


Interactions between Algebra and Computer Science / Intéractions entre la science informatique et l'algèbre Org: Olga Kharlampovich (McGill), Alexei G. Myasnikov (McGill) and/et Vladimir Shpilrain (CUNY)
 PETER ACKERMANN, Dortmund University
Gröbner Basis Cryptosystems

We extend and generalize Gröbner basis theory to submodules of free
right modules over monoid rings. We then show how this theory can be
used to introduce a new class of Gröbner basis cryptosystems and how
one can connect these to classical well known cryptosystems.
 ARKADY BERENSTEIN, University of Oregon
Geometric Key Establishment

The aim of the talk is to present a new class of key establishment
schemes which are based on geometric generalizations of the classical
DiffieHellman. The simplest of our schemesbased on the geometry
of the unit circleuses only multiplication of rational numbers by
integers and addition of rational numbers in its key creation.
Preliminary estimations show that our schemes are resistant to
attacks. This resistance follows the pattern of the discrete
logarithm problem and hardness of multidimensional lattice problems.
 ANDREW DUNCAN, Newcastle University
Quantum computation in finite groups: extending the
DeutschJozsaHøyer algorithm

This talk reports a generalisation of the quantum algorithm of the
title. One reason for interest in this algorithm is
complexitytheoretic: it may help to show whether or not the class of
bounded error probabilistic quantum problems BQP is, relative to
some oracle, contained in the class of problems in the polynomial
hierarchy PH, or not.
Let f be a function f : {0,1}^{n} ® {0,1}. We
say f is balanced if f^{1}(0) = f^{1}(1) = 2^{n1}.
Given that f is known either to be a constant function or a balanced
function the DeutschJozsa algorithm can determine which: in a single
call to a quantum blackbox which evaluates the function. By contrast
a classical algorithm would need, in the worst case, at least
2^{n1}+1 evaluations of the function f to perform the same task.
Høyer has shown how this quantum algorithm generalises to functions
f : G® H, where G and H are finite groups and f
is known to be a function of one of two particular types. In this
talk we show how to extend the types of function the algorithm can
cope with to a much wider class. Conditions rbalanced and
rconstant, on such functions f, are defined which depend on
choices of columns of the matrices of the irreducible (unitary)
representations of H. We show that we can distinguish between
rbalanced and rconstant functions in one call to a quantum
blackbox for f. In the case where H is cyclic this condition
also has a characterisation in terms of cyclotomic polynomials, which
will be described.
If time allows the possible complexity theory implications of this
generalisation will be discussed.
 BEN FINE, Fairfield University, Fairfield, CT 06430
Some Suggestions in Algebraic Cryptography

Because of the increased power of computers the security of
cryptosystems, especially public key cryptosystems, is being
reexamined. To develop more secure cryptosystems various encryption
methods using combinatorial group theory, such as AnshelGoldfeld and
KoLee, have been developed. However several security problems have
been pointed out relative to these methods, so that additional work is
necessary. In this talk we present some further suggestions for
cryptosystems, both public key and classical, based on the use of
different representations of free groups. The decryption methods
depend upon on group theoretic rewriting systems, such as
ReidemeisterSchreier. The difficulty of determining Shreier
transversals as well as the introduction of some random "noise"
provides the oneway function necessary for a secure cryptosystem.
Several experiments based on the computing feasibility of these
suggestions will also be mentioned.
This is joint work with G. Baumslag and X. Xu.
 LOTHAR GERRITZEN, University of Bochum, D44780 Bochum, NA 2/33
Planar Calculus

A planar monomial is by definition an isomorphism class of a finite,
planar, reduced rooted tree. On the set P of planar monomials there
are mary grafting operations for m greater or equal to 2. A
planar polynomial over a field K is a Klinear combination over
P. A planar power series is an infinite formal expression in P
over K. If x denotes the tree with a single vertex, any planar
power series is an infinite sum over Kmultiples of nonassociative
products in x. The system of mary multiplications turn the
Kvectorspace A of planar power series into an algebra over the
operand freely generated by a sequence of mary operations for
m > 1. One can substitute g(x) in A for x in f(x) in A to
obtain a planar power series f ( g(x) ) in case the order
of g(x) is > 0. The derivative d/dx is a derivation on A with
d/dx(x) = 1.
There exists a unique planar power series Exp(k,x) in A for any
k > 1 such that the order of Exp(k,x)  1  x is > 1 and that the
kth power of Exp(k,x) is equal to Exp(k,k*x). Then one can
show that d/dx ( Exp(k,x) ) = Exp(k,x). It is called
the kary planar exponential series. Its coefficients depend
algebraically on k. There are also planar series Log(k,x),
Sin(k,x), ArcSin(k,x) etc. One can define planar
differentials and one can prove a planar chain rule. These examples
are instances of a theory which can be called planar calculus. One can
expect applications to number theory and cominatorics.
 DENNIS HOFHEINZ, Universität Karlsruhe, Fakultät für Informatik, Am
Fasanengarten 5, 76128 Karlsruhe, Germany
From hard problems to cryptographic security

