Harmonic Analysis of Binary Functions Jean-Claude Belfiore Yi Hong Emanuele Viterbo Communications and Electronics Dept. ECSE Department ECSE Department Telecom ParisTech Monash University Monash University Paris, France Clayton, Australia Clayton, Australia jean-claude.belfi[email protected] [email protected] [email protected] Abstract—In this paper we introduce the two-modular Fourier of C2. We then introduce the corresponding inverse Fourier transform of a binary function f : R!R defined over a finite transform, and prove the corresponding convolution theorem. R F F commutative ring = 2[X]/ϕ(X), where 2[X] is the ring of Our two-modular Fourier transform is based on the two- polynomials with binary coefficients and ϕ(X) is a polynomial of degree n, which is not a multiple of X. We also introduce modular indecomposable representations of the group C2 in the corresponding inverse Fourier transform. We then prove the dimension two and defines n + 1 “spectral coefficients” as corresponding convolution theorem. matrices over R of size 2k × 2k, for k = 0 : : : ; n. Our two- Index Terms—two-modular Fourier transform, binary func- modular Fourier transform preserves the structure of the ring tions, binary groups R, which is lost if the characteristic zero Fourier transform were used. This Fourier transform may have broad applications I. INTRODUCTION to problems, where binary functions need to be reliably computed or for classification of binary functions. Harmonic analysis of binary functions is a powerful tool which allows to derive deep results in computational com- II. BACKGROUND plexity (cf. the PCP problem in [1, Chap. 22]). When the function takes scalar binary values, then, this scalar value can A. Fourier Transform of f : G ! C be mapped to the set f+1; −1g and we get a classical problem of harmonic analysis of complex-valued functions. When the The classical notion of Fourier transform over arbitrary function takes values in a binary vector space, then we can finite groups is based on the degree-n representations of × C no more use the same method and we have to find a Fourier group elements by complex n n matrices in GL(n; ). It transform which complies with the characteristics 2. generalizes the well known discrete Fourier transform, which The Fourier transforms of functions over finite abelian is naturally defined over a cyclic group, which has n distinct groups f : G ! C (complex field) or f : G ! Z (ring scalar representations ρk, given by the powers of the n-th roots of integers) have been extensively studied [3]. For complex of unity valued functions, and when the group is cyclic, the Fourier ρ = fexp(i2πk`=n); ` = 0; : : : n − 1g transform is the well-known discrete Fourier Transform [3]. k For complex valued functions, and if the group G is Cn 2 for k = 0; : : : n−1. In the general case where G is not cyclic, where C is the cyclic group of order 2, the Fourier transform 2 the group representations are matrices and we have is the well known Hadamard transform, commonly used for Definition 1: ([3]) Given a finite group G, the Fourier analysing Boolean functions [2]. transform of a function f : G ! C evaluated for a given A less common case occurs for f : G !R, where R representation ρ : G ! GL(d ; C) of G, of degree d , is is a ring of characteristics p. For any prime p co-prime with ρ ρ given by the d × d matrix the order of G, the p-modular Fourier transform is defined ρ ρ X in a way similar to the case where f is complex-valued. f^(ρ) = f(a)ρ(a) However, when p is not coprime with the order of the group, a2G and especially in the case of p = 2 and the order of G is 2n, the Fourier transform cannot be handled in the same way. The complete Fourier transform is obtained by considering the In this paper, we introduce the two-modular Fourier trans- set fρkg of all inequivalent representations of G. The matrix form of a binary function f : R!R defined over a entries of ρk are mutually orthogonal matrix functions over finite commutative ring R = F2[X]/ϕ(X), where F2[X] is G and fρkg form the Fourier basis. Since the dimension of the ring of polynomials with binary coefficients and ϕ(X) the ‘frequency domain’ space is equal to the dimension of the is a polynomial of degree n, which is not a multiple of X. ‘time domain’ space we have The function f can be viewed as a binary function over the X R !R d2 = jGj elements of the additive group of , i.e., f : G , where