The Identity Correspondence Problem and its Applications

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: "Is it decidable for a finitely generated semigroup S of square integral matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a finite set of matrices generates a group is also undecidable. We also answer several question for matrices over different number fields. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words.Comment: We have made some proofs clearer and fixed an important typo from the published journal version of this article, see footnote 3 on page 1

Distances on Lozenge Tilings

International audienceIn this paper, a structural property of the set of lozenge tilings of a 2n-gon is highlighted. We introduce a simple combinatorial value called Hamming-distance, which is a lower bound for the flipdistance (i.e. the number of necessary local transformations involving three lozenges) between two given tilings. It is here proven that, for n5, We show that there is some deficient pairs of tilings for which the flip connection needs more flips than the combinatorial lower bound indicates

On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a nite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: \Is it decidable for a nitely generated semigroup S of integral square matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a nite set of matrices generates a group is also undecidable. We also answer several questions for matrices over di erent number elds. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words

Monads and Quantitative Equational Theories for Nondeterminism and Probability

The monad of convex sets of probability distributions is a well-known tool for modelling the combination of nondeterministic and probabilistic computational effects. In this work we lift this monad from the category of sets to the category of extended metric spaces, by means of the Hausdorff and Kantorovich metric liftings. Our main result is the presentation of this lifted monad in terms of the quantitative equational theory of convex semilattices, using the framework of quantitative algebras recently introduced by Mardare, Panangaden and Plotkin

New method of verifying cryptographic protocols based on the process model

A cryptographic protocol (CP) is a distributed algorithm designed to provide a secure communication in an insecure environment. CPs are used, for example, in electronic payments, electronic voting procedures, database access systems, etc. Errors in the CPs can lead to great financial and social damage, therefore it is necessary to use mathematical methods to justify the correctness and safety of the CPs. In this paper, a new mathematical model of a CP is introduced, which allows one to describe both the CPs and their properties. It is shown how, on the base of this model, it is possible to solve the problems of verification of CPs

ADDING PERFECT FORWARD SECRECY TO KERBEROS

Kerberos system is a powerful and widely implemented authentication system. Despite this fact it has several problems such as the vulnerability to dictionary attacks which is solved with the use of public key cryptography. Also an important security feature that is not found in Kerberos is perfect forward secrecy. In this work the lack of this feature is investigated in Kerberos in its original version. Also a public key based modification to Kerberos is presented and it is shown that it lacks the prefect forward secrecy too. Then some extensions are proposed to achieve this feature. The extensions are based on public key concepts (Diffie-Hellman) with the condition of keeping the password based authentication; this requires little modifications to the original Kerberos. Four extensions are proposed; two of them modify the (Client-Authentication Server) exchange achieving conditional perfect forward secrecy, while the remaining two modify the Client-Server exchange achieving perfect forward secrecy but with increased overhead and delay

ON THE UNDECIDABILITY OF THE IDENTITY CORRESPONDENCE PROBLEM AND ITS APPLICATIONS FOR WORD AND MATRIX SEMIGROUPS

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: "Is it decidable for a finitely generated semigroup S of integral square matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a finite set of matrices generates a group is also undecidable. We also answer several questions for matrices over different number fields. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words