Digital signature method and system with security only depending on discrete logarithms
A digital signature and discrete logarithm technology, applied in the field of information security, can solve problems such as long signature data, and achieve the effects of short signature data, high efficiency, and low computational complexity
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Embodiment 1
[0081] This embodiment is a digital signature method based on elliptic curve discrete logarithm, and the process is as follows:
[0082] System parameter generation: finite field is F q , the elliptic curve y 2 =x 3 +ax+b in the finite field F q The two parameters above are a, b. point group on elliptic curve The generator of is G, and its order is a large prime number n, where n>2 160 , the system parameters are:
[0083]
[0084] Key generation: choose a random number As private key SK=(α,β);
[0085] Calculated as follows
[0086]
[0087]
[0088] Then the public key is PK=(A, B).
[0089] Signature: choose message m ∈ Z n ,random number Using the formula (x 1 ,y 1 )=k·G to determine the new coordinate point (x 1 ,y 1 ), G is the coordinate point, after calculation with k, a new coordinate point (x 1 ,y 1 ).
[0090] According to the abscissa x of the new coordinate point 1 , using the formula σ 1 =x 1 modn calculates the first parameter σ...
Embodiment 2
[0103] This embodiment is a digital signature based on discrete logarithm, and the specific process is as follows:
[0104] System parameter generation: select a large prime number q,p, where q≥2 160 ,p≥2 1024 , and q|p-1. The generator of the cyclic group G is g, and its order is q, then the system parameters are SP=(G,g,q,p)
[0105] Key generation: choose a random number As the private key SK=(α,β), it is calculated as follows
[0106] A=g α mod p∈G
[0107] B=g β mod p∈G
[0108] Then the public key is PK=(A, B).
[0109] Signature: choose message m ∈ Z n ,random number calculate
[0110] X=g k mod p
[0111] σ 1 =X mod q
[0112] σ 2 =k -1 (α+m+β·σ 1 ) mod q
[0113] If σ 1 = 0 or σ 2 = 0, reselect the random number k and recalculate σ 1 ,σ 2 ;
[0114] If σ 1 ≠0 and σ 2 ≠0, then the signature is determined to be (σ 1 ,σ 2 ).
[0115] Verification: check σ 1 ,σ 2 Whether they all belong to [1,q-1], if not, the signature verification is rej...
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