Research and publications

  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.   I occasionally looked also 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work with free Lie algebras is easier, so I occasionally turned to free Lie algebras to try one or another conjecture originally made for free groups.


Statistical and asymptotic group theory and cryptography

  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 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.

These are my papers on statistical and asymptotic group theory:

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.
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.

These are my papers on cryptography:

book:  A.G.Myasnikov, V. Shpilrain and A.Ushakov, Group-based cryptography, Birkhauser, 2008.
book:  A.G.Myasnikov, V. Shpilrain and A.Ushakov, Non-commutative Cryptography and Complexity of Group-theoretic Problems, Amer. Math. Soc. Surveys and Monographs, 2011.

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 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), 252260.
V. Shpilrain, Cryptanalysis of Stickel's key exchange scheme, in: Computer Science in Russia 2008, Lecture Notes Comp. Sc. 5010 (2008), 283288.
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), 194200.
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, Contemp. Math., Amer. Math. Soc., to appear.
D. Grigoriev and V. Shpilrain, Yao's millionaires' problem and decoy-based public key encryption by classical physics, Journal of Foundations of Computer Science 25 (2014), 409417.
D. Kahrobaei, H. Lam, V. Shpilrain, Public key exchange using extensions by endomorphisms and matrices over a Galois field, preprint.
B. Cavallo, D. Kahrobaei, V. Shpilrain, Decoy-based secure delegation of computation, with application to RSA encryption, preprint.
L. Bromberg, V. Shpilrain, A. Vdovina, Navigating in the Cayley graph of SL_2(F_p) and applications to hashing, preprint.


Combinatorial group theory

  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.

These are my papers on combinatorial group theory:

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.
V. Shpilrain, Search and witness problems in group theory, Groups, Complexity, and Cryptology 2 (2010), 231246.


Affine algebraic geometry

  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.

These are my papers on affine algebraic geometry:

V. Shpilrain and J.-T. Yu, Polynomial automorphisms and Groebner reductions, J. Algebra 197 (1997), 546--558.
A. van den Essen, V. Shpilrain, Some combinatorial questions about polynomial mappings, J. Pure Appl. Algebra 119 (1997), 47-52.
V. Shpilrain, Combinatorial methods: from groups to polynomial algebras, Groups '97  Bath/St. Andrews, Vol. 2  (Bath, 1997), 679-688,  London Math. Soc. Lecture Note Ser. 261, Cambridge Univ. Press, Cambridge, 1999.
V. Shpilrain and J.-T. Yu, Embeddings of curves in the plane, J.Algebra 217 (1999), 668-678.
V. Shpilrain and J.-T. Yu, Polynomial retracts and the Jacobian conjecture, Trans. Amer. Math. Soc. 352 (2000), 477-484.
V.Drensky, V. Shpilrain and J.-T. Yu, On the density of the set of generators of a polynomial algebra, Proc. Amer. Math. Soc. 128 (2000), 3465-3469.
V. Shpilrain and J.-T. Yu, Peak reduction technique in commutative algebra: a survey. Contemp. Math., Amer. Math. Soc. 264 (2000), 237-247.
V. Shpilrain and J.-T. Yu, Embeddings of hypersurfaces in affine spaces,  J.Algebra 239 (2001), 161-173.
V. Shpilrain and J.-T. Yu, Non-extendable isomorphisms between affine varieties,  J. Pure Appl. Algebra 172 (2002), 285-291.
V. Shpilrain and J.-T. Yu, Affine varieties with equivalent cylinders,  J. Algebra 251 (2002), 295-307.
V. Shpilrain and J.-T.Yu, Birational morphisms of the plane, Proc. Amer. Math. Soc. 132 (2004), 2511-2515.
L. Makar-Limanov, P. van Rossum, V. Shpilrain and J.-T.Yu, The stable equivalence and cancellation problems, Comment. Math. Helv. 79 (2004), 341-349.
V. Shpilrain and J.-T.Yu, Test polynomials, retracts, and the Jacobian conjecture, Contemp. Math., Amer. Math. Soc. 369 (2005), 253-259.
L. Makar-Limanov, V. Shpilrain and J.-T.Yu, Equivalence of polynomials under automorphisms of K[x,y], J. Pure Appl. Algebra 209 (2007), 71-78.
C. M. Lam, V. Shpilrain, and J.-T.Yu, Recognizing and parametrizing curves isomorphic to a line, J. Symb. Comput. 42 (2007), 751-756.