Often, the security of a cryptographic system is reduced to a
mathematical problem. However, there are various notions of security
for, say, a public key cryptosystem, and it is not always clear what
notion of security is implied by the hardness of a given mathematical
problem. Assuming that factoring of integers is hard, what can we say
about the security of plain RSA?
This talk uses the example of cryptography based on braid groups to
illustrate in which different ways a mathematical problem can be used
for cryptography and what the respective consequences are. In
particular, several variations of problems in braid groups are related
to security properties of public key encryption and signature
schemes.
 STEPAN HOLUB, Charles University, Sokolovska 83, Prague
Binary GPCP is in PTIME

Post Correspondence Problem (PCP) consists in determining whether for
two morphisms g,h : A^{*}® B^{*}, the equality g(w)=h(w) can be
obtained for some nonempty word w Î A^{*}. The input of the
generalized version of the problem (GPCP) contains in addition to
morphisms g and h words p_{1}, p_{2}, s_{1}, s_{2}, and asks
whether there is a word w such that p_{1} g(w)s_{1}=p_{2} h(w)s_{2}.
The importance of the problem is given by the fact that it is one of
the most famous undecidable problems, reduction to which has been used
to prove undecidability of other problems. There are however special
cases, in which the problem is decidable. For example, in the binary
case, i.e., if the cardinality of A is two, GPCP is well known
to be decidable. The decision algorithm described in the literature
admits exponential complexity.
With help of an additional insight into the structure of the solutions
we show that binary GPCP is in PTIME.
 JOINT PANEL DISCUSSION
Participating Sessions

This is a joint Panel Discussion of the following sessions:
 Interactions between Algebra and Computer Science
 Combinatorial and Geometric Group Theory
 Groups, Equations, Noncommutative Algebraic Geometry
 ILYA KAPOVICH, University of Illinois at UrbanaChampaign
Subadditivity and its consequences

We investigate the consequences of Kingman's Subadditive Ergodic
Theorem for interpreting the results of various grouptheoretic
experiments. This applies to generic subgroup distortion, random
elements in regular languages, the behaviour of various normal forms
algorithms, etc. The talk is based on a portion of a joint paper
with Paul Schupp and Vadim Kaimanovich.
 BILAL KHAN, John Jay College (CUNY), 899 Tenth Avenue, New York, NY
On positive theories of fully residually free groups

Let f be a standardform firstorder sentence in the language of
group theory extended by constants from a group G. Specifically,
take
f: "x_{1} $y_{1} "x_{2} $y_{2} ¼"x_{k} $y_{k} f_{0} (x_{1}, y_{1}, x_{2}, y_{2}, ..., x_{k}, y_{k}), 

where f_{0} is a quantifierfree conjunction of disjunctions of
equations and inequalities.
We say that f lifts in G, if G \models f and
witness values for each existentially quantified variable y_{i} are
expressible as elements of G[X] = G*F(X), i.e., there
is a Skolemization of f using elements of G[X].
The sentence f is called positive if it is logically
equivalent to a standardform sentence in which f_{0} contains
no inequalities. The set of all positive sentences that are
true in G is called the positive theory of G.
The seminal result in this area is due to Merzlykov who showed (though
not in this terminology) that if G is a free nonabelian group, then
the positive theory of G lifts. We extend Merzlykov's result to a
large class of finitely generated fully residually free groups,
including surface groups, extensions of centralizers, and others.
This is joint work with Alexei G. Miasnikov and Denis Serbin.
 MARTIN KREUZER, Fachbereich Mathematik, Universitaet Dortmund, D44221
Dortmund, Germany
Some Applications of Groebner Bases to Group Theory

Given a finitely presented group G, we construct the universal
representation F: G ® Mat_{n}(R) where R = K[x_{1},...,x_{n}]/I is an affine algebra. Based on Groebner basis
computations for ideals in R, we can solve the word problem, the
subgroup problem, etc., for the image F(G). In particular,
using a new algorithm for computing minimal polynomials of elements in
noncommutative algebras, we can show that some elements of F(G)
have finite resp. infinite order. This yields a new approach to
checking the finiteness of G computationally. Moreover, we shall
see examples where Groebner basis computations lead to new insights
into the structure of the group G.
 STUART MARGOLIS, BarIlan University, Queens College (NY) and Hunter College
Representation Theory and the Theory of Automata and Regular
Languages

