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After some excitement generated by recently suggested public key exchange protocols due to Anshel–Anshel–Goldfeld and Ko–Lee et al., it is a prevalent opinion now that the conjugacy search problem is unlikely to provide sufficient level of security if a braid group is used as the platform. In this paper we address the following questions: (1) whether(More)
We prove that Whitehead's algorithm for solving the au-tomorphism problem in a fixed free group F k has strongly linear time generic-case complexity. This is done by showing that the " hard " part of the algorithm terminates in linear time on an exponentially generic set of input pairs. We then apply these results to one-relator groups. We obtain a(More)
In this paper we present a new key establishment protocol based on the decomposition problem in non-commutative groups which is: given two elements w, w1 of the platform group G and two subgroups A, B ⊆ G (not necessarily distinct), find elements a ∈ A, b ∈ B such that w1 = awb. Here we introduce two new ideas that improve the security of key establishment(More)
We offer a public key exchange protocol in the spirit of Diffie-Hellman, but we use (small) matrices over a group ring of a (small) symmetric group as the platform. This " nested structure " of the platform makes computation very efficient for legitimate parties. We discuss security of this scheme by addressing the Decision Diffie-Hellman (DDH) and(More)
Let K[x, y] be the polynomial algebra in two variables over a field K of characteristic 0. In this paper, we contribute toward a classification of two-variable polynomials by classifying (up to an automor-phism of K[x, y]) polynomials of the form ax n + by m + im+jn≤mn c ij x i y j , a, b, c ij ∈ K (i.e., polynomials whose Newton polygon is either a(More)
The conjugacy search problem in a group G is the problem of recovering an $$x \in G$$ from given $$g \in G$$ and h = x −1 gx. This problem is in the core of several recently suggested public key exchange protocols, most notably the one due to Anshel, Anshel, and Goldfeld, and the one due to Ko, Lee et al. In this note, we make two observations that seem to(More)
Motivated by cryptographic applications, we study subgroups of braid groups Bn generated by a small number of random elements of relatively small lengths compared to n. Our experiments show that " most " of these subgroups are equal to the whole Bn, and " almost all " of these subgroups are generated by positive braid words. We discuss the impact of these(More)
One of the possible generalizations of the discrete logarithm problem to arbitrary groups is the so-called conjugacy search problem (sometimes erroneously called just the conjugacy problem) : given two elements a, b of a group G and the information that a x = b for some x ∈ G, find at least one particular element x like that. Here a x stands for xax −1. The(More)
We propose an authentication scheme where forgery (a.k.a. impersonation) seems infeasible without finding the prover's long-term private key. The latter would follow from solving the conjugacy search problem in the platform (noncommutative) semigroup, i.e., to recovering X from X −1 AX and A. The platform semigroup that we suggest here is the semigroup of n(More)