If you have a general idea about my research, you can go straight to my preprints. Or, perhaps, you would like to see a (more or less) complete list of my papers since 1993. You can download most of them.
If you are here for the first time, I hope the following
brief description of my research might be of interest.
I have done research in several areas of mathematics and theoretical computer science;
currently I am mostly working on what is usually called statistical and
asymptotic group theory
and on cryptography (more specifically, on using non-commutative groups
in public key cryptography). These areas are naturally related because the emerging area of non-commutative
cryptography requires rigorous mathematical justification to be accepted by the cryptographic community, and
this can be provided by the tools of statistical and asymptotic group theory.
Previously, I have been splitting most of my time between
combinatorial group theory and
affine algebraic geometry.
These areas may look unrelated to the naked eye but, as it turns out,
they have a lot in common. It is remarkable that
the exchange of ideas goes both ways: ideas from
algebraic geometry have found their way into combinatorial group
theory through the work of G.Baumslag, O.Kharlampovich,
A.G.Myasnikov, V.Remeslennikov, and, essentially independently, of
B.I.Plotkin, whereas ideas from
combinatorial group theory have found their way into affine algebraic
geometry through the work of J.-T.Yu and myself.
Here you will find a more detailed description of what
affine algebraic geometry is all about, together with a list of
my papers on the subject.
Finally, I have to say that I sometimes look at free associative and Lie algebras for inspiration. Many combinatorial properties of free Lie algebras are very similar to those of free groups, but to work with free Lie algebras is easier, so I often turn to free Lie algebras to try one or another conjecture originally made for free groups.
During 2001-2007, I have participated in several projects in statistical and asymptotic group theory,
starting with an attempt to expand the very definition
of a probability measure from finite to infinite groups (see paper #1 on the list
below).
Together with I. Kapovich, A.G.Myasnikov, and P. Schupp,
I have applied probabilistic methods to the study of complexity of various
decision problems in group theory. This direction of research brings
together mathematics, statistics,
and theoretical computer science by providing statistical analysis
and, at the same time, rigorous mathematical justification of the
successful performance of various non-deterministic algorithms widely
used in real-life applications, in particular, to cryptography.
A.Borovik, A.G.Myasnikov,
V. Shpilrain,
Measuring sets in infinite groups,
Contemp. Math., Amer. Math. Soc. 298 (2002), 21-42.
I. Kapovich, A. G. Myasnikov, P.
Schupp,
and V.Shpilrain,
Generic-case complexity, decision problems in group theory and random
walks, J. Algebra 264 (2003), 665-694.
pdf
I. Kapovich, A. G. Myasnikov,
P. Schupp,
and V.Shpilrain,
Average-case complexity and decision
problems in group theory, Advances in Math. 190 (2005), 343-359.
V. Shpilrain,
Counting primitive elements of a free group,
Contemp. Math., Amer. Math. Soc. 372 (2005).
I. Kapovich, P.
Schupp,
and V.Shpilrain,
Generic properties of Whitehead's algorithm
and isomorphism rigidity of random one-relator groups, Pacific J. Math.
223 (2006), 113-140.
A.G.Myasnikov,
V. Shpilrain,
Some metric properties of automorphisms of groups, J. Algebra 304
(2006), 782-792.
I. Kapovich, I. Rivin, P. Schupp,
and V.Shpilrain,
Densities in free groups and Z^k, visible points and
test elements, Math. Res. Lett. 14 (2007), 263-284.
V. Shpilrain,
Assessing security of some
group based cryptosystems, Contemp. Math., Amer. Math. Soc.
360 (2004), 167-177.
V. Shpilrain and A.Ushakov,
Thompson's group
and public key cryptography, Lecture Notes Comp. Sc.
3531 (2005), 151-164.
A. G. Myasnikov,
V. Shpilrain and A.Ushakov,
A practical attack on some braid group based cryptographic
protocols, in CRYPTO 2005, Lecture Notes Comp. Sc. 3621 (2005), 86-96.
A. G. Myasnikov,
V. Shpilrain and A.Ushakov,
Random subgroups of braid groups:
an approach to cryptanalysis of a braid group based cryptographic
protocol, in PKC 2006, Lecture Notes Comp. Sc. 3958 (2006), 302-314.
V. Shpilrain and A.Ushakov,
The conjugacy search problem in
public key cryptography: unnecessary and insufficient, Applicable Algebra in Engineering,
Communication
and Computing 17 (2006), 285-289.
V. Shpilrain and G.Zapata, Combinatorial group theory and public key
cryptography, Applicable Algebra in Engineering,
Communication and Computing 17 (2006), 291-302.
