On sums of binomial coefficients and their applications

In this paper we study recurrences concerning the combinatorial sum $[n,r]_m=\sum_{k\equiv r (mod m)}\binom {n}{k}$ and the alternate sum $\sum_{k\equiv r (mod m)}(-1)^{(k-r)/m}\binom{n}{k}$, where m>0, $n\ge 0$ and r are integers. For example, we show that if $n\ge m-1$ then $$\sum_{i=0}^{\lfloor(m-1)/2\rfloor}(-1)^i\binom{m-1-i}i [n-2i,r-i]_m=2^{n-m+1}.$$ We also apply such results to investigate Bernoulli and Euler polynomials. Our approach depends heavily on an identity established by the author [Integers 2(2002)]

Mixed Hodge polynomials of character varieties

We calculate the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,F_q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,C)-character variety. The calculation also leads to several conjectures about the cohomology of M_n: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n = 2.Comment: with an appendix by Nicholas M. Katz; 57 pages. revised version: New definition for homogeneous weight in Definition 4.1.6, subsequent arguments modified. Some other minor changes. To appear in Invent. Mat

Completed K-theory and Equivariant Elliptic Cohomology

Kitchloo and Morava give a strikingly simple picture of elliptic cohomology at the Tate curve by studying a completed version of $S^1$-equivariant $K$-theory for spaces. I present a $G$-equivariant version of their construction, which is a completed version of the Freed-Hopkins-Teleman model of $K$-theory for local quotient groupoids and resolves the issues concerning twisting and degree that arise in a first attempt to relate their work to elliptic cohomology.Comment: 23 page

Revisit the Concept of PEKS: Problems and a Possible Solution

Since Boneh et al. propose the concept, non-interactive\ud Public-key Encryption with Keyword Search (PEKS) has attracted lots of attention from cryptographers. Non-interactive PEKS enables a third party to test whether or not a tag, generated by the message sender, and a trapdoor, generated by the receiver, contain the same keyword without revealing further information. In this paper we investigate a non-interactive PEKS application proposed by Boneh et al. and show our observations, especially that privacy is\ud not protected against a curious server. We propose the notion of interactive PEKS, which, in contrast to non-interactive PEKS, requires the tag to be generated interactively by the message sender and the receiver. For this new primitive, we identify two types of adversaries, namely a curious user and a curious server, and provide\ud security formulations for the desirable properties. We propose a construction for interactive PEKS and prove its security in the proposed security model