Picard Groups and Refined Discrete Logarithms

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.

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...

On the Relation between Brauer Groups and Discrete Logarithms

In this article we want to make evident that Brauer groups of local and global fields play an important role in public key cryptopgraphy. In the first section we show that all ideal class groups resp. divisor class groups attached to curves over finite fields Fq taken into consideration for DL-systems can be mapped to such Brauer groups via the Tate-Lichtenbaum pairing. It is well known that th...

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...

On the Computation of Discrete Logarithms in Class Groups