A New Approach to Constructing Quadratic Pseudo-Planar Functions Over $ {\mathbb F}_{2^{n}}$

Abstract

Planar functions over finite fields give rise to finite projective planes. They were also used in the constructions of DES-like iterated ciphers, error-correcting codes, and codebooks. They were originally defined only in finite fields with odd characteristic, but recently Zhou introduced pesudo-planar functions in even characteristic which yields similar applications. All known pesudo-planar functions are quadratic and hence they give presemifields. In this paper, a new approach to constructing quadratic pseudo-planar functions is given. Then five explicit families of pseudo-planar functions are constructed, one of which is a binomial, two of which are trinomials, and the other two are quadrinomials. All known pesudo-planar functions are revisited, some of which are generalized. These functions not only lead to projective planes, relative difference sets and presemifields, but also give optimal codebooks meeting the Levenstein bound, complete sets of mutually unbiased bases (MUB) and compressed sensing matrices with low coherence.

Further Study of Planar Functions in Characteristic Two

“…One class over F q 3 is an extension of f a,b (x) investigated in [3], while our proofs seem to be much simpler. In addition, although the planar binomial over F q 2 of our results is finally a known planar monomial, we also answer the necessity at the same time and solve partially an open problem for the binomial case proposed in [11].…”

“…These new planar functions over F 2 n , as an analogue of planar functions in odd characteristic, also bring about finite projective planes and have the same types of applications as odd-characteristic planar functions do. Note that such functions are still called 'planar' by Y. Zhou [15], while they are called 'pseudo-planar' in [1,11] to avoid confusion with planar functions in odd characteristic. In the rest of the paper, we still call them planar functions without ambiguity.…”

“…, where c ∈ F * 2 m and Tr F 2 m /F 2 (c) = 0, where Tr F 2 m /F 2 (c) = c + c 2 + • • • + c 2 m−1 is the absolutely trace function over F 2 m (see [13,Theorem 3.1]), and it was generalized by [11,Theorem 26] that c ∈ F * 2 2m and Tr F 2 m /F 2 (c 2 m +1 ) = 0;…”

“…They are defined over finite fields of odd characteristic originally and generalized by Y. Zhou [15] in even characteristic. In 2016, L. Qu [11] proposed a new approach to constructing quadratic planar functions over F 2 n .Very recently, D. Bartoli and M. Timpanella [3] characterized the condition on coefficients a, bIn this paper, together with the Hasse-Weil Theorem and the new approach introduced in [11], we completely characterize the necessary and sufficient conditions on coefficients of four classes of planar functions over finite fields in characteristic two. The first and the last class of them are over F q 2 and F q 4 respectively, while the other two classes are over F q 3 , where q = 2 m .…”

“…They are defined over finite fields of odd characteristic originally and generalized by Y. Zhou [15] in even characteristic. In 2016, L. Qu [11] proposed a new approach to constructing quadratic planar functions over F 2 n .…”