Composition collisions and projective polynomials

Abstract

The functional decomposition of polynomials has been a topic of great interest and importance in pure and computer algebra and their applications. The structure of compositions of (suitably normalized)is well understood in many cases, but quite poorly when the degrees of both components are divisible by the characteristic p. This work investigates the decomposition of polynomials whose degree is a power of p. An (equal-degree) icollision is a set of i distinct pairs (g, h) of polynomials, all with the

Compositions and collisions at degreep2

“…Theorem 5.1 and Proposition 5.4 of von zur Gathen, Giesbrecht & Ziegler (2010) determine the number c…”

“…Fact 3.5 (von zur Gathen, Giesbrecht & Ziegler (2010), Proposition 6.2). Let r be a power of p, and u, s, ε, m and u * , s * , ε * , m * satisfy the conditions of Fact 3.1.…”

A characterization of the number of roots of linearized and projective polynomials in the field of coefficients

“…The original motivation was to find polynomials with a given Galois group. However they have also been studied over F q n , for example by Bluher [2], and in [6], due to their interesting possible number of roots, and their connection to calculating composition collisions. Projective polynomials with maximum number of roots have been used in [9] for attacking the discrete logarithm problem in cryptography; see Section 6 for further details.…”

“…If H n−1 = H n−2 = 0 then using the second recursion (6), this automatically implies that −wzH σ 2 n−4 − H σ n−3 = 0. Hence, by the form of the matrix in the statement of Theorem 10, A L is a diagonal matrix if and only if H n−1 = H n−2 = 0.…”

A polynomial time algorithm for computing all minimal decompositions of a polynomial

“…Even though the bounds are sharp, in many cases there are much fewer minimal decomposition than stated in Corollary 3.1. For more details on the number of decomposition of polynomials see von zur Gathen (2009), von zur Gathen,Giesbrecht & Ziegler (2010), andBlankertz, von zur Gathen & Ziegler (2012).…”