Generalized Learning Problems and Applications to Non-commutative Cryptography

Abstract. We propose a generalization of the learning parity with noise (LPN) and learning with errors (LWE) problems to an abstract class of group-theoretic learning problems that we term learning homomorphisms with noise (LHN). This class of problems contains LPN and LWE as spe-cial cases, but is much more general. It allows, for example, instantiations based on non-abelian groups, resulting in a new avenue for the applica-tion of combinatorial group theory to the development of cryptographic primitives. We then study a particular instantiation using relatively free groups and construct a symmetric cryptosystem based upon it

Topology and Non-Deterministic Polynomial Time Computation : Avoidance of The Misbehaviour of Hub-Free Diagrams and Consequences

To study groups with small Dehn's function, Olshanskii and Sapir developed a new invariant of bipartite chords diagrams and applied it to hub-free realization of S-machines. In this paper we consider this new invariant together with groups constructed from S-machines containing the hub relation. The idea is to study the links between the topology of the asymptotic cones and polynomial time computations. Indeed it is known that the topology of such metric space depends on diagrams without hubs that do not correspond to the computations of the considered S-machine. This work gives sufficient conditions that avoid this misbehaviour, but as we shall see the method has a significant drawback

Describing Periodicity in Two-Way Deterministic Finite Automata Using Transformation Semigroups

Abstract. A framework for the study of periodic behaviour of two-way deterministic finite automata (2DFA) is developed. Computations of 2DFAs are represented by a two-way analogue of transformation semi-groups, every element of which describes the behaviour of a 2DFA on a certain string x. A subsemigroup generated by this element represents the behaviour on strings in x+. The main contribution of this paper is a description of all such monogenic subsemigroups up to isomorphism. This characterization is then used to show that transforming an n-state 2DFA over a one-letter alphabet to an equivalent sweeping 2DFA re-quires exactly n+1 states, and transforming it to a one-way automaton requires exactly max06`6nG(n − `) + ` + 1 states, where G(k) is the maximum order of a permutation of k elements.

Hairpin structures in DNA words

Abstract. We formalize the notion of a DNA hairpin secondary structure, examining its mathematical properties. Two related secondary structures are also investigated, taking into the account imperfect bonds (bulges, mismatches) and multiple hairpins. We characterize maximal sets of hairpin-forming DNA sequences, as well as hairpin-free ones. We study their algebraic properties and their computational complexity. Related polynomial-time algorithms deciding hairpin-freedom of regular sets are presented. Finally, effective methods for design of long hairpinfree DNA words are given.