The ElGamal encryption scheme can be used for both digital signatures and encryption, and its security results from the difficulty of calculating discrete logarithms in a finite field. This algorithm usually works in a multiplicative group of GF (p) and in this case the progress in the discrete logarithm problem forces the users of such a basic ElGamal public key cryptosystem to permanently inc...

This series of three lectures of one hour each was preceded by two introductory lectures by Henk van Tilborg about applications of discrete logarithms (in multiplicative groups as well as elliptic curves) to cryptography. Those introductory lectures should now serve as motivation for the coming three lectures. Large cyclic subgroups of prime order in elliptic curves or in Jacobians of higher ge...

In this paper, we obtain an analytical Lyapunovbased stability conditions for scalar linear and nonlinear stochastic systems with discrete time-delay. The Lyapunov– Krasovskii and Lyapunov–Razumikhin methods are applied with techniques from stochastic calculus to obtain the regions of mean square asymptotic stability in the parameter space. Both delay-independent and delay-dependent stability c...

In this paper, we describe a new algorithm for discrete logarithms in small characteristic. This algorithm is based on index calculus and includes two new contributions. The first is a new method for generating multiplicative relations among elements of a small smoothness basis. The second is a new descent strategy that allows us to express the logarithm of an arbitrary finite field element in ...

In cryptanalysis, solving the discrete logarithm problem (DLP) is key to assessing the security of many public-key cryptosystems. The index-calculus methods, that attack the DLP in multiplicative subgroups of finite fields, require solving large sparse systems of linear equations modulo large primes. This article deals with how we can run this computation on GPUand multi-core-based clusters, fe...

We examine ‘weak-distributivity’ as a rewriting rule ; WD defined on multiplicative proof-structures (so, in particular, on multiplicative proof-nets: MLL). This rewriting does not preserve the type of proofsnets, but does nevertheless preserve their correctness. The specific contribution of this paper, is to give a direct proof of completeness for ; WD : starting from a set of simple generator...

We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allows us to address the various interactions between forms and vector fields (suc...

We consider a classical (propositional) version, CBI, of O’Hearn and Pym’s logic of bunched implications (BI) from a modeland prooftheoretic perspective. We present a class of classical models of BI which extend the usual BI-models, based on partial commutative monoids, with an algebraic notion of “resource negation”. This class of models gives rise to natural definitions of multiplicative fals...

The calculus of structures is a proof theoretical formalism which generalizes the sequent calculus with the feature of deep inference: In contrast to the sequent calculus, inference rules can be applied at any depth inside a formula, bringing shorter proofs than any other formalisms supporting analytical proofs. However, deep applicability of the inference rules causes greater nondeterminism th...