My research interests include: cryptography, complexity of algorithms, data mining, combinatorial and computational group theory, affine algebraic geometry.

I split most of my time between combinatorial group theory and cryptography (more specifically, public key cryptography). In particular, I am trying to apply various ideas from combinatorial group theory and from combinatorial commutative algebra to see whether cryptography based on combinatorial algebra can be considered a serious alternative to more "traditional" number-theoretic cryptography.

I am also interested in problems of computational nature, especially in the complexity of various algorithms. This has recently brought me to the area of statistical group theory where, together with several colleagues (I.Kapovich, A.G.Myasnikov, 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.

http://shpilrain.ccny.cuny.edu/res.html