Discrete logarithms in free groups

Weak discrete logarithms in non-abelian groups

The intractability of the traditional discrete logarithm problem (DLP) forms the basis for the design of numerous cryptographic primitives. In [2] M. Sramka et al. generalize the DLP to arbitrary finite groups. One of the reasons mentioned for this generalization is P. Shor’s quantum algorithm [4] which solves efficiently the traditional DLP. The DLP for a nonabelian group is based on a particu...

Discrete Logarithms and Mordell-Weil Groups

Let Ep be an elliptic curve over a prime finite field Fp, p ≥ 5, and Pp, Qp ∈ Ep(Fp). The elliptic curve discrete logarithm problem, ECDLP, on Ep is to find mp ∈ Fp such that Qp = mpPp if Qp ∈ 〈Pp〉. We propose an algorithm to attack the ECDLP relying on a Hasse principle detecting linear dependence in Mordell-Weil groups of elliptic curves via a finite number of reductions.

On the Computation of Discrete Logarithms in Class Groups

Solving Discrete Logarithms in Smooth-Order Groups with CUDA

This paper chronicles our experiences using CUDA to implement a parallelized variant of Pollard’s rho algorithm to solve discrete logarithms in groups with cryptographically large moduli but smooth order using commodity GPUs. We first discuss some key design constraints imposed by modern GPU architectures and the CUDA framework, and then explain how we were able to implement efficient arbitrary...

Structure computation and discrete logarithms in finite abelian p-groups

We present a generic algorithm for computing discrete logarithms in a finite abelian p-group H, improving the Pohlig–Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for H without using a relation matrix. The problem of computing a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian group G is addres...