Tame logarithmic signatures of abelian groups

2017 ◽  
Vol 11 (4) ◽  
Author(s):  
Dominik Reichl
Keyword(s):  
Abelian Groups ◽  
Logarithmic Signatures ◽  
Asymmetric Cryptosystem

AbstractThe security of the asymmetric cryptosystem MST

scholarly journals LOGARITHMIC SIGNATURES FOR ABELIAN GROUPS AND THEIR FACTORIZATION

2013 ◽  
Vol 57 (1) ◽  
pp. 21-33 ◽  
Author(s):  
Pavol Svaba ◽  
Tran van Trung ◽  
Paul Wolf
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Efficient Algorithms ◽  
Finite Abelian Groups ◽  
Logarithmic Signatures ◽  
Vital Importance ◽  
Essential Component ◽  
Factorization Algorithms ◽  
Factoring Algorithm

ABSTRACT Factorizable logarithmic signatures for finite groups are the essential component of the cryptosystems MST1 and MST3. The problem of finding efficient algorithms for factoring group elements with respect to a given class of logarithmic signatures is therefore of vital importance in the investigation of these cryptosystems. In this paper we are concerned about the factorization algorithms with respect to transversal and fused transversal logarithmic signatures for finite abelian groups. More precisely we present algorithms and their complexity for factoring group elements with respect to these classes of logarithmic signatures. In particular, we show a factoring algorithm with respect to the class of fused transversal logarithmic signatures and also its complexity based on an idea of Blackburn, Cid and Mullan for finite abelian groups.

scholarly journals Strongly aperiodic logarithmic signatures

2013 ◽  
Vol 7 (2) ◽  
pp. 147-179
Author(s):  
Reiner Staszewski ◽  
Tran van Trung
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Public Key ◽  
Logarithmic Signatures ◽  
Public Key Cryptosystems ◽  
Private Key ◽  
Main Component

Abstract. Logarithmic signatures for finite groups are the essential constituent of public key cryptosystems and . Especially they form the main component of the private key of . Regarding the use of , it has become a vital issue to construct new classes of logarithmic signatures having features that do not share with the well-known class of transversal or fused transversal logarithmic signatures. For this purpose Baumeister and de Wiljes recently presented an interesting method of constructing aperiodic logarithmic signatures for abelian groups. In this paper we introduce the concept of strongly aperiodic logarithmic signatures and show their constructions for abelian p-groups on the basis of the Baumeister–de Wiljes method.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

2021 ◽  
pp. 1-36
Author(s):  
ARIE LEVIT ◽  
ALEXANDER LUBOTZKY
Keyword(s):  
Ergodic Theorem ◽  
Abelian Groups ◽  
Wreath Products ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Amenable Groups ◽  
Pointwise Ergodic Theorem ◽  
Lamplighter Group ◽  
Invariant Random Subgroups

Abstract We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

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