Quasi-subfield Polynomials and the Elliptic Curve Discrete Logarithm Problem

Abstract

<jats:title>Abstract</jats:title><jats:p>We initiate the study of a new class of polynomials which we call quasi-subfield polynomials. First, we show that this class of polynomials could lead to more efficient attacks for the elliptic curve discrete logarithm problem via the index calculus approach. Specifically, we use these polynomials to construct factor bases for the index calculus approach and we provide explicit complexity bounds. Next, we investigate the existence of quasi-subfield polynomials.</jats:p>

New results on quasi-subfield polynomials

“…Proof. Type 1 is proven in [1,Lemma 4.3]. Moreover it is obvious that L X−1 is a quasi-subfield polynomial over F p n for any p prime and n (tolerating here = 0).…”

“…In Our results. We expand the study of quasi-subfield polynomials initiated in [1], focusing on polynomials whose roots form a subgroup of either the additive or the multiplicative group of finite fields. In the additive case this amounts to searching for a linearized polynomial…”

“…In this section we first recall known properties of linearized polynomials, including a characterization of linearized polynomials that split completely over their field of definition. We then proceed to prove Theorem 1, and we compare the bound it provides with the bounds given in [1] and [2].…”

“…We now study another family of quasi-subfield polynomials considered in [1], namely polynomials whose roots form a multiplicative group of F p n .…”

“…The polynomial X p n − X splits over F p n and its roots are exactly all the elements of F p n . Quasi-subfield polynomials, introduced by Huang et al [1], naturally generalize this polynomial.…”