David Garber

Eli Bagno, David Garber

A symmetric function is called Schur-positive if it admits an expansion in the Schur basis with nonnegative coefficients. In this paper, we study the Schur-positivity of symmetric functions naturally associated with set partitions, with respect to a descent set function that considers i as descent, if i and i+1 share a block in the partition. The Schur expansion involves hook-shaped Young diagrams, and the corresponding coefficients are given by Touchard-Riordan polynomials, which enumerate matchings by their number of crossings.

David Garber, Delaram Kahrobaei, Ha T. Lam

After the Anshel-Anshel-Goldfeld (AAG) key-exchange protocol was introduced in 1999, it was implemented and studied with braid groups and with the Thompson group as its underlying platforms. The length-based attack, introduced by Hughes and Tannenbaum, has been used to extensively study AAG with the braid group as the underlying platform. Meanwhile, a new platform, using polycyclic groups, was proposed by Eick and Kahrobaei. In this paper, we show that with a high enough Hirsch length, the polycyclic group as an underlying platform for AAG is resistant to the length-based attack. In particular, polycyclic groups could provide a secure platform for any cryptosystem based on conjugacy search problem such as non-commutative Diffie-Hellman, ElGamal and Cramer-Shoup key exchange protocols.