Claims
- 1. An encrypting device comprising:
key generation means for generating two prime numbers p and q of which product is n=pq as a private key and generating as a public key g1 and g2 respectively given by the following Equations (1) and (2) using two random numbers s and t and a maximal generator g in a multiplicative group of integers modulo n; and encrypting arithmetic means for, in response to receipt of a plaintext m, generating a ciphertext C=(C1, C2) respectively given by the following Equations (3) and (4) using the public key {g1, g2}, a private key n, and random numbers r1 and r2, g1=gs(p−1)(modn), (1) g2=gt(q−1)(modn), (2) C1=m·g1r1(modn), (3) C2=m·g2r2(modn), (4) where gcd{s, q−1}=1 and gcd{t, p−1}=1.
- 2. An encrypting device comprising:
key generation means for generating prime numbers p and q of which product is n=pq, where p is a private key, and generating as a public key g1 given by the following Equation (1) using a random number s and a maximal generator g in a multiplicative group of integers modulo n; and encrypting arithmetic means for, in response to receipt of a plaintext m, generating a ciphertext C given by the following Equation (3)′ using the public key g1, a private key n, and a random number r, g1=gs(p−1)(modn), (1) C=m·g1r(modn), (3)′where when information b is a size of p (bits), 0<m<2b−1 and gcd{s, q−1}=1.
- 3. The encrypting device according to claim 1, wherein:
e given by the following equation: e=h(d) (h is one-way hash function), where d=(C1+C2)/m (mod n), is added to the ciphertext C=(C1, C2) so as to be a ciphertext C=(C1, C2, e).
- 4. The encrypting device according to claim 1, further comprising:
a database for saving data resulting from calculation of a random number portion of the ciphertext C.
- 5. The encrypting device according to claim 1, wherein:
the encrypting arithmetic means encrypt only a plaintext element m1, which is a first element in the plaintext m, to the ciphertext element C1=(C11, C12), and ciphertext elements following the ciphertext element C1 are generated using a received plaintext mi, bit information of the plaintext m1, and two random numbers R1 or R2 which are contained in the ciphertext C1.
- 6. A decrypting device wherein included are decrypting arithmetic means for receiving a ciphertext C=(C1, C2), which is an encrypted plaintext m, respectively given by the following Equations (3) and (4) using a public key {g1, g2}, a private key n, and random numbers r1 and r2, the private key n being n=pq where p and q are prime numbers generated as a private key, g1 and g2 being respectively given by the Equations (1) and (2) using two random numbers s and t and a maximal generator g in a multiplicative group of integers modulo n, and
performing decryption in such a manner so as to generate received ciphertexts a and b respectively given by the following Equations (5) and (6) using the Fermat's little theorem and then derive the plaintext m satisfying the following Equation (7) from the received ciphertexts a and b using the Chinese remainder theorem, g1=gs(p−1)(modn), (1) g2=gt(q−1)(modn), (2) C1=m·g1r1(modn), (3) C2=m·g2r2(modn), (4) a=C1(modp)=m(modp), (5) b=C2(modq)=m(modq), (6) m=aAq+bBp(modn), (7) where gcd{s, q−1}=1, gcd{t, p−1}=1, Aq (mod p)=1, and Bp (mod q)=1.
- 7. A decrypting device wherein included are decrypting arithmetic means for receiving a ciphertext C of an inputted plaintext m, given by the following Equation (3)′ using a public key g1, a private key n, and a random number r, the private key n being n=pq where p and q are prime numbers, p being generated as a private key, g1 being given by the following Equation (1) using a random number s and a maximal generator g in a multiplicative group of integers modulo n, and
performing decryption in such a manner so as to derive the plaintext m satisfying the following Equation (8) using the Fermat's little theorem, g1=gs(p−1)(modn), (1) C=m·g1r(modn), (3)′m=C(modp), (8) where gcd{s, q−1}=1.
- 8. A cryptosystem comprising:
an encrypting device including: key generation means for generating two prime numbers p and q of which product is n=pq as a private key and generating as a public key g1 and g2 respectively given by the following Equations (1) and (2) using two random numbers s and t and a maximal generator g in a multiplicative group of integers modulo n; and encrypting arithmetic means for, in response to receipt of a plaintext m, generating a ciphertext C=(C1, C2) respectively given by the following Equations (3) and (4) using the public key {g1, g2}, a private key n, and random numbers r1 and r2; and a decrypting device including decrypting arithmetic means for receiving ciphertext elements C1 and C2 calculated by the encrypting device and performing decryption in such a manner so as to generate received ciphertexts a and b respectively given by the following Equations (5) and (6) using the Fermat's little theorem and then derive the plaintext m satisfying the following Equation (7) from the received ciphertexts a and b using the Chinese remainder theorem, g1=gs(p−1)(modn), (1) g2=gt(q−1)(modn), (2) C1=m·g1r1(modn), (3) C2=m·g2r2(modn), (4) a=C1(modp)=m(modp), (5) b=C2(modq)=m(modq), (6) m=aAq+bBp(modn), (7) where gcd{s, q−1}=1, gcd{t, p−1}=1, Aq (mod p)=1, and Bp (mod q)=1.
