Construction of bent functions from two known bent functions

A (I, -1 )-matrix will be called a bent type matrix if each row and each column are bent sequences. A similar description can be found in Carlisle M. Adams and Stafford E. Tavares, Generating and counting binary sequences, IEEE Trans. Inform. Theory, vol. 36, no. 5, pp. 1170-1173, 1990, in which the authors use the properties of bent type matrices to construct a class of bent functions. In this paper we give a general method to construct bent type matrices and show that the bent sequence obtained from a bent type matrix is a generalized result of the Kronecker product of two known bent sequences. Also using two known bent sequences of length 22 we can construct 2k 2 bent sequences of length 2 2k , more than in the ordinary construction, which gives construct 10 bent sequences of length 22k from two known bent sequences of length length 22k2 • Let Vn be the vector space of n tuples of elements from GF(2). Let a, {3 E Vn. Write a = (al,··· I an), {3 = (bI1 ···, bn ), where ai, bi E GF(2). Write (a,(3) = "LJ=1 ajbj for the scalar product of a and (3. Definition 1 We call the function hex) = alxl + ... + anxn + c, aj, C E GF(2), an affine function, in particular I h( x) will be called a linear function if C = o. Australasian Journal of Combinatorics 9(1994), 00.21-35 Definition 2 Let f(x) be a function from Vn to GF(2) (simply, a function on Vn). If :z:EV" for every f3 E Vn . We call f(x) a bent function on Vn . From Definition 2, bent functions on Vn only exist for even n. Bent functions were first introduced and studied by Rothaus [13]. Further properties, constructions and equivalence bounds for bent functions can be found in [2], [5], [7], [12], [16]. Kumar, Scholtz and Welch [6] defined and studied the bent functions from Z; to Zq. Bent functions are useful for digital communications, coding theory and cryptography [3], [1], [4], [7], [8], [10], [9], [11], [12]. We say a = (al,"', an) < f3 = (bl ,···, bn) if there exists k, 1 ~ k ~ 2 , such that al = bl, ... , ak-l = bk l and ak = 0, bk = 1. Hence we can order all vectors in Vn by the relation < where ao (0"",0), al (0,,,,,1), a2,,-1_1 (0, I, ... ,1), a2,,-1 (1,0, .. ·,0), a2Y1.-1 (1,1,,,,,1). Definition 3 Let f(x) be a function from Vn to GF(2). We call (_1)!(ao), (_l)!(ad , ... , (_1)!(a2 Y1.) the sequence of f(x). We call the sequence of f(x) a bent sequence if f(x) is bent. A (1, -I)-sequence will be called an affine sequence a (linear sequence) if it is the sequence of an affine function (a linear function). Definition 4 A (1, -1 )-matrix H of order h will be called an Hadamard matrix if HHT = hJn. If h is the order of an Hadamard matrix then his 1,2 or divisible by 4 [IS). A special kind of Hadamard matrix defined as following will be relevant. Definition 5 The Sylvester-Hadamard matrix (or Walsh-Hadamard matrix) of order 2n, denoted by Hn, is generated by the recursive relation H [ Hn-l H n 1 1 1 2 IT 1 n = H ,n = , , ... , .l.lO = . n-l n-l Let f(x) be a function from Vn to GF(2), < be the sequence (regarded as a row vector) of f( x). Then the following three conditions are equivalent (i) f(x) is bent, (ii) 2-~n Hn<T is a (I, -1 )-row vector, (iii) for any affine sequence 1 «, l) = ±2~n. The equivalence of (i) and (ii) can be found in many references, for example, [2], [16]. Note that any affine sequence of length 2 is a row of ±Hn (see subsection 2.3) thus (ii) and (iii) are equivalent. Definition 6 We call a (1, -I)-matrix of order 2 X 2n a bent type matrix if each row is a bent sequence of length of 2n and each column is a bent sequence of length of 2m .