Band 4, Heft 1

Abstract. Key establishment protocols based on hardness assumptions, such as the discrete logarithm problem and the integer factorization problem, are vulnerable to quantum computer attacks, whereas the protocols based on other hardness assumptions, such as the conjugacy search problem and the decomposition search problem, can resist such attacks. The existing protocols based on the hardness assumptions which can resist quantum computer attacks are only passively secure. Compilers are used to convert a passively secure protocol to an actively secure protocol. Compilers involve some tools such as a signature scheme and a collision-resistant hash function. If there are only passively secure protocols but not a signature scheme based on the same assumption, then the application of existing compilers requires the use of such tools based on different assumptions. But the introduction of new tools, based on different assumptions, makes the new actively secure protocol rely on more than one hardness assumption. We offer an approach to derive an actively secure two-party protocol from a passively secure two-party protocol without introducing further hardness assumptions. This serves as a useful formal tool to transform any basic algebraic method of public key cryptography to the real world applicable cryptographic scheme.

Abstract. We consider the problem of constructing continuous cryptographic primitives. We present several candidates for continuous hard-to-invert functions. To formulate these candidates, we introduce constructions based on tropical and supertropical circuits.

Abstract. Direct products of finite groups are a simple method to construct new groups from old ones. A difficult problem by comparison is to prove a generic group is indecomposable, or locate a proper nontrivial direct factor. To solve this problem it is shown that in most circumstances has a proper nontrivial subgroup such that every maximal direct product decomposition of induces a unique set of subgroups of where and for each , the nonabelian direct factors of are direct factors of . In particular, is indecomposable if and is contained in the Frattini subgroup of . This “local-global” property of direct products can be applied inductively to and so that the existence of a proper nontrivial direct factor depends on the direct product decompositions of the chief factors of . Chief factors are characteristically simple groups and therefore a direct product of isomorphic simple groups. Thus a search for proper direct factors of a group of size is reduced from the global search through all normal subgroups to a search of local instances induced from chief factors. There is one family of groups where no subgroup admits the local-global property just described. These are -groups of nilpotence class 2. There are isomorphism types of class 2 groups with order , which prevents a case-by-case study. Also these groups arise in the course of the induction described above so they cannot be ignored. To identify direct factors for nilpotent groups of class 2, a functor is introduced to the category of commutative rings. The result being that indecomposable -groups of class 2 are identified with local commutative rings. This relationship has little to do with the typical use of Lie algebras for -groups and is one of the essential and unexpected components of this study. These results are the by-product of an efficient polynomial-time algorithm to prove indecomposability or locate a proper nontrivial direct factor. The theorems also explain how many isomorphism types of indecomposable groups exists of a given order and how many direct factors a group can have. These two topics are explained in a second part to this paper.

Abstract. For every odd prime and every integer , there is a Heisenberg group of order that has pairwise nonisomorphic quotients of order . Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most . They are also directly and centrally indecomposable and of the same indecomposability type. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.

Abstract. In 2003 Cohn and Umans introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group satisfying the triple product property (TPP) as a means to bound the exponent of matrix multiplication. We present two new characterizations of the TPP, which are used for theoretical considerations and for TPP test algorithms. We describe the algorithms for all known TPP tests and present the runtime differences between their GAP implementations. We prove that the search for non-trivial-sized TPP triples of subgroups of a given group can be restricted to the set of its non-normal subgroups, and apply this, together with other preconditions, to describe brute-force search algorithms for largest-sized TPP triples of subgroups and subsets. In addition we present the results of the subset brute-force search for all groups of order up to 32 and selected results of the subgroup brute-force search for 2-groups, and . Our results for the groups and suggest tentative answers to certain questions posed by Cohn and Umans.

Abstract. We consider the multiple conjugacy search problem over a subclass of partially commutative groups and experimentally attack it with a genetic algorithm hybridised with a “length attack”. We detail symbolic computation of words over the groups, constructing functions which measure certain statistics of those words. By experimentation, the hybrid algorithm is shown to be effective, showing that the standard conjugacy search problem is harder than the multiple conjugacy search problem for our groups. Moreover, some intuitive methods of increasing problem difficulty are overcome by the algorithm, and in fact make the problem easier to solve. We show our algorithm is efficient, comparing well with traditional approaches in groups that are statistically similar. Finally, via “approximation” of braid groups by our subclass, we consider implications of the attack on certain cryptosystems, pointing to further work in the discipline of group-theoretic cryptography.

Abstract. We consider a key exchange procedure whose security is based on the difficulty of computing discrete logarithms in a group, and where exponentiation is hidden by a conjugation. We give a platform-dependent cryptanalysis of this protocol. Finally, to take full advantage of this procedure, we propose a group of matrices over a noncommutative ring as platform group.

Abstract. We propose a class of authentication schemes that are literally zero-knowledge, as compared to what is formally defined as “zero-knowledge” in cryptographic literature. We call this “no-leak” authentication to distinguish from an established “zero-knowledge” concept. The “no-leak” condition implies “zero-knowledge” (even “perfect zero-knowledge”), but it is actually stronger, as we illustrate by examples. The principal idea behind our schemes is: the verifier challenges the prover with questions that he (the verifier) already knows answers to; therefore, even a computationally unbounded verifier who follows the protocol cannot possibly learn anything new during any number of authentication sessions. This is therefore also true for a computationally unbounded passive adversary.