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 cryptography (more specifically, on using non-commutative groups in public key cryptography). A naturally related area is algorithmic group theory, in particular complexity of group-theoretic problems.
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. 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 can say that I occasionally looked 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work with free Lie algebras is easier, so I occasionally turned to free Lie algebras to try one or another conjecture originally made for free groups.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
starting with an attempt to expand the very definition
of a probability measure from finite to infinite groups (see paper #1 on the list
Together with I. Kapovich, A.G.Myasnikov, and P. Schupp, I have applied probabilistic methods to the study of generic- and average-case 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.
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.
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, Sublinear time algorithms in the theory of groups and semigroups, Illinois J. Math. 54 (2011), 187-197.
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 A.Ushakov, An authentication scheme based on the twisted conjugacy problem, in: ACNS 2008, Lecture Notes Comp. Sc. 5037 (2008), 366-372.
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. 5010 (2008), 252–260.
V. Shpilrain, Cryptanalysis of Stickel's key exchange scheme, in: Computer Science in Russia 2008, Lecture Notes Comp. Sc. 5010 (2008), 283–288.
V. Shpilrain and G. Zapata, Using decision problems in public key cryptography, Groups, Complexity, and Cryptology 1 (2009), 33-49.
D. Grigoriev and V. Shpilrain, Authentication from matrix conjugation, Groups, Complexity, and Cryptology 1 (2009), 199-206.
D. Grigoriev and V. Shpilrain, Zero-knowledge authentication schemes from actions on graphs, groups, or rings, Ann. Pure Appl. Logic 162 (2010), 194–200.
G. Baumslag, N. Fazio, A. Nicolosi, V. Shpilrain, W. E. Skeith III, Generalized learning problems and applications to non-commutative cryptography, in: ProvSec 2011, Lecture Notes Comp. Sc. 6980 (2011), 324-339.
D. Grigoriev and V. Shpilrain, No-leak authentication by the Sherlock Holmes method, Groups, Complexity, and Cryptology 4 (2012), 177-189.
M. Habeeb, D. Kahrobaei, and V. Shpilrain, A secret sharing scheme based on group presentations and the word problem, Contemp. Math., Amer. Math. Soc. 582 (2012), 143-150.
D. Grigoriev and V. Shpilrain, Secrecy without one-way functions, Groups, Complexity, and Cryptology 5 (2013), 31-52.
D. Kahrobaei, C. Koupparis, and V. Shpilrain, Public key exchange using matrices over group rings, Groups, Complexity, and Cryptology 5 (2013), 97-115.
M. Habeeb, D. Kahrobaei, C. Koupparis, and V. Shpilrain, Public key exchange using semidirect product of (semi)groups, in: ACNS 2013, Lecture Notes Comp. Sc. 7954 (2013), 475-486.
D. Grigoriev and V. Shpilrain, Secure information transmission based on physical principles, in: UCNC 2013, Lecture Notes Comp. Sc. 7956 (2013), 113-124.
D. Grigoriev and V. Shpilrain, Tropical cryptography, Comm. Algebra 42 (2014), 2624-2632.
D. Kahrobaei, C. Koupparis, and V. Shpilrain, A CCA secure cryptosystem using matrices over group rings, preprint.
D. Grigoriev and V. Shpilrain, Yao's millionaires' problem and decoy-based public key encryption by classical physics, 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
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.
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.
V. Shpilrain, Search and witness problems in group theory, Groups, Complexity, and Cryptology 2 (2010), 231–246.
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.