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Prime order elliptic curve generation method and system for signature and key exchange
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An elliptic curve and key exchange technology, which is applied in the field of prime number order elliptic curve generation method and system, can solve the problems of low efficiency and achieve the effect of high security
Active Publication Date: 2022-04-12
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[0004] With the continuous improvement of computing speed and the gradual improvement of algorithms, the ability to crack elliptic curve cryptography will also be strengthened, so the elliptic curve currently in use in practice may need to be gradually replaced in the future, replaced by a larger prime number field elliptic curve of prime number order, and the inventor found that the current method of constructing a safe elliptic curve is mainly a random selection method, that is, first generating a pseudo-random elliptic curve, then calculating the order of the elliptic curve, and finally judging whether it satisfies the condition of primality or approximate primality. This method is a violent search method with low efficiency
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Embodiment 1
[0064] The prime number order elliptic curve generation method used for signature and key exchange in this embodiment, the prime number order elliptic curve is used to sign the file together with the private key of the sender, and the receiver uses the shared public key and prime number order Elliptic curve to verify the validity of the signature.
[0065] In specific implementation, in addition to solving the quadratic remaining 2 roots, the present embodiment only needs to find the 3 k + 1 type prime numbers of the 3 remaining 3 roots, that is, gcd (n, p-1)=3, so only It is necessary to give an algorithm for solving the three remaining cubic roots.
[0066] ① Algorithm for finding the 2nd root of the quadratic remainder:
[0067] Input: prime numbers p and
[0068] output: minimal make x 2 ≡a(modp) or "a is not a quadratic remainder modulo p".
[0069] Step 1: Input prime numbers p and a.
[0070] Step 2: Use Euclidean algorithm and Miller algorithm to check whether...
Embodiment 2
[0152] This embodiment provides a prime number order elliptic curve generation system for signature and key exchange. The prime number order elliptic curve is used to sign files together with the sender's private key, and the receiver uses the shared public key and prime number order elliptic curve to verify the validity of the signature; the prime number order elliptic curve generation system for signature and key exchange includes:
[0153] A large prime number generation module, which is used to generate a large prime number p of the 3k+1 type with a specified number of digits n and a sparse representation;
[0154] Large prime factorization module for decomposing p into p=c 2 -cd+d 2 of the form, where c≡2(mod 3), d≡0(mod 3);
[0155] A prime number judging module, which is used to judge the following n i , whether there is a prime number in 1≤i≤6, if not, regenerate p:
[0156] no 1 :=p+1-d+2c,n 2 :=p+1+c+d, n 3 :=p+1-c+2d,
[0157] no 4 :=p+1+d-2c,n 5 :=p+1-c-d...
Embodiment 3
[0162] This embodiment provides a computer-readable storage medium, on which a computer program is stored, and when the program is executed by a processor, the method for generating a prime order elliptic curve for signature and key exchange as described in the first embodiment above is implemented in the steps.
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Abstract
The invention belongs to the field of computer applications and provides a method and system for generating a prime number order elliptic curve for signature and key exchange. Wherein, the prime number order elliptic curve is used to sign the file together with the private key of the sender, and the validity of the signature is verified by the receiver using the shared public key and the prime number order elliptic curve; or the prime number order elliptic curve is used to on the basis of finding enough secure elliptic curves, the prime order elliptic curve generation method can further select a Koblitz curve with better mathematical properties, and realize the cryptography contained in the signature and encryption on it The purpose of faster computing speed and higher security.
Description
technical field [0001] The invention belongs to the field of computer applications, in particular to a method and system for generating prime order elliptic curves for signature and key exchange. Background technique [0002] The statements in this section merely provide background information related to the present invention and do not necessarily constitute prior art. [0003] The secure encryption and decryption algorithms and signature algorithms used in data transmission, storage and identity authentication play a vital role in building a safe and stable network environment. Traditional public key cryptosystems are mainly divided into two categories: public key cryptosystems based on integer factorization, such as RSA; public key cryptosystems based on discrete logarithms, such as DSA, DH; public key cryptosystems based on elliptic curves (ECC) also belongs to the public key cryptosystem based on discrete logarithm. Among them, the public-key cryptosystem based on ell...
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