Maximally Nonlinear Functions and Bent Functions

We give a construction of bent functions in 2n variables, here n is odd, by using maximally nonlinear functions on GF(2 ⁿ ).     

Some Properties of Bent Functions

First, this paper discusses and sums up some properties of a pair of functions p（x）, q（x） that makes （y ＋ 1）p（x） ＋ yq（x） into a bent function. Then it discusses the properties of bent functions. Also, the upper and lower bounds of the number of bent functions on GF（2）2k are discussed.     

Minimal cyclic codes of length pq

Explicit expressions for all the 3n+2 primitive idempotents in the ring R_pⁿq=GF(ℓ)[x]/(x^pnq−1), where p,q,ℓ are distinct odd primes, ℓ is a primitive root modulo pⁿ and q both, , are obtained. The dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length pⁿq over GF(ℓ) are also discussed.     

Some results concerning cryptographically significant mappings over GF(2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

In this paper we investigate the existence of permutation polynomials of the form F(x) = x ^d  + L(x) over GF(2 ⁿ ), L being a linear polynomial. The results we derive have a certain impact on the long-term open problem on the nonexistence of APN permutations over GF(2 ⁿ ), when n is even. It is shown that certain choices of exponent d cannot yield APN permutations for even n. When n is odd, an infinite class of APN permutations may be derived from Gold mapping x ³ in a recursive manner, that is starting with a specific APN permutation on GF(2 ^k ), k odd, APN permutations are derived over GF(2 ^k+2i ) for any i ≥ 1. But it is demonstrated that these classes of functions are simply affine permutations of the inverse coset of the Gold mapping x ³. This essentially excludes the possibility of deriving new EA-inequivalent classes of APN functions by applying the method of Berveglieri et al. (approach proposed at Asiacrypt 2004, see [3]) to arbitrary APN functions.     

Trace Representation of Legendre Sequences

In this paper, a Legendre sequence of period p for any odd prime p is explicitely represented as a sum of trace functions from GF(2 ⁿ ) to GF(2), where n is the order of 2 mod p.     

Bundles, presemifields and nonlinear functions

Bundles are equivalence classes of functions derived from equivalence classes of transversals. They preserve measures of resistance to differential and linear cryptanalysis. For functions over GF(2 ⁿ ), affine bundles coincide with EA-equivalence classes. From equivalence classes (“bundles”) of presemifields of order p ⁿ , we derive bundles of functions over GF(p ⁿ ) of the form λ(x)*ρ(x), where λ, ρ are linearised permutation polynomials and * is a presemifield multiplication. We prove there are exactly p bundles of presemifields of order p ² and give a representative of each. We compute all bundles of presemifields of orders p ⁿ ≤ 27 and in the isotopism class of GF(32) and we measure the differential uniformity of the derived λ(x)*ρ(x). This technique produces functions with low differential uniformity, including PN functions (p odd), and quadratic APN and differentially 4-uniform functions (p = 2).     

On “bent” functions

Let P(x) be a function from GF(2ⁿ) to GF(2). P(x) is called “bent” if all Fourier coefficients of (−1)^P(x) are ±1. The polynomial degree of a bent function P(x) is studied, as are the properties of the Fourier transform of (−1)^P(x), and a connection with Hadamard matrices.     

Patterson-Wiedemann construction revisited

In 1983, Patterson and Wiedemann constructed Boolean functions on n=15 input variables having nonlinearity strictly greater than 2^n-1-2^(n-1)/2. Construction of Boolean functions on odd number of variables with such high nonlinearity was not known earlier and also till date no other construction method of such functions are known. We note that the Patterson-Wiedemann construction can be understood in terms of interleaved sequences as introduced by Gong in 1995 and subsequently these functions can be described as repetitions of a particular binary string. As example we elaborate the cases for n=15,21. Under this framework, we map the problem of finding Patterson-Wiedemann functions into a problem of solving a system of linear inequalities over the set of integers and provide proper reasoning about the choice of the orbits. This, in turn, reduces the search space. Similar analysis also reduces the complexity of calculating autocorrelation and generalized nonlinearity for such functions. In an attempt to understand the above construction from the group theoretic view point, we characterize the group of all GF(2)-linear transformations of GF(2^ab) which acts on PG(2,2^a).     

On counting absolute trace of powers inGF(p m )

There are a few results of Welch (1967) and O. Moreno (1980) that count the number of solutions ofTr(y ^l )=0 inGF(2^m), for certain values ofl. This paper counts the number of solutions ofTr(y ^l)=h inGF(p ^m), for further values ofl. Then O. Moreno's question is answered.     

Crooked binomials

A function f : GF(2 ^r ) → GF(2 ^r ) is called crooked if the sets {f(x) + f(x + a)|x ∈ GF(2 ^r )} is an affine hyperplane for any nonzero a ∈ GF(2 ^r ). We prove that a crooked binomial function f(x) = x ^d + ux ^e defined on GF(2 ^r ) satisfies that both exponents d, e have 2-weights at most 2.