The Discrete Logarithm (DL Sample Clauses

The Discrete Logarithm (DL. Assumption In a cyclic group G1 with prime order q and a generator P ∈ G1, the problem given P and Y = yP with random integer y ∈ Z∗ to compute y is hard. Theorem 1 Based on the DL assumption, the problem, given ⟨q, X0, X0, xˆ, X, xX ⟩ q where y is a random integer in Z∗, to find integer v and b such that eˆ(P, P )v = eˆ(P, P )by is hard. Proof: The proof is straightforward. If there exists a randomized polynomial algorithm A to find the integer v and b with non-negligible advantage, we can simply use this algorithm to solve the DL problem by returning y = v/b. However the above theorem is only useful to evaluate some attacks presented in the following part. In fact, an adversary does not need to find both b and v such that eˆ(P, P )v = eˆ(P, P )by at the same time to launch an attack. To break the protocol, an adversary simply needs to find v and bP (instead of b) satisfying equation eˆ(vP, P ) = eˆ(bP, yP ). Fortunately it seems to be hard to solve this problem formally defined as following. Assumption 3 Given ⟨q, eˆ, G , G , P, yP ⟩ with y ←R− Z∗, it is hard to find v ∈ 1 2 q Z∗ and Q = bP ∈ G∗ such that eˆ(P, P )v = eˆ(Q, yP ) = eˆ(P, P )by.
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