a device for reading the bivector X out of the first storage memory, and the parameter A out of the second storage memory, and, when the bivector X is supposed to be coordinate value row of points in a point-set Q on the curve defined with the parameter A, operating a bivector Y comprised of coordinate value row of points in a point-set R equal to a doubled Q in Jacobian group of the curve defined with the parameter A.

There is yet further provided a recording medium readable by a computer, storing a program therein for causing a computer to act as an apparatus for doubling a bivector including a plurality of pairs of elements selected from a predetermined finite field, the apparatus including (a) a device for inputting a bivector X, and a parameter A for defining a curve therethrough, (b) a first storage memory for storing the bivector X therein, (c) a second storage memory for storing a bivector Xa which is a copy of the bivector X, (d) a third storage memory for storing the parameter A therein, (e) a bivector adding device for reading the bivector X out of the first storage memory, the bivector Xa out of the second storage memory, and the parameter A out of the third storage memory, and adding the bivector X to the bivector Xa to thereby have a sum of 2X, and (f) a device for outputting the bivector 2X operated by the bivector adding device.

There is still yet further provided an apparatus for multiplying a bivector by an integer, the bivector having a plurality of pairs of elements selected from a predetermined finite field, the apparatus including (a) a device for inputting therethrough an integer N, a bivector X, and a parameter A for defining a curve, (b) a first storage memory for storing the integer N therein, (c) a second storage memory for storing the bivector X therein, (d) a third storage memory for storing the parameter A therein, and (e) a device for reading the integer N out of the first storage memory, the bivector X out of the second storage memory, and the parameter A out of the third storage memory, and, when the bivector X is supposed to be coordinate value row of points in a point-set Q on the curve defined with the parameter A, operating a bivector Z comprised of coordinate value row of points in a point-set R equal to the point-set Q multiplied by the integer N in Jacobian group of the curve defined with the parameter A.

There is further provided an apparatus for multiplying a bivector by an integer, the bivector having a plurality of pairs of elements selected from a predetermined finite field, the including (a) a device for inputting therethrough an integer N, a bivector X, and a parameter A for defining a curve, (b) a first storage memory for storing the integer N therein, (c) a second storage memory for storing the bivector X therein, (d) a third storage memory for storing a bivector Y which is a copy of the bivector X, (e) a fourth storage memory for storing a bivector Z therein, (f) a fifth storage memory for storing the parameter A therein, (g) a sixth storage memory for storing an integer R therein, (h) a bivector adding device for summing bivectors, (i) a bivector doubling device for doubling a bivector, (j) a device for reading the integer N out of the first storage memory, calculating a remainder R obtained when the integer N is divided by 2, and storing the thus obtained R in the sixth storage memory, (k) a device for reading the integer N out of the first storage memory, calculating a quotient by dividing the integer N by 2, and storing the thus obtained quotient in the first storage memory as a renewed integer N, (l) a device for reading the integer R out of the sixth storage memory, if the integer R is equal to 1, reading the bivector Y out of the third storage memory, the bivector Z out of the fourth storage memory, and the parameter A out of the fifth storage memory, inputting the bivectors Y and Z and the parameter A into the bivector adding device, calculating a sum of the bivectors Y and Z, and storing the thus calculated sum in the fourth storage memory, and (m) a device for reading the integer N out of the first storage memory, if the thus read-out integer N is greater than 0, reading the bivector Y out of the third storage memory and the parameter A out of the fifth storage memory, inputting the bivector Y and the parameter A into the bivector doubling device, doubling the bivector Y, and storing the thus doubled bivector Y in the third storage memory, and if the integer N is equal to 0, reading the bivector Z out of the fourth storage memory.

There is still further provided a recording medium readable by a computer, storing a program therein for causing a computer to act as an apparatus for multiplying a bivector by an integer, the bivector including a plurality of pairs of elements selected from a predetermined finite field, the apparatus including (a) a device for inputting therethrough an integer N, a bivector X, and a parameter A for defining a curve, (b) a first storage memory for storing the integer N therein, (c) a second storage memory for storing the bivector X therein, (d) a third storage memory for storing the parameter A therein, and (e) a device for reading the integer N out of the first storage memory, the bivector X out of the second storage memory, and the parameter A out of the third storage memory, and, when the bivector X is supposed to be coordinate value row of points in a point-set Q on the curve defined with the parameter A, operating a bivector Z comprised of coordinate value row of points in a point-set R equal to the point-set Q multiplied by the integer N in Jacobian group of the curve defined with the parameter A.

