Encrypting device, decrypting device, cryptosystem including the same devices, encrypting method, and decrypting method
View Patent ↗A cryptosystem includes an encrypting device, a communication path, and a decrypting arithmetic device. Key generation means in the encrypting device generate a public key {g 1 , g 2 } as random numbers respectively including the power of (p−1) and the power of (q−1) and decrypt a message m using the Fermat's little theorem and the Chinese remainder theorem. This makes it possible to suggest an extremely simple cryptosystem, which is simplified by reducing the amount of computations for encryption and decryption and enables encryption and decryption by simple calculations, while maintaining a security equivalent to the RSA encryption scheme.
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,
g 1 =g s(p−1) (mod n), (1)
g 2 =g t(q−1) (mod n), (2)
C 1 =m·g 1 r1 (mod n), (3)
C 2 =m·g 2 r2 (mod n), (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,
g 1 =g s(p−1) (mod n), (1)
C=m·g 1 r (mod n), (3)′
where when b is a size of p (bits), 0<m<2 b−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=(C 1 +C 2 )/m (mod n), is added to the ciphertext C=(C 1 , C 2 ) so as to be a ciphertext C=(C 1 , C 2 , 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 C 1 =(C 11 , C 12 ), and ciphertext elements following the ciphertext element C 1 are generated using a received plaintext m i , bit information of the plaintext m 1 , and two random numbers R 1 or R 2 which are contained in the ciphertext C 1 .
6. A decrypting device wherein included are decrypting arithmetic means for receiving a ciphertext C=(C 1 , C 2 ), which is an encrypted plaintext m, respectively given by the following Equations (3) and (4) using a public key {g 1 , g 2 }, 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,
g 1 =g s(p−1) (mod n), (1)
g 2 =g t(q−1) (mod n), (2)
C 1 =m·g 1 r1 (mod n), (3)
C 2 =m·g 2 r2 (mod n), (4)
a=C 1 (mod p)= m (mod p), (5)
b=C 2 (mod q)= m (mod q), (6)
m=aAq+bBp (mod n), (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,
g 1 =g s(p−1) (mod n), (1)
C=m·g 1 r (mod n), (3)′
m=C (mod p), (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=(C 1 , C 2 ) respectively given by the following Equations (3) and (4) using the public key {g 1 , g 2 }, a private key n, and random numbers r1 and r2; and
a decrypting device including decrypting arithmetic means for receiving ciphertext elements C 1 and C 2 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,
g 1 =g s(p−1) (mod n), (1)
g 2 =g t(q−1) (mod n), (2)
C 1 =m·g 1 r1 (mod n), (3)
C 2 =m·g 2 r2 (mod n), (4)
a=C 1 (mod p)= m (mod p), (5)
b=C 2 (mod q)= m (mod q), (6)
m=aAq+bBp (mod n), (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,
g 1 =g s(p−1) (mod n), (1)
C=m·g 1 r (mod n), (3)′
m=C (mod p), (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 g 1 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 Cl 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,
g 1 =g s(p−1) (mod n), (1)
g 2 =g t(q−1) (mod n), (2)
C 1 =m·g 1 r1 (mod n), (3)
C 2 =m·g 2 r2 (mod n), (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 g 1 , a private key n, and a random number r,
g 1 =g s(p−1) (mod n), (1)
C=m·g 1 r (mod n), (3)′
where when b is a size of p (bits), 0<m<2 b−1 and gcd{s, q−1}=1.
12. A decrypting method comprising the steps of:
receiving a ciphertext C=(C 1 , C 2 ), 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,
g 1 =g s(p−1) (mod n), (1)
g 2 =g t(q−1) (mod n), (2)
C 1 =m·g 1 r1 (mod n), (3)
C 2 =m·g 2 r2 (mod n), (4)
a=C 1 (mod p)= m (mod p), (5)
b=C 2 (mod q)= m (mod q), (6)
m=aAq+bBp (mod n), (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, g 1 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,
g 1 =g s(p−1) (mod n), (1)
C=m·g 1 r (mod n), (3)
m=C (mod p), (8)
where gcd{s, q−1}=1.