On symmetric association schemes and associated quotient-polynomial graphs

A Note on Strongly Quotient Graphs

Verifiable symmetric polynomial-based key distribution schemes

Symmetric polynomial-based key distribution scheme has been widely adopted in various communication applications. This type of key distribution consists of a server and a set of users, where the server is responsible to distribute shares for each user via a symmetric polynomial. Based on the property of symmetry of this polynomial, each pair of users can compute a common secret key using their ...

Polynomial reproduction by symmetric subdivision schemes

We first present necessary and sufficient conditions for a linear, binary, uniform, and stationary subdivision scheme to have polynomial reproduction of degree d and thus approximation order d + 1. Our conditions are partly algebraic and easy to check by considering the symbol of a subdivision scheme, but also relate to the parameterization of the scheme. After discussing some special propertie...

Imprimitive Q-polynomial Association Schemes

It is well known that imprimitive P -polynomial association schemes X = (X, {Ri}0≤i≤d) with k1 > 2 are either bipartite or antipodal, i.e., intersection numbers satisfy either ai = 0 for all i, or bi = cd−i for all i 6= [d/2]. In this paper, we show that imprimitiveQ-polynomial association schemesX = (X, {Ri}0≤i≤d) with d > 6 and k∗ 1 > 2 are either dual bipartite or dual antipodal, i.e., dual ...

Deterministic Polynomial Factoring and Association Schemes

The problem of finding a nontrivial factor of a polynomial f(x) over a finite field Fq has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the art by focusing on prime degree polynomials; let n be the degree. If (n − 1) has a ‘larg...