There is still yet further provided a recording medium readable by a computer, storing a program therein for causing a computer to act as an apparatus for multiplying a bivector by an integer, the bivector including a plurality of pairs of elements selected from a predetermined finite field, the apparatus including (a) a device for inputting therethrough an integer N, a bivector X, and a parameter A for defining a curve, (b) a first storage memory for storing the integer N therein, (c) a second storage memory for storing the bivector X therein, (d) a third storage memory for storing a bivector Y which is a copy of the bivector X, (e) a fourth storage memory for storing a bivector Z therein, (f) a fifth storage memory for storing the parameter A therein, (g) a sixth storage memory for storing an integer R therein, (h) a bivector adding device for summing bivectors, (i) a bivector doubling device for doubling a bivector, (j) a device for reading the integer N out of the first storage memory, calculating a remainder R obtained when the integer N is divided by 2, and storing the thus obtained R in the sixth storage memory, (k) a device for reading the integer N out of the first storage memory, calculating a quotient by dividing the integer N by 2, and storing the thus obtained quotient in the first storage memory as a renewed integer N, (l) a device for reading the integer R out of the sixth storage memory, if the integer R is equal to 1, reading the bivector Y out of the third storage memory, the bivector Z out of the fourth storage memory, and the parameter A out of the fifth storage memory, inputting the bivectors Y and Z and the parameter A into the bivector adding device, calculating a sum of the bivectors Y and Z, and storing the thus calculated sum in the fourth storage memory, and (m) a device for reading the integer N out of the first storage memory, if the thus read-out integer N is greater than 0, reading the bivector Y out of the third storage memory and the parameter A out of the fifth storage memory, inputting the bivector Y and the parameter A into the bivector doubling device, doubling the bivector Y, and storing the thus doubled bivector Y in the third storage memory, and if the integer N is equal to 0, reading the bivector Z out of the fourth storage memory.

There is further provided a system for distributing a public key wherein a parameter A defining a curve, and a bivector Q including a plurality of pairs of elements selected from a predetermined finite field are in advance informed of to all users, a user terminal U randomly selects an integer Nu and keeps the thus selected integer Nu secret, a user terminal V randomly selects an integer Nv and keeps the thus selected integer Nv secret, the user terminal U transmits a bivector Qu (Qu=Nu.times.Q) to the user terminal V, the bivector Qu being obtained by multiplying the bivector Q by the integer Nu through the use of the integer Nu, the bivector Q, and the parameter A, the user terminal V transmits a bivector Qv (Qv=Nv.times.Q) to the user terminal U, the bivector Qv being obtained by multiplying the bivector Q by the integer Nv through the use of the integer Nv, the bivector Q, and the parameter A, the user terminal U multiplies the bivector Qv by the integer Nu through the use of the bivector Qv having been transmitted from the user terminal V, the integer Nu, and the parameter A, to thereby obtain a bivector K (K=Nu.times.Qv=Nu.times.Nv.times.Q) as a common key K, and the user terminal V multiplies the bivector Qu by the integer Nu through the use of the bivector Qu having been transmitted from the user terminal U, the integer Nv, and the parameter A, to thereby obtain a bivector K (K=Nv.times.Qu=Nv.times.Nu.times.Q) as a common key K.

There is further provided a system for distributing a public key, including a center and a plurality of user terminals, the center including (a) a device for receiving a request for a parameter A defining a curve, and a bivector Q including a plurality of pairs of elements selected from a predetermined finite field, and (b) a device for disclosing the bivector Q and the parameter A to a user terminal making a request, the user terminal including (a) a first device for requesting the center to transmit the bivector Q and parameter A both made open, (b) a second device for receiving and retaining the bivector Q and parameter A, and transmitting them to a later mentioned device for multiplying a bivector by an integer, (c) a third device for randomly selecting an integer Nu, keeping the thus selected integer Nu secret, and transmitting the integer Nu to a later mentioned device for multiplying a bivector by an integer, (d) a device for multiplying a bivector by an integer, for receiving the bivector Q and the parameter A from the second device, receiving the integer Nu from third device, and calculating a bivector Qu by multiplying the bivector Q by the integer Nu, (e) a fourth device for transmitting the bivector Qu to other user terminals, (f) a fifth device for receiving a bivector Qv transmitted from other user terminals, and transmitting the bivector Qv to the device, the device receiving the bivector Qv transmitted from the other user terminals, the integer Nu stored in the third device, and the parameter A retained in the second device, and multiplying the bivector Qv by the integer Nu to thereby have a bivector K, and (g) a sixth device for storing the bivector K as a secret key.