Representation theory is one of the most classical areas of finite
semigroup theory. It was developed by Clifford, Munn, Ponizovsky and
others in the 1950s and has been resurrected by Putcha, Renner and
Okninski and their students in the last 15 years. On the other hand,
the main thrust of research in finite semigroups for the last 40 years
has been its connection with the theory of automata and formal
languages following the KrohnRhodes Theorem and the theory of
pseudovarieties developed originally by Eilenberg and Schutzenberger.
Up to now, except for some work of Rhodes in the 1960s, there has
been almost no use of representation theory in the automata theoretic
aspects of finite semigroup theory. In this talk, we show that there
are some very deep connections between the two fields. We show that
the collection of finite semigroups triangularizable and
unitriangularizable over a finite field forms a pseudovariety and give
both a syntactic and semantic description of this collection. We also
generalize a Theorem of Rhodes by calculating the congruence obtained
by restricting the Jacobson radical of the semigroup algebra KS to
S for all fields K and all finite semigroups S. Finally, we show
a strong relationship between the language theoretic operation of
counting subwords over a field K to the irreducible representations
of the syntactic monoid M(L) of languages L over the field K.
This is joint work of Jorge Almeida, Benjamin Steinberg and Mikhail
Volkov.
 ALEX D. MIASNIKOV, The Graduate Center of CUNY
Heuristics for The Whitehead Minimization Problem

In this talk we will discuss several heuristic strategies which allow
one to solve the Whitehead's minimization problem much faster (on most
inputs) than the classical Whitehead algorithm. The mere fact that
these strategies work in practice leads to several interesting
mathematical conjectures. In particular, we conjecture that the
length of most nonminimal elements in a free group can be reduced by
a Nielsen automorphism which can be identified by inspecting the
structure of the corresponding Whitehead Graph.
 GERHARD ROSENBERGER, Fachbereich Mathematik, University of Dortmund, 44221 Dortmund
Interactions between commutation transitive, conjugately
separated and restricted Gromov groups

Power commutative, commutation transitive, power transitive,
conjugately separated abelian and restricted Gromov groups appear in
various contexts. We investigate relations between these groups and
show that there are many equivalences under certain additional
conditions.
 PAUL SCHUPP, University of Illinois
Groups with polynomial time word problem and large Dehn
function

This is joint work with Ilya Kapovich. We consider a sequence of
groups G_{n} which are iterated HNN extensions and the onerelator
BaumslagGersten group B = áa,t : a^{(at)} = a^{2} ñ
where a^{t} = t a t^{1}. Gersten proved that the Dehn function of
G_{n} is a tower of nexponentials and that the Dehn function of B
is not bounded by any fixed tower of exponentials. We show that the
word problem for each G_{n} is solvable in polynomial time and that
the word problem of B is solvable in at worst single exponential
time.
 LEV SHNEERSON, Hunter College (The City University of New York), Department
of Mathematics and Statistics, 695 Park Avenue, New York, NY
10021, USA
Growth and infinite words in semigroup varieties

We study the structure of geodesic infinite words in the varieties
where every finitely generated semigroup has polynomial growth.
 VLADIMIR SHPILRAIN, The City College of New York, New York, NY 10031
Combinatorial group theory and public key cryptography

After some excitement generated by recently suggested public key
exchange protocols due to AnshelAnshelGoldfeld and KoLee et
al., it is a prevalent opinion now that the conjugacy search
problem is unlikely to provide sufficient level of security no matter
what group is used as the platform. In this talk, we reflect on
whether some other "hard" problem from combinatorial group theory
can be used, instead of the conjugacy search problem, in a public key
exchange protocol.
 JIRI TUMA, Charles University in Prague, Faculty of Mathematics and
Physics, Sokolovska 83, 186 00 Prague 8, Czech Republic
On Rejewski's reconstruction of Enigma

In December 1932 a Polish mathematician Marian Rejewski formulated a
complicated system of equations in the symmetric group over 26 letters
describing the internal structure of the Enigma machine. By
exploiting errors of the machine operators he was able to solve the
system and this allowed a reconstruction of the Enigma machine. I
will review the mathematical solution and suggest how to fill in some
points missing in published reports of Rejewski's work.
 ALEXANDER USHAKOV, Graduate Center/CUNY 365, 5th Avenue, Manhattan, NY 10016, USA
Random annuli diagrams and generic

Let G = áX ; R ñ be a finite presentation and u,w be
words in X^{±1}. The Search Conjugacy problem for G is an
algorithmic question if there exists an algorithm which for any pair
of conjugated words u ~ _{G} w finds an actual conjugator
(i.e., a word x Î F(X) such that x^{1} u x = _{G} w).
The hardness of this problem lies in the basis of several groupbased
cryptographic schemes.
In this talk I will describe an effective algorithm for solving the
Search Conjugacy problem for finitely presented groups and show that
the algorithm has polynomial time genericcase complexity.
This work is a continuation of our joint research with Alexei
Miasnikov, "Random van Kampen Diagrams and algorithmic problems in
groups".

