116 research outputs found
On modular k-free sets
Let and be integers. A set is
-free if for all in , . We determine the maximal
cardinality of such a set when and are coprime. We also study several
particular cases and we propose an efficient algorithm for solving the general
case. We finally give the asymptotic behaviour of the minimal size of a
-free set in which is maximal for inclusion
Hard Instances of the Constrained Discrete Logarithm Problem
The discrete logarithm problem (DLP) generalizes to the constrained DLP,
where the secret exponent belongs to a set known to the attacker. The
complexity of generic algorithms for solving the constrained DLP depends on the
choice of the set. Motivated by cryptographic applications, we study sets with
succinct representation for which the constrained DLP is hard. We draw on
earlier results due to Erd\"os et al. and Schnorr, develop geometric tools such
as generalized Menelaus' theorem for proving lower bounds on the complexity of
the constrained DLP, and construct sets with succinct representation with
provable non-trivial lower bounds
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