There is further provided a recording medium readable by a computer, storing a program therein for accomplishing a system for distributing a public key, the system including a center and a plurality of user terminals, the program causing a computer to act as the center including (a) a device for receiving a request for a parameter A defining a curve, and a bivector Q including a plurality of pairs of elements selected from a predetermined finite field, and (b) a device for disclosing the bivector Q and the parameter A to a user terminal making a request, the program causing a computer to act as the user terminal including (a) a first device for requesting the center to transmit the bivector Q and parameter A both made open, (b) a second device for receiving and retaining the bivector Q and parameter A, and transmitting them to a later mentioned device for multiplying a bivector by an integer, (c) a third device for randomly selecting an integer Nu, keeping the thus selected integer Nu secret, and transmitting the integer Nu to a later mentioned device for multiplying a bivector by an integer, (d) a device for multiplying a bivector by an integer, for receiving the bivector Q and the parameter A from the second device, receiving the integer Nu from third device, and calculating a bivector Qu by multiplying the bivector Q by the integer Nu, (e) a fourth device for transmitting the bivector Qu to other user terminals, (f) a fifth device for receiving a bivector Qv transmitted from other user terminals, and transmitting the bivector Qv to the device, the device receiving the bivector Qv transmitted from the other user terminals, the integer Nu stored in the third device, and the parameter A retained in the second device, and multiplying the bivector Qv by the integer Nu to thereby have a bivector K, and (g) a sixth device for storing the bivector K as a secret key.

There is further provided an El-Gamal type encryption system wherein a parameter A defining a curve, and a bivector Q comprising a plurality of pairs of elements selected from a predetermined finite field are in advance informed of to all users, a user terminal U randomly selects an integer Nu, and keeps the thus selected integer Nu secret, the user terminal U transmits a bivector Qu (Qu=Nu.times.Q) to other users as a public key, the bivector Qu being obtained by multiplying the bivector Q by the integer Nu through the use of the integer Nu, the bivector Q, and the parameter A, the user terminal U encrypts a text through the use of the integer Nu and a public key Qv of a user terminal V to which the user terminal U intends to transmit the text, and the user terminal V having received the thus encrypted text decrypts the encrypted text through the use of an integer Nv which the user terminal V retains in secret.

There is further provided an El-Gamal type encryption system including a center and a plurality of user terminals, the center including (a) a first device for receiving public keys disclosed by the user terminals, (b) a second device for receiving a request to transmit a parameter A defining a curve, a bivector Q including a plurality of pairs of elements selected from a predetermined finite field, a public key Qu to a user terminal, and (c) a third device for disclosing the bivector Q, the parameter A, and the public key Qu to the user terminal making the request, when the second means receives the request, the user terminal as a transmitter, including (a) a fourth device for requesting the center to transmit the bivector Q, the parameter A, and the public keys Qv of other user terminals, (b) a fifth device for receiving and retaining the bivector Q, the parameter A, and the public key Qv which have been disclosed by the center in accordance with a request from the fourth means, and transmitting them to a later mentioned first device for multiplying a bivector by an integer, (c) a sixth device for randomly selecting an integer Nu, keeping the thus selected integer Nu secret, and transmitting the integer Nu to a later mentioned first device for multiplying a bivector by an integer, (d) a first apparatus for multiplying a bivector by an integer, for receiving the bivector Q and the parameter A from the second device, receiving the integer Nu from the sixth device, and calculating a bivector Qu by multiplying the bivector Q by the integer Nu, (e) a seventh device for receiving the bivector Qu from the first device, and transmitting the bivector Qu to the center for disclosing as a public key, (f) a second apparatus for selecting a random number Ru and keeping the thus selected random number Ru secret, and transmitting the random number Ru to the first device, the first device receiving the bivector Q transmitted from the second device, the parameter A, and the random number Ru stored in the second apparatus, and multiplying the bivector Q by the random number Ru to thereby have a bivector C1 as a cipher, and storing the thus made cipher in first storage memory, the first apparatus receiving the public key Qv of other user terminals stored in the second device, the parameter A, and the random number Ru, stored in the second apparatus, and multiplying the bivector Qv by the random number Ru to thereby have a bivector T1, and transmitting the thus made bivector T1 to eighth device, (g) an eighth device for calculating a sum t1 of first elements in each of groups included in the bivector T1, and making a cipher C2 to which a correspondence M is added, and (h) a ninth device for cooperating with the eighth means to transmit the ciphers C1 and C2 to other user terminals, the user terminal as a receiver, including (a) a tenth device for receiving and retaining the ciphers C1 and C2 transmitted from the user terminals as a transmitter, the first apparatus receiving the cipher C1, an integer Nv retained in the sixth means, and the parameter A, and calculating a bivector T2 by multiplying the bivector C1 by the integer Nv, and (b) an eleventh device for receiving the cipher C2 and the bivector T2, calculating a sum t2 of first elements in each of groups included in the bivector T2, and decrypting the correspondence M by subtracting the sum t2 from the cipher C2.

