‐Relative Difference Sets and Their Representations

Abstract

Abstract. We show that every (2 n , 2 n , 2 n , 1)-relative difference set D in Z n 4 relative to Z n 2 can be represented by a polynomial f (x) ∈ F 2 n [x], where f (x + a) + f (x) + xa is a permutation for each nonzero a. We call such an f a planar function on F 2 n . The projective plane Π obtained from D in the way of Ganley and Spence [15] is coordinatized, and we obtain necessary and sufficient conditions of Π to be a presemifield plane. We also prove that a function f on F 2 n with exactly two elements in its image set and f (0) = 0 is planar, if and only if, f (x + y) = f (x) + f (y) for any x, y ∈ F 2 n .

Perfect nonlinear functions and cryptography

“…When m = n, bent functions do not exist, see Theorem 1, which also holds under some additional assumptions for generalized q-ary bent functions. For odd n and q = 2, the lower bound for L(F ) is 2 (n+1)/2 and functions that achieve this bound are known to exist [14]. Such functions are called almost bent (AB).…”

“…While probably without direct application in cryptography a new definition of planar function has recently been introduced for vectorial Boolean function. In [13], a function F : Z n 2 → Z n 2 is called planar if for all non-zero a ∈ Z n 2 the functions x → F (x) + F (x + a) + ax are bijective. This notion which is not related to the definitions of PN and APN functions studied in this paper will not be further discussed.…”

“…One of the main tools Chabaud and Vaudenay used in [14] for proving useful and nontrivial results on lower bounds of linearity was the following theorem, which links the derivatives and linear approximations of a vectorial Boolean functions. Recently this result was used in [17,18] in the cryptographic context to show relations between statistical distinguishers in the differential and linear cryptanalysis contexts.…”

Cayley Graphs of Diameter Two from Difference Sets

“…Proof. The proof that D is a (q, q, q, 1)-relative difference set can be found in [21,22,26]. Actually D corresponds to the trivial planar function f (x) = 0 defined over F 2 m , which gives rise to the Desarguesian plane of order 2 m .…”

On homogeneous planar functions

“…Most recently, the notion of planar functions have been defined and studied in the even characteristic in a series of papers, c.f. [15,19,20,25,26].…”