Volume 5, Issue 1

Abstract. We extend results of [Proc. Lond. Math. Soc. 104 (2012), 486–512] and prove shortlex automaticity and regularity of geodesics in a family of Artin groups that includes all groups of large type but also allows some commuting pairs of generators.

Abstract. The generic-case approach to algorithmic problems was suggested by Myasnikov, Kapovich, Schupp and Shpilrain in 2003. This approach studies the behavior of an algorithm on “most” or “typical” inputs. The remaining inputs form the so-called black hole of the algorithm. In the present paper we consider Hilbert's tenth problem and use arithmetic circuits for the representation of Diophantine equations. We prove that this Diophantine problem is generically hard in the following sense. For every generic polynomial algorithm deciding this problem, there exists a polynomial algorithm for random generation of inputs from the black hole.

Abstract. We show that some problems in information security can be solved without using one-way functions. The latter are usually regarded as a central concept of cryptography, but the very existence of one-way functions depends on difficult conjectures in complexity theory, most notably on the notorious “” conjecture. This is why cryptographic primitives that do not employ one-way functions are often called “unconditionally secure”. In this paper, we suggest protocols for secure computation of the sum, product, and some other functions of two or more elements of an arbitrary constructible ring, without using any one-way functions. A new input that we offer here is that, in contrast with other proposals, we conceal “intermediate results” of a computation. For example, when we compute the sum of k numbers, only the final result is known to the parties; partial sums are not known to anybody. Other applications of our method include voting/rating over insecure channels and a rather elegant and efficient solution of the “two millionaires problem”. Then, while it is fairly obvious that a secure (bit) commitment between two parties is impossible without a one-way function, we show that it is possible if the number of parties is at least 3. We also show how our unconditionally secure (bit) commitment scheme for three parties can be used to arrange an unconditionally secure (bit) commitment between just two parties if they use a “dummy” (e.g., a computer) as the third party. We explain how our concept of a “dummy” is different from the well-known concept of a “trusted third party”. Based on a similar idea, we also offer an unconditionally secure k - n oblivious transfer protocol between two parties who use a “dummy”. We also suggest a protocol, without using a one-way function, for the so-called “mental poker”, i.e., a fair card dealing (and playing) over distance. Finally, we propose a secret sharing scheme where an advantage over Shamir's and other known secret sharing schemes is that nobody, including the dealer, ends up knowing the shares (of the secret) owned by any particular player. It should be mentioned that computational cost of our protocols is negligible to the point that all of them can be executed without a computer.

Abstract. We construct a provably pseudo-free family of finite computational groups under the general integer factoring intractability assumption. This family has exponential size. But each element of a group in our pseudo-free family is represented by infinitely many bit strings.

Abstract. We present a new algorithm to find apartments in coset geometries. It permits to obtain some missing results in a paper by Buekenhout and Leemans. We compare the performance of our algorithm to the one given by Buekenhout and Leemans.

Abstract. In [Groups Complex. Cryptol. 3 (2011), 349–355] we showed that any hyperbolic limit group can be faithfully represented in . The proof was constructive in that given a fixed JSJ decomposition for the given limit group the representation can be constructed. The proof depended on showing that certain amalgams of groups admitting faithful representations into also admit such faithful representations. In this short note we give an elegant proof that the restriction to the hyperbolic case can be removed.

Abstract. 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 Computational Diffie–Hellman (CDH) problems for our platform.