There is further provided a recording medium readable by a computer, storing a program therein for accomplishing an El-Gamal type encryption system including a center and a plurality of user terminals, the program causing a computer to act as the center including (a) a first device for receiving public keys disclosed by the user terminals, (b) a second device for receiving a request to transmit a parameter A defining a curve, a bivector Q including a plurality of pairs of elements selected from a predetermined finite field, a public key Qu to a user terminal, and (c) a third device for disclosing the bivector Q, the parameter A, and the public key Qu to the user terminal making the request, when the second device receives the request, the program causing a computer to act as the user terminal as a transmitter, the user terminal including (a) a fourth device for requesting the center to transmit the bivector Q, the parameter A, and the public keys Qv of other user terminals, (b) a fifth device for receiving and retaining the bivector Q, the parameter A, and the public key Qv which have been disclosed by the center in accordance with a request from the fourth device, and transmitting them to a later mentioned first apparatus for multiplying a bivector by an integer, (c) a sixth device for randomly selecting an integer Nu, keeping the thus selected integer Nu secret, and transmitting the integer Nu to a later mentioned first apparatus for multiplying a bivector by an integer, (d) a first apparatus for multiplying a bivector by an integer, for receiving the bivector Q and the parameter A from the third device, receiving the integer Nu from the sixth device, and calculating a bivector Qu by multiplying the bivector Q by the integer Nu, (e) a seventh device for receiving the bivector Qu from the first apparatus, and transmitting the bivector Qu to the center for disclosing as a public key, (f) a second apparatus for selecting a random number Ru and keeping the thus selected random number Ru secret, and transmitting the random number Ru to the first apparatus, the first apparatus receiving the bivector Q transmitted from the second device, the parameter A, and the random number Ru stored in the second apparatus, and multiplying the bivector Q by the random number Ru to thereby have a bivector C1 as a cipher, and storing the thus made cipher in a first storage memory, the first apparatus receiving the public key Qv of other user terminals stored in the second device, the parameter A, and the random number Ru, stored in the second apparatus, and multiplying the bivector Qv by the random number Ru to thereby have a bivector T1, and transmitting the thus made bivector T1 to an eighth device, (g) an eighth device for calculating a sum t1 of first elements in each of groups included in the bivector T1, and making a cipher C2 to which a correspondence M is added, and (h) a ninth device for cooperating with the eighth device to transmit the ciphers C1 and C2 to other user terminals, the program causing a computer to act as the user terminal as a receiver, the user terminal including (a) a tenth device for receiving and retaining the ciphers C1 and C2 transmitted from the user terminals as a transmitter, the first device receiving the cipher C1, an integer Nv retained in the sixth device, and the parameter A, and calculating a bivector T2 by multiplying the bivector C1 by the integer Nv, and (b) an eleventh device for receiving the cipher C2 and the bivector T2, calculating a sum t2 of first elements in each of groups included in the bivector T2, and decrypting the correspondence M by subtracting the sum t2 from the cipher C2.

There is further provided an El-Gamal type signature system wherein a parameter A defining a curve, and a bivector Q including a plurality of pairs of elements selected from a predetermined finite field are in advance informed of to all users, a certifier terminal U randomly selects an integer Nu as a signature key, and keeps the thus selected integer Nu secret, the certifier terminal U discloses a bivector Qu (Qu=Nu.times.Q) as a verification key, the bivector Qu being obtained by multiplying the bivector Q by the integer Nu, the certifier terminal U makes a signature text for a correspondence M through the use of any integer and the signature key Nu, and transmits the thus made signature text to a verification terminal V together with the correspondence M, and the verification terminal V verifies the correspondence M through the use of the signature text and the verification key Qu of the certifier terminal U.

