arXiv subjects
Publications and source records attributed to Vladimir Shpilrain.
We develop new tools for constructing redactable authenticated data structures with post-quantum security. In our construction, inverting the proposed one-way function means solving a polynomial equation (or a system of polynomial equations) in more than one variable. This is presently considered quantum-safe, i.e., there is no known quantum algorithm that could solve this problem efficiently if parameters are chosen wisely.
The problems that we consider in this paper are as follows. Let $A_1, \ldots, A_k$ be square matrices (over reals). Let $W=w(A_1, \ldots, A_k)$ be a random product of $n$ matrices. What is the expected growth rate of the largest (in the absolute value) entry in such a random product? What is the (maximal) Lyapunov exponent for a random matrix product like that? We give an answer to the first question under some mild restrictions on the entries of $A_i$. For the second question, we offer a very simple and efficient method to produce an upper bound on the Lyapunov exponent.
In this survey, we describe recent progress on asymptotic properties of various automorphic orbits in free groups. In particular, we address the problem of counting potentially positive elements of a given length. We also discuss complexity (worst-case, average-case, and generic-case) of Whitehead's automorphism problem and relevant properties of automorphic orbits, including orbit-blocking words.
The problems that we consider in this paper are as follows. Let A and B be 2x2 matrices (over reals). Let w(A, B) be a word of length n. After evaluating w(A, B) as a product of matrices, we get a 2x2 matrix, call it W. What is the largest (by the absolute value) possible entry of W, over all w(A, B) of length n, as a function of n? What is the expected absolute value of the largest (by the absolute value) entry in a random product of n matrices, where each matrix is A or B with probability 0.5? What is the Lyapunov exponent for a random matrix product like that? We give partial answer to the first of these questions and an essentially complete answer to the second question. For the third question (the most difficult of the three), we offer a very simple method to produce an upper bound on the Lyapunov exponent in the case where all entries of the matrices A and B are nonnegative.
By strengthening known results about primitivity-blocking words in free groups, we prove that for any nontrivial element w of a free group of finite rank, there are words that cannot be subwords of any cyclically reduced automorphic image of w. This has implications for the average-case complexity of a variant of Whitehead's problem.
We describe an alternative way of computing Alexander polynomials of knots/links, based on the Artin representation of the corresponding braids by automorphisms of a free group. Then we apply the same method to other representations of braid groups discovered by Wada and compare the corresponding isotopic invariants to Alexander polynomials.
Cayley hash functions are based on a simple idea of using a pair of semigroup elements, A and B, to hash the 0 and 1 bit, respectively, and then to hash an arbitrary bit string in the natural way, by using multiplication of elements in the semigroup. The main advantage of Cayley hash functions compared to, say, hash functions in the SHA family is that when an already hashed document is amended, one does not have to hash the whole amended document all over again, but rather hash just the amended part and then multiply the result by the hash of the original document. In this article, we survey some of the previously proposed Cayley hash functions and single out a very simple hash function whose security has not been compromised up to date.
The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case time complexity of the word problem in several classes of groups and show that it is often the case that the average-case complexity is linear with respect to the length of an input word. The classes of groups that we consider include groups of matrices over rationals (in particular, polycyclic groups), some classes of solvable groups, as well as free products. Along the way, we improve several bounds for the worst-case complexity of the word problem in groups of matrices, in particular in nilpotent groups. For free products, we also address the average-case complexity of the subgroup membership problem and show that it is often linear, too. Finally, we discuss complexity of the identity problem that has not been considered before.
Let F_2 denote the free group of rank 2. Our main technical result of independent interest is: for any element u of F_2, there is g in F_2 such that no cyclically reduced image of u under an automorphism of F_2 contains g as a subword. We then address computational complexity of the following version of the Whitehead automorphism problem: given a fixed u in F_2, decide, on an input v in F_2 of length n, whether or not v is an automorphic image of u. We show that there is an algorithm that solves this problem and has constant (i.e., independent of n) average-case complexity.
The isomorphism problem for infinite finitely presented groups is probably the hardest among standard algorithmic problems in group theory. Classes of groups where it has been completely solved are nilpotent groups, hyperbolic groups, and limit groups. In this short paper, we address the problem of isomorphism to particular groups, including free groups. We also address the algorithmic problem of embedding a finitely presented group in a given limit group.
Cayley hash functions are based on a simple idea of using a pair of semigroup elements, A and B, to hash the 0 and 1 bit, respectively, and then to hash an arbitrary bit string in the natural way, by using multiplication of elements in the semigroup. The main advantage of Cayley hash functions compared to, say, hash functions in the SHA family is that when an already hashed document is amended, one does not have to hash the whole amended document all over again, but rather hash just the amended part and then multiply the result by the hash of the original document. Some authors argued that this may be a security hazard, specifically that this property may facilitate finding a second preimage by splitting a long bit string into shorter pieces. In this paper, we offer a way to get rid of this alleged disadvantage and keep the advantages at the same time. We call this method ``Cayley hashing with cookies" using terminology borrowed from the theory of random walks in a random environment. For the platform semigroup, we use 2x2 matrices over F_p.
We use tropical algebras as platforms for a very efficient digital signature protocol. Security relies on computational hardness of factoring one-variable tropical polynomials; this problem is known to be NP-hard.
In this survey, we address the worst-case, average-case, and generic-case time complexity of the word problem and some other algorithmic problems in several classes of groups and show that it is often the case that the average-case complexity of the word problem is linear with respect to the length of an input word, which is as good as it gets if one considers groups given by generators and defining relations. At the same time, there are other natural algorithmic problems, for instance, the geodesic (decision) problem or Whitehead's automorphism problem, where the average-case time complexity can be sublinear, even constant.
We offer two very transparent digital signature schemes: one using non-square matrices and the other using scrap automorphisms. The former can be easily converted to a public key encryption scheme.
We offer a digital signature scheme using Boolean automorphisms of a multivariate polynomial algebra over integers. Verification part of this scheme is based on the approximation of the number of zeros of a multivariate Boolean function.
The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case complexity (i.e., the expected runtime) of algorithms that solve a well-known problem, the Whitehead problem in a free group, which is: given two elements of a free group, find out whether there is an automorphism that takes one element to the other. First we address a special case of the Whitehead problem, namely deciding if a given element of a free group is part of a free basis. We show that there is an algorithm that, on a cyclically reduced input word, solves this problem and has constant (with respect to the length of the input) average-case complexity. For the general Whitehead problem, we show that the classical Whitehead algorithm has linear average-case complexity if the rank of the free group is 2. We argue that the same should be true in a free group of any rank but point out obstacles to establishing this general result.
We use matrices over bit strings as platforms for Diffie-Hellman-like public key exchange protocols. When multiplying matrices like that, we use Boolean OR operation on bit strings in place of addition and Boolean AND operation in place of multiplication. As a result, (1) computations with these matrices are very efficient; (2) standard methods of attacking Diffie-Hellman-like protocols are not applicable.
We offer a public key exchange protocol based on a semidirect product of two cyclic (semi)groups of matrices over Z_p. One of the (semi)groups is additive, the other one multiplicative. This allows us to take advantage of both operations on matrices to diffuse information. We note that in our protocol, no power of any matrix or of any element of Z_p is ever exposed, so all standard attacks on Diffie-Hellman-like protocols (including Shor's quantum algorithm attack) are not applicable.