Defeating the Kalka--Teicher--Tsaban linear algebra attack on the Algebraic Eraser

The Algebraic Eraser (AE) is a public key protocol for sharing information over an insecure channel using commutative and noncommutative groups; a concrete realization is given by Colored Burau Key Agreement Protocol (CBKAP). In this paper, we describe how to choose data in CBKAP to thwart an attack by Kalka--Teicher--Tsaban

A converse theorem for double Dirichlet series and Shintani zeta functions

The main aim of this paper is to obtain a converse theorem for double Dirichlet series and use it to show that the Shintani zeta functions which arise in the theory of prehomogeneous vector spaces are actually linear combinations of Mellin transforms of metaplectic Eisenstein series on GL (2). The converse theorem we prove will apply to a very general family of double Dirichlet series which we now define

The cubic Pell equation L-function

For $d > 1$ a cubefree rational integer, we define an $L$-function (denoted $L_d(s)$) whose coefficients are derived from the cubic theta function for $\mathbb Q\left(\sqrt{-3}\right)$. The Dirichlet series defining $L_d(s)$ converges for $\text{Re}(s) > 1$, and its coefficients vanish except at values corresponding to integral solutions of $mx^3 - dny^3 = 1$ in $\mathbb Q\left(\sqrt{-3}\right)$, where $m$ and $n$ are squarefree. By generalizing the methods used to prove the Takhtajan-Vinogradov trace formula, we obtain the meromorphic continuation of $L_d(s)$ to $\text{Re}(s) > \frac{1}{2}$ and prove that away from its poles, it satisfies the bound $L_d(s) \ll |s|^{\frac{7}{2}}$ and has a possible simple pole at $s = \frac{2}{3}$, possible poles at the zeros of a certain Appell hypergeometric function, with no other poles. We conjecture that the latter case does not occur, so that $L_d(s)$ has no other poles with $\text{Re}(s) > \frac{1}{2}$ besides the possible simple pole at $s = \frac{2}{3}$