There is further provided an El-Gamal type signature system including a center and a plurality of certifier terminals and verifier terminals, the center including (a) a first device for receiving verification keys disclosed by the certifier terminals, (b) a second device for receiving a request from one of the certifier and verifier terminals to transmit a bivector Q including a plurality of pairs of elements selected from a predetermined finite field, a parameter A defining a curve, and a verification key Qu, and (c) a third device for disclosing the bivector Q, the parameter A, and the verification key Qu to the one of the certifier and verifier terminals making the request, when the second device receives the request, the certifier terminal including (a) a fourth device for requesting the center to transmit the bivector Q and the parameter A, (b) a fifth device for receiving and retaining the bivector Q and the parameter A which have been disclosed by the center in accordance with a request from the fourth device, and transmitting them to a later mentioned first apparatus for multiplying a bivector by an integer, (c) a sixth device for randomly selecting an integer Nu, keeping the thus selected integer Nu secret as a signature key, and transmitting the signature key Nu to a later mentioned first apparatus for multiplying a bivector by an integer, (d) a first apparatus for multiplying a bivector by an integer, for receiving the bivector Q and the parameter A from the second device, receiving the signature key Nu from the sixth device, and calculating a bivector Qu by multiplying the bivector Q by the integer Nu, (e) a seventh device for receiving the bivector Qu from the first apparatus, and transmitting the bivector Qu to the center for disclosing as a verification key, (f) a second apparatus for selecting a random number K, keeping the thus selected random number K secret, and transmitting the random number K to the first device, the first device receiving the bivector Q and the parameter A transmitted from the second device, and the random number K stored in the second apparatus, multiplying the bivector Q by the random number K to thereby have a bivector R as a signed text, and storing the signed text R in an eighth device, (g) a ninth device for receiving a correspondence M, the signed text R (R=k.times.Q) from the first apparatus, the random number K from the second apparatus, and the signature key Nu from the sixth device, and calculating S (S=(M-Nu.times.x(R))K.sup.-1 mod O(Q), wherein x(R) indicates a sum of first elements in each of groups included in a bivector R, and O(Q) indicates an order of the bivector Q) as a signed text, the signed text R, the signed text S, and the correspondence M being transmitted to the verifier terminal from the eighth, tenth and ninth devices, the verifier terminal including (a) an eleventh device for requesting the center to transmit the bivector Q, the parameter A, and the verification key all of which have been disclosed, (b) a twelfth device for receiving and retaining the bivector Q, the parameter A, and the verification key Qu, and transmitting them to a later mentioned third apparatus for multiplying a bivector by an integer, (c) a third apparatus for multiplying a bivector by an integer, the third apparatus receiving the bivector Q, the parameter A, and the correspondence M, calculating M.times.Q by multiplying the bivector Q by the M to thereby have a bivector T1 as a result, and storing the thus calculated bivector T1 in a first storage memory, the third apparatus receiving a sum x(R) of first elements in each of groups included the bivector R having been received from the eighth device, receiving the verification key Qu and the parameter A from the twelfth device, calculating x(R).times.Qu to thereby have a bivector T2 as a result, and storing the thus calculated bivector T2 in second storage device, the third apparatus receiving the bivector R, the signed text S, and the parameter A, calculating S.times.R by multiplying the bivector R by the S to thereby have a bivector R3 as a result, and storing the bivector T3 in third storage device, (d) a fourth apparatus for summing bivectors, the fourth apparatus receiving the bivectors T2 and T3, and the parameter A, calculating (T2+T3) to thereby have a bivector T4 as a result, and storing the thus calculated bivector T4 in a fourth storage memory, and (e) a verification device for confirming whether the bivector T1 stored in the first storage memory is identical with the bivector T4 stored in the fourth storage memory, to thereby verify whether the correspondence M is made by the certifier terminal U.

