Bent Functions and Units in Group Algebras Sugata Gangopadhyay and Deepika Saini Department of Mathematics Indian Institute of Technology Roorkee, Roorkee–247667, INDIA. {gsugata, deepikasainidma}@gmail.com
Abstract. Let Gn be an abelian 2-group having n generators and C be the field of complex numbers. Then CGn is said to be the complex group algebra of Gn . Also let Zn 2 be the set of n-tuples of Z2 . Any function from Zn 2 to Z2 is said to be a Boolean function. In this paper we consider representation of Boolean functions as elements in CGn . We demonstrate that bent Boolean functions on n variables correspond to a particular type of unitary units in CGn .
1
Introduction
Let C be the field of complex numbers and let Gn be an abelian 2-group with n generators presented as follows: Gn = ha1 , . . . , an : a2i = 1, ai aj = aj ai , i, j ∈ {1, . . . , n}i.
(1)
The group algebra CGn is defined as the finite formal sums of the form: a=
X
a(g)g, where a(g) ∈ C, for all g ∈ Gn .
(2)
g∈Gn
Suppose a = defined as
P
g∈Gn
a(g)g and b =
P
g∈Gn
a+b=
b(g)g are any two elements in CGn . Addition is
X
(a(g) + b(g))g,
(3)
g∈Gn
and multiplication is defined as ab =
X X
a(g)b(h)gh
g∈Gn h∈Gn
=
X
X
g∈Gn
h∈Gn
=
X
X
g∈Gn
a(h)b(h−1 g) g
h∈Gn
a(h)b(hg) g.
(4)
The complement of any z ∈ C is denoted by z. The involution of a ∈ CGn is X a∗ = a(g)g −1 g∈Gn
=
X
(5) a(g)g,
g∈Gn
since g 2 = 1 for all g ∈ Gn . Any invertible element in CGn is said to be a unit. Definition 1. An element u ∈ CG is said to be a unitary unit if uu∗ = 1. Let Z2 be the field consisting of two elements. We define Zn2 = {x = (x1 , . . . , xn ) : xi ∈ Z2 , i ∈ {1, . . . , n}}.
(6)
Any function from Zn2 to Z2 is said to be a Boolean function on n variables and the set of Boolean functions on n variables is denoted by Bn . Alternatively, Boolean functions can be thought of as functions from Zn2 to the set {−1, 1}. All through this paper we shall consider the latter description of a Boolean function if not stated otherwise. For detailed study on bent functions we refer to [1, 2]. We note that there is a natural isomorphism from Zn2 to Gn . Thus, a Boolean function f ∈ Bn can also be thought of as a function from Gn to {−1, 1} and hence as the following element of the group algebra CGn : X f (g)g. f= (7) g∈Gn
2
Bent functions and units in CGn
The bent functions form a very important class of Boolean functions defined on even number of n variables. It is known that bent functions possess maximum nonlinearity namely 2n−1 − 2 2 −1 , where n is the number of input variables. Theorem 1. Let n be an even positive integer. Suppose f ∈ Bn is a bent function. Then n 2− 2 f ∈ CGn is a unitary unit. Proof. Suppose f ∈ Bn . The autocorrelation of f at a point u ∈ Zn2 is defined as: Cf (u) =
X
f (x)f (x + u).
(8)
x∈Zn 2
It is known that if f ∈ Bn is a bent function then X Cf (u) = f (x)f (x + u) = 0, x∈Zn 2
(9)
for all nonzero u ∈ Zn2 and Cf (0) = 2n . If we convert this expression in the group algebra domain, supposing that u 7→ g then X f (h)f (hg) = 0, (10) Cf (g) = h∈Gn
for all nonzero g ∈ Gn and Cf (1) = 2n . Let the bent function when written as an element of CGn is X f (h)h. (11) f= h∈Gn
By using (4) and (10) we obtain f f = 2n .1 −n 2
(2 n
n
f )(2− 2 f ) = 1.
This implies that 2− 2 f is a unitary unit in CGn .
3
(12) t u
Conclusion
This is our work in progress. We demonstrate above a connection between units in complex group algebras and bent functions. We have not come across this connection before. However the connection between group rings and Boolean functions is investigated by Pott [4]. It is an open question to check whether this connection can be utilized to study the structure of bent functions and whether generalized bent functions as described by Tokareva [5] can be put in the same framework.
References 1. C. Carlet, Boolean functions for cryptography and error correcting codes, Chapter of the monograph, “Boolean Models and Methods in Mathematics, Computer Science and Engineering,” Cambridge Univ. Press, Y. Crama, P. Hammer (eds.), pp. 257–397, 2010. 2. T. W. Cusick, P. St˘ anic˘ a, Cryptographic Boolean functions and applications, Elsevier – Academic Press, 2009. 3. C.P.Milies and S.K.Sehgal, An Introduction of Group Ring, Kluwer Academic Publisher, 2002. 4. A. Pott, Nonlinear function in abelian groups and relative difference sets, Discrete Applied Mathematics 138 (2004), pp. 177-193. 5. N. N. Tokareva, Generalization of Bent functions. A Survey, Journal of Applied and Industrial Mathematics, 5 (2011), no. 1, pp. 110-129.
