Partially APN functions with APN-like polynomial representations

Abstract

In this paper we investigate several families of monomial functions with APN-like exponents that are not APN, but are partially 0-APN for infinitely many extensions of the binary field F 2 . We also investigate the differential uniformity of some binomial partial APN functions. Furthermore, the partial APN-ness for some classes of multinomial functions is investigated. We show also that the size of the pAPN spectrum is preserved under CCZ-equivalence.

New infinite classes of 0-APN power functions over F2n

“…For two polynomials F (x, y), G(x, y) ∈ F q n [x, y] of positive degree in y, the resultant R(F, G, y) of F and G with respect to y is the resultant of F and G when considered as polynomials in the single variable y. In this case, R(F, G, y) ∈ F q n [x]∩ F, G , where F, G is the ideal generated by F and G. Thus any pair (a, b) with Up to now, these datas are not yet used by others which come from Table 1 in [17].…”

“…The last column (d, n) denotes the examples of 0-APN (but not APN) power functions x d over F 2 n (n ≤ 11), which come from Table 1 in [17].…”

“…Very rencently, some infinite classes of 0-APN power functions over F 2 n were constructed in [20,21]. To further investigate the new 0-APN functions, some pairs of (d, n) are not yet explained in [17], seeing Table 1. In this paper, we give the new infinite classes of 0-APN functions by using multivariate method and resultant elimination, the main results are listed in Table 2.…”

New Infinite Classes of 0-APN Power Functions over 𝔽_2<i>ⁿ</i>

“…Furthermore, F is a 0-APN function if and only if the equation F (x + 1) + F (x) + 1 = 0 has no solution in F 2 n \F 2 . In [5], [6] Budaghyan et al explicitly constructed some 0-APN power functions f (x) = x d over F 2 n , and they further gave the exponents of all power functions over F 2 n for 1 ≤ n ≤ 11 that are 0-APN but not APN functions. Moreover, Pott proved that for any n ≥ 3, there are partial 0-APN permutations on F 2 n in [16].…”

“…Very recently, some infinite classes of 0-APN power functions over F 2 n were constructed in [10], [13]. To further investigate the new 0-APN functions, we list some pairs of (d, n) that are not yet "explained" in [5], seeing Table 1. In this paper, we give new infinite classes of 0-APN functions using the multivariate method and resultant elimination.…”