There is further provided a recording medium readable by a computer, storing a program therein for accomplishing an El-Gamal type signature system including a center and a plurality of certifier terminals and verifier terminals, the program causing a computer to act as the center including (a) a first device for receiving verification keys disclosed by the certifier terminals, (b) a second device for receiving a request from one of the certifier and verifier terminals to transmit a bivector Q including a plurality of pairs of elements selected from a predetermined finite field, a parameter A defining a curve, and a verification key Qu, and (c) a third device for disclosing the bivector Q, the parameter A, and the verification key Qu to the one of the certifier and verifier terminals making the request, when the second means receives the request, the program causing a computer to act as the certifier terminal including (a) a fourth device for requesting the center to transmit the bivector Q and the parameter A, (b) a fifth device for receiving and retaining the bivector Q and the parameter A which have been disclosed by the center in accordance with a request from the fourth device, and transmitting them to a later mentioned first apparatus for multiplying a bivector by an integer, (c) a sixth device for randomly selecting an integer Nu, keeping the thus selected integer Nu secret as a signature key, and transmitting the signature key Nu to a later mentioned first device for multiplying a bivector by an integer, (d) a first apparatus for multiplying a bivector by an integer, for receiving the bivector Q and the parameter A from the second device, receiving the signature key Nu from the sixth device, and calculating a bivector Qu by multiplying the bivector Q by the integer Nu, (e) a seventh device for receiving the bivector Qu from the first apparatus, and transmitting the bivector Qu to the center for disclosing as a verification key, (f) a second apparatus for selecting a random number K, keeping the thus selected random number K secret, and transmitting the random number K to the first apparatus, the first apparatus receiving the bivector Q and the parameter A transmitted from the second device, and the random number K stored in the second apparatus, multiplying the bivector Q by the random number K to thereby have a bivector R as a signed text, and storing the signed text R in an eighth device, (g) a ninth device for retaining a correspondence M therein, and (h) a tenth device for receiving the correspondence M from the tenth device, the signed text R (R=k.times.Q) from the first apparatus, the random number K from the second apparatus, the signature key Nu from the sixth device, and the correspondence M from the tenth device, and calculating S (S=(M-Nu.times.x(R))K.sup.-1 mod O(Q), wherein x(R) indicates a sum of first elements in each of groups included in a bivector R, and O(Q) indicates an order of the bivector Q) as a signed text, the signed text R, the signed text S, and the correspondence M being transmitted to the verifier terminal from the eighth, tenth and ninth devices, the program causing a computer to act as the verifier terminal including (a) an eleventh device for requesting the center to transmit the bivector Q, the parameter A, and the verification key all of which have been disclosed, (b) a twelfth device for receiving and retaining the bivector Q, the parameter A, and the verification key Qu, and transmitting them to a later mentioned third apparatus for multiplying a bivector by an integer, (c) a third apparatus for multiplying a bivector by an integer, the third apparatus receiving the bivector Q, the parameter A, and the correspondence M, calculating M.times.Q by multiplying the bivector Q by the M to thereby have a bivector T1 as a result, and storing the thus calculated bivector T1 in a first storage memory, the third apparatus receiving a sum x(R) of first elements in each of groups included the bivector R having been received from the eighth device, receiving the verification key Qu and the parameter A from the twelfth device, calculating x(R).times.Qu to thereby have a bivector T2 as a result, and storing the thus calculated bivector T2 in second storage memory, the third apparatus receiving the bivector R, the signed text S, and the parameter A, calculating S.times.R by multiplying the bivector R by the S to thereby have a bivector R3 as a result, and storing the bivector T3 in third storage memory, (d) a fourth apparatus for summing bivectors, the fourth apparatus receiving the bivectors T2 and T3, and the parameter A, calculating (T2+T3) to thereby have a bivector T4 as a result, and storing the thus calculated bivector T4 in fourth storage memory, and (e) a verification device for confirming whether the bivector T1 stored in the first storage memory is identical with the bivector T4 stored in the fourth storage memory, to thereby verify whether the correspondence M is made by the certifier terminal U.

The above and other objects and advantageous features of the present invention will be made apparent from the following description made with reference to the accompanying drawings, in which like reference characters designate the same or similar parts throughout the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an apparatus for summing bivectors, in accordance with an embodiment of the present invention.

FIG. 2 is a flow chart of the apparatus for summing bivectors, illustrated in FIG. 1.

FIG. 3 is a block diagram of a device for converting a point-set, employed in the apparatus for summing bivectors, illustrated in FIG. 1.

FIG. 4 is a flow chart of the device for converting a point-set, illustrated in FIG. 3.

FIG. 5 illustrates a data format of a parameter defining a curve.

FIG. 6 is a flow chart of a device for operating a common curve.

FIG. 7 is a flow chart of a device for operating an intersection-set.

FIG. 8 is a block diagram of an apparatus for doubling a bivector, in accordance with an embodiment of the present invention.

FIG. 9 is a flow chart of the apparatus for doubling a bivector, illustrated in FIG. 8.

FIG. 10 is a block diagram of an apparatus for multiplying a bivector by an integer, in accordance with an embodiment of the present invention.

FIG. 11 is a flow chart of the apparatus for multiplying a bivector by an integer, illustrated in FIG. 10.

FIG. 12 is a block diagram of a system for distributing a public key, in accordance with an embodiment of the present invention.

FIG. 13 is a block diagram of an example of a center partially constituting the system for distributing a public key, illustrated in FIG. 12.

FIG. 14 is a block diagram of an example of a user terminal partially constituting the system for distributing a public key, illustrated in FIG. 12.

FIG. 15 is a flow chart of the system for distributing a public key, illustrated FIG. 12.

FIG. 16 is a block diagram of an El-Gamal type encryption system in accordance with an embodiment of the present invention.