- 9. A cryptosystem comprising:
an encrypting device including: key generation means for generating prime numbers p and q of which product is n=pq, where p is a private key, and generating as a public key g1 given by the following Equation (1) using a random number s and a maximal generator g in a multiplicative group of integers modulo n; and encrypting arithmetic means for, in response to receipt of a plaintext m, generating a ciphertext C given by the following Equation (3)′ using the public key g1, a private key n, and a random number r; and a decrypting device including decrypting arithmetic means for receiving the ciphertext C from the encrypting device and performing decryption in such a manner so as to derive the plaintext m satisfying the following Equation (8) using the Fermat's little theorem, g1=gs(p−1)(modn), (1) C=m·g1r(modn), (3)′m=C(modp), (8) where gcd{s, q−1}=1.
- 10. An encrypting method comprising the steps of:
generating two prime numbers p and q of which product is n=pq as a private key and generating as a public key g1 and g2 respectively given by the following Equations (1) and (2) using two random numbers s and t and a maximal generator g in a multiplicative group of integers modulo n; and in response to receipt of a plaintext m, generating ciphertext elements C1 and C2 respectively given by the following Equations (3) and (4) using the public key {g1, g2}, a private key n, and random numbers r1 and r2, g1=gs(p−1)(modn), (1) g2=gt(q−1)(modn), (2) C1=m·g1r1(modn), (3) C2=m·g2r2(modn), (4) where gcd{s, q−1}=1 and gcd{t, p−1}=1.
- 11. An encrypting method comprising the steps of:
generating prime numbers p and q of which product is n=pq, where p is a private key, and generating as a public key g1 given by the following Equation (1) using a random number s and a maximal generator g in a multiplicative group of integers modulo n; and in response to receipt of a plaintext m, generating a ciphertext C given by the following Equation (3)′ using the public key g1, a private key n, and a random number r, g1=gs(p−1)(modn), (1) C=m·g1r(modn), (3)′where when information b is a size of p (bits), 0<m<2b−1 and gcd{s, q−1}=1.
- 12. A decrypting method comprising the steps of:
receiving a ciphertext C=(C1, C2), which is an encrypted plaintext m, respectively given by the following Equations (3) and (4) using a public key {g1, g2}, a private key n, and random numbers r1 and r2, the private key n being n=pq where p and q are prime numbers generated as a private key, g1 and g2 being respectively given by the Equations (1) and (2) using two random numbers s and t and a maximal generator g in a multiplicative group of integers modulo n; and performing decryption in such a manner so as to generate received ciphertexts a and b respectively given by the following Equations (5) and (6) using the Fermat's little theorem and then derive the plaintext m satisfying the following Equation (7) from the received ciphertexts a and b using the Chinese remainder theorem, g1=gs(p−1)(modn), (1) g2=gt(q−1)(modn), (2) C1=m·g1r1(modn), (3) C2=m·g2r2(modn), (4) a=C1(modp)=m(modp), (5) b=C2(modq)=m(modq), (6) m=aAq+bBp(modn), (7) where gcd{s, q−1}=1, gcd{t, p−1}=1, Aq (mod p)=1, and Bp (mod q)=1.
- 13. A decrypting method comprising the steps of:
receiving a ciphertext C of an inputted plaintext m, given by the following Equation (3)′ using a public key g1, a private key n, and a random number r, the private key n being n=pq where p and q are prime numbers, p being generated as a private key, g1 being given by the following Equation (1) using a random number s and a maximal generator g in a multiplicative group of integers modulo n; and performing decryption in such a manner so as to derive the plaintext m satisfying the following Equation (8) using the Fermat's little theorem, g1=gs(p−1)(modn), (1) C=m·g1r(modn), (3)′m=C(modp), (8) where gcd{s, q−1}=1.
Priority Claims (2)
Number |
Date |
Country |
Kind |
2003-16761 |
Jan 2003 |
JP |
|
2004-013401 |
Jan 2004 |
JP |
|
Parent Case Info
[0001] This Nonprovisional application claims priority under 35 U.S.C. § 119(a) on Patent Application No. 16761/2003 filed in Japan on Jan. 24, 2003, and Patent Application No. 13401/2004 filed in Japan on Jan. 21, 2004, the entire contents of which are hereby incorporated by reference.