V. Shpilrain and G. Zapata, Using the
subgroup membership search problem in public key cryptography, Contemp. Math., Amer. Math. Soc.
418 (2006), 169-179.
V. Shpilrain and A.Ushakov,
A new key exchange protocol based on the decomposition problem, Contemp.
Math., Amer. Math. Soc. 418 (2006), 161-167.
V. Shpilrain,
Hashing with polynomials, in: ICISC 2006, Lecture Notes Comp. Sc. 4296 (2006),
22-28.
V. Shpilrain and G. Zapata, Using decision
problems in public key cryptography, preprint.
D. Osin and
V. Shpilrain,
Public key encryption and encryption emulation attacks, version for group
theorists
version for cryptographers, in: Computer Science
in Russia 2008, Lecture Notes Comp. Sc., to appear.
V. Shpilrain, Cryptanalysis of
Stickel's key exchange scheme, in: Computer Science
in Russia 2008, Lecture Notes Comp. Sc., to appear.
V. Shpilrain and A.Ushakov,
An authentication scheme based on the twisted conjugacy problem, in:
ACNS 2008, Lecture Notes Comp. Sc. 5037 (2008), 366-372.
D. Grigoriev and
V. Shpilrain, Zero-knowledge authentication schemes
from actions on graphs, groups, or rings, preprint.
During 1993-2000, my research in group theory was primarily focused on
free groups and their automorphisms; in particular, on various properties
of orbits under the action of the group of automorphisms of a free
group. Especially fruitful and inspiring to many (at least 30) people
turned out to be the concept of a test element introduced in paper #2 on the list
below. The idea was to
distinguish, for example, automorphisms among arbitrary endomorphisms by means
of their action on a single element, a test element. The same goal of
distinguishing automorphisms, but in a different context, led me to
introducing non-commutative determinants (see paper #5 on the list
below).
I also have a thing for braid groups. I find the class
of braid groups fascinating because it brings together many different areas of
mathematics (and physics!): algebra, topology, differential equations,
to name just a few. Recently, braid groups have been also used in cryptography.
N. Gupta, V.
Shpilrain,
Nielsen's commutator test for two-generator groups,
Math. Proc. Cambridge Phil. Soc. 114 (1993), 295-301.
V. Shpilrain,
Recognizing
automorphisms of the free groups, Arch. Math. 62 (1994), 385-392.
V. Shpilrain,
Test elements
for endomorphisms of free groups and algebras, Israel J. Math. 92
(1995), 307-316.
V. Shpilrain,
On monomorphisms of free groups, Arch. Math. 64 (1995), 465-470.
V. Shpilrain,
Non-commutative
determinants and automorphisms of groups, Comm. Algebra 25 (1997),
559-574.
V. Shpilrain,
Fixed
points of endomorphisms of a free metabelian group, Math. Proc.
Cambridge Phil. Soc. 123 (1998), 77-85.
V. Shpilrain,
Generalized
primitive elements of a free group, Arch. Math. 71 (1998),
270-278.
V. Shpilrain,
Automorphisms
of one-relator groups, Math. Proc. Cambridge Phil. Soc. 26
(1999), 499--504.
V. Shpilrain, Representing
braids by automorphisms, Internat. J. Algebra and
Comput. 11 (2001), 773-778.
A.D.Myasnikov,
A.G.Myasnikov and V.Shpilrain,
On
the Andrews-Curtis equivalence, Contemp. Math., Amer. Math. Soc.
296 (2002), 183-198.
G.Baumslag, A.G.Myasnikov
and V.Shpilrain,
Open
problems in combinatorial group theory. Second edition,
Contemp. Math., Amer. Math. Soc. 296 (2002), 1-38.
A.G.Myasnikov,
V. Shpilrain,
Automorphic orbits in free groups, J.
Algebra 269 (2003), 18-27.
V. Bardakov,
V. Shpilrain, V. Tolstykh,
On the palindromic and primitive widths of a free
group, J. Algebra 285 (2005), 574-585.
I. Kapovich, G. Levitt, P. Schupp,
and V.Shpilrain,
Translation equivalence in free groups, Trans. Amer. Math. Soc.
359 (2007), 1527-1546.
Affine algebraic geometry is a fascinating area of mathematics that studies polynomials and polynomial mappings. An interesting thing about this area is that most of the research here is focused on five or six outstanding problems. The statements of these problems are rather elementary and can be understood by an average high school student. However, some methods that have been employed so far for attacking these problems are rather sophisticated, and, more importantly, they come from several different areas of mathematics, which stimulates additional interest. If you would like to learn more about these problems, you can download this file.