FIG. 17 is a block diagram of an example of a center partially constituting the El-Gamal type encryption system illustrated in FIG. 16.

FIG. 18 is a block diagram of an example of a user terminal partially constituting the El-Gamal type encryption system illustrated in FIG. 16.

FIG. 19 is a flow chart of the El-Gamal type encryption system illustrated in FIG. 16.

FIG. 20 is a block diagram of an El-Gamal type signature system in accordance with an embodiment of the present invention.

FIG. 21 is a block diagram of an example of a center partially constituting the El-Gamal type signature system illustrated in FIG. 20.

FIG. 22 is a block diagram of an example of a certifier terminal partially constituting the El-Gamal type signature system illustrated in FIG. 20.

FIG. 23 is a block diagram of an example of a verifier terminal partially constituting the El-Gamal type signature system illustrated in FIG. 20.

FIG. 24 is a flow chart of the El-Gamal type signature system illustrated in FIG. 20.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the preferred embodiments in accordance with the present invention, there is employed Jacobian group of algebraic curves having a genus G in a finite field GF(p), in place of multiplicative group in a finite field. As mentioned earlier, Jacobian group of algebraic curves having a genus G in a finite field GF(p) are comprised of point-sets {Q.sub.1, Q.sub.2, - - - , Q.sub.G } each composed of G number of points on a curve, and is represented as the following row where coordinate values of points are arranged.

[(x(Q.sub.1), y(Q.sub.1)), (x(Q.sub.2), Y(Q.sub.2)), - - - , (x(Q.sub.G), Y(Q.sub.G))] wherein x(Q.sub.i) indicates an X-axis coordinate of a point Qi, and y(Q.sub.i) indicates a Y-axis coordinate of a point Qi.

Thus, if a vector comprising a plurality of pairs of elements selected from a finite field is called a bivector, each of elements in Jacobian group can be represented with a bivector or bivectors. A public key encryption system in accordance with the present invention is comprised of an apparatus for operating a bivector in a finite field.

[Apparatus for Summing Bivectors]

Hereinbelow is explained an apparatus for summing bivectors, in accordance with an embodiment of the present invention.

An apparatus for summing bivectors in accordance with the embodiment operates and outputs a bivector X3, when bivectors X1 and X2 each comprising a plurality of pairs of elements selected from a predetermined finite field are supposed to be coordinate value rows of points in point-sets Q1 and Q2 on a curve defined with a parameter A, respectively, comprised of coordinate value row of points in a point-set Q3 equal to a sum of the point-sets Q1 and Q2 in Jacobian group of the curve defined with the parameter A.

This apparatus can be accomplished in a computer.

FIG. 1 is a block diagram of the apparatus for summing bivectors in accordance with the embodiment. FIG. 2 is a flow chart of the apparatus for summing bivectors, illustrated in FIG. 1. FIG. 3 is a block diagram of an example of a point-set conversion apparatus. FIG. 4 is a flow chart of the point-set conversion apparatus illustrated in FIG. 3.

The apparatus for summing bivectors, illustrated in FIG. 1, is comprised of a first device 11 for calculating a union-set, a second device 12 for converting a point-set, a memory 13, a fourth device 14 for inputting data therethrough, a fifth device 15 for outputting results therethrough, and a central processing unit (CPU) 16.

The fourth device 14 receives bivectors X1 and X2, and a parameter A defining a curve, and transmits them to CPU 16.

The memory 13 includes a first storage file for storing the bivector X1 therein, a second storage file for storing the bivector X2 therein, a third storage file for storing the parameter A therein, a fourth storage file for storing a bivector T1 therein, and a fifth storage file for storing a bivector T2 therein.

The first device 11 for calculating a union-set receives the bivector X1 from the first storage file, the bivector X2 from the second storage file, and the parameter A from the third storage file, and operates the bivector T1 which, when the bivectors X1 and X2 are supposed to be coordinate value rows of points in point-sets on a curve defined with the parameter A, respectively, is comprised of coordinate value row of points in a union-set of those point-sets indicated by X1 and X2.

The second device 12 for converting a point-set receives the bivector T1 from the fourth storage file, and the parameter A from the third storage file, and operates the bivector T2 which, when the bivector T1 is supposed to be coordinate value row of points on a curve defined with the parameter A, is comprised of coordinate value row of points in the point-set representing the inverse of the point-set indicated by T1 in Jacobian group of the curve defined with the parameter A.

The second device 12 further receives the bivector T2 from the fifth storage file, and the parameter A from the third storage file, and operates a bivector X3 which, when the bivector T2 is supposed to be coordinate value row of points on a curve defined with the parameter A, is comprised of coordinate value row of points in the point-set representing the inverse of the point-set indicated by T2 in Jacobian group of the curve defined with the parameter A.

The fifth device 15 for outputting results outputs the thus operated bivector X3.

The central processing unit 16 controls the first device 11, the second device 12, the memory 13, the fourth device 14, and the fifth device 15.

As illustrated in FIG. 3, the second device 12 for converting a point-set is comprised of a first device 21 for operating a common curve, a second device 22 for operating an intersection-set, a third device 23 for operating a difference-set, a memory 24, a fifth device 25 for inputting signals therethrough, a sixth device 26 for outputting operation results therethrough, and a central processing unit (CPU) 27.

The fifth device 25 receives the bivector T1, and the parameter A for defining a curve, and transmits them to the central processing unit 27.

The memory 24 includes a first storage file for storing the bivector T1 input through the fifth device 25, a second storage file for storing the parameter A input through the fifth device 25, a third storage file for storing an operated parameter B therein, and a fourth storage file for storing an operated bivector S1 therein.

The first device 21 for operating a common curve receives the bivector T1 from the first storage file, and the parameter A from the second storage file, and operates a parameter B of a curve which, when the bivector T1 is supposed to be coordinate value row of points on a curve defined with the parameter A, passes through all points belonging to the point-sets.

The second device 22 for operating an intersection-set receives the parameter B from the third storage file, and the parameter A from the second storage file, and operates a bivector S1 comprised of coordinate value row of points in an intersection between a curve defined with the parameter A and a curve defined with the parameter B.

The third device 23 for operating a difference-set receives the bivector T1 from the first storage file, and the bivector S1 from the fourth storage file, and operates a bivector T2 which, when the bivectors T1 and S1 are supposed to be coordinate value rows of points on a curve defined with said parameter A, respectively, is comprised of coordinate value row of points in a point-set obtained by subtracting a point-set indicated by the bivector T1 from a point-set indicated by the bivector S1.

The sixth device 26 outputs the thus operated bivector T2.

The central processing unit 27 controls the first device 21, the second device 22, the third device 23, the memory 24, the fifth device 25, and the sixth device 26.

The central processing unit 16, the fourth device 14, the fifth device 15, and the memory 13 all illustrated in FIG. 1 may double as the central processing unit 27, the fifth device 25, the sixth device 26, and the memory 24, respectively. In addition, the bivector T1 and the parameter A both stored in the memory 13 illustrated in FIG. 1 may be employed without employing the bivector T1 and the parameter A which have been input through the fifth device 25 illustrated in FIG. 3.

Hereinbelow is explained an operation of the apparatus for summing bivectors, illustrated in FIGS. 1 and 3. Herein, it is supposed that a curve defined by the equation y.sup.3 =x.sup.4 +1 on a finite field GF(17) is employed.

In the apparatus for summing bivectors, illustrated in FIG. 1, it is supposed that the fourth device 14 receives two bivectors X1 and X2 on a finite field GF(17) having an order number of 17, and a parameter A defining the curve y.sup.3 =F(x)=x.sup.4 +1 on the finite field GF(17). Data format of the parameter A is as shown in FIG. 5. The bivectors X1 and X2, and the parameter A are represented as follows.

X1=((0, 1), (1, 8), (2, 0))

X2=((3, 10), (4, 8), (5, 10))

A=(3, 4, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1)

The central processing unit 16 temporarily stores the bivectors X1 and X2, and the parameter A transmitted from the fourth device 14, in the memory 13. Then, the central processing unit 16 receives the bivectors X1 and X2 from the first and second storage files, respectively, and transmits them to the first device 11 for calculating a union-set.

The first device 11 for calculating a union-set considers the bivectors X1 and X2 as coordinate value rows of points on the curve, respectively, and calculates a union-set of them. That is, the first device 11 considers the bivector X1 as a set composed of three points (0, 1), (1, 8) and (2, 0), and the bivector X2 as a set composed of three points (3, 10), (4, 8) and (5, 10), and calculates a union-set of the bivectors X1 and X2 to thereby obtain a set, {(0, 1), (1, 8), (2, 0), (3, 10), (4, 8), (5, 10)}. Then, the first device 11 outputs the bivector T1 associated the thus obtained set, namely, the bivector T1 represented with ((0, 1), (1, 8), (2, 0), (3, 10), (4, 8), (5, 10)).

Then, the central processing unit 16 temporarily stores the bivector T1 transmitted from the first device, in the memory 13.

Then, the central processing unit 16 receives the bivector T1 from the fourth storage file, and the paramete