ρk n × · · · × G = C2 = C2 C2 is the direct product of n copies k Definition 2: ([3]) The inverse Fourier transform evaluated Then, we can represent the n-fold direct product G = C2 × 2 · · · × n at a G is given by C2 = C2 as tensor product of the representations of C2 X ( ) [4], i.e., 1 −1 ^ f(a) = dρk Tr ρk(a )f(ρk) n−1 jGj π (G) , π (C )⊗· · ·⊗π (C ) = π − (C )⊗π (C ) (2) k n 1 2 1 2 n 1 2 1 2 where Tr(·) is the trace of the matrix. Note that the elements of G are n-bit binary vectors b and the The above Fourier transform is well defined for complex group operation ⊕ is bitwise exclusive-OR. The corresponding valued functions over finite groups G and can be used to representation matrix of an element b = (b1; : : : ; bn) is ([4]) transform convolution in the ‘time-domain’ defined as [3] , ⊗ · · · ⊗ X Eb π1(b1) π1(bn) : (3) (f ∗ f )(a) = f (ab−1)f (b) 1 2 1 2 (1) We also define the representation of the trivial group f0g as b2G π0(f0g) , 1. into the product in the ‘frequency domain’ i.e., [3] Lemma 1: The representations πk for k = 0; : : : ; n are faithful. \ ^ ^ (f1 ∗ f2)(ρ) = f1(ρ)f2(ρ) Proof: For k = 0 and 1 it is straightforward. For k ≥ 2 we prove it by induction using the recursion (2). Thus it is B. Fourier Transform of f : G ! K enough to consider the case k = 2 and show that π2 is a 2 2 If we are interested in computing the Fourier transform of group homomorphism between C2 and π2(C2 ), i.e., that functions over a finite group G taking values in a finite field K π (b ⊕ c ; b ⊕ c ) = π (b b ) · π (c c ) of prime characteristic p we need to modify the Definition 1 2 1 1 2 2 2 1 2 2 1 2 and work with p-modular representations of the group G. The or equivalently, representations ρ(a) are now in GL(n; K) i.e., n×n matrices · with entries in K. Eb1⊕c1;b2⊕c2 = Eb1b2 Ec1c2 : Definition 3: A p-modular representation of a group G From (2) we have π = π ⊗ π then over a field K of prime characteristic p is a group homomor- 2 1 1 phism π which associates group elements to k × k matrices · ⊗ · ⊗ Eb1b2 Ec1c2 = (Eb1 Eb2 ) (Ec1 Ec2 ) over K, i.e., π : G 7! GL(k; K), such that the addition of = (E E ) ⊗ (E E ) two group elements corresponds to the matrix multiplication b1 c1 b2 c2 ⊕ ⊕ of the corresponding representation matrices. = Eb1 c1;b2 c2 (4) The case where K = C has characteristic zero and yields the well known representation theory [4]. Definitions 1 and 2 We can think of the the Fourier transform as a unitary provide the Fourier transform pair only if p is co-prime with transformation of a vector representing the values of the jGj. function f on a given ordered set of elements of G. In the following, we assume that such order is determined by the III. THE TWO-MODULAR FOURIER TRANSFORM decimal representation D(g) 2 f0;:::; jGj−1g corresponding In this paper, we focus on binary functions f : R!R to the binary representation of g 2 G. defined over a finite commutative ring R = F2[X]/ϕ(X), Next, we specialize the definition of convolution in (1) for where ϕ(X) is a polynomial of degree n, which is not a the case of additive groups. multiple of X. The elements of R can be represented as Definition 4: Given a pair of functions f1 and f2 : G !R binary coefficient polynomials of degree less than n, where we define the convolution product f3 : G !R as the ring operations are polynomial addition and multiplication X ∗ mod ϕ(X). f3(g) = f1(g) f2(g) = f1(u + g)f2(u) u2G Note that G = C2 × · · · × C2 (the direct product of n copies cyclic groups of order two) coincides with the additive group of R, where the elements can also be represented as It can be easily shown that the convolution product is com- binary vectors b of length n with mod two addition (or bitwise mutative. exclusive-OR). This means that we are actually working with The aim is to project the function f : G !R on a specific !R functions f : G . In the special case where ϕ(X) is an ‘Fourier basis’ f kg where k is the ‘frequency index’. The irreducible polynomial R is the finite field K = F2n . Fourier basis vectors are made up of all the two-modular k k Here, we consider the two-modular representation of C2 = representations (2 ×2 matrices) of the elements of the nested f g × f g n j j 0; 1 as 2 2 binary matrices, i.e., π1(C2) = E0;E1 , subgroups of G = C2 ,(n = log2 G ), namely where ( ) ( ) f0g C H1 C ··· C Hk C ··· C Hn−1 C G 1 0 1 1 π1(0) = E0 = and π1(1) = E1 = 0 1 0 1 where H1 = C2, H2 = C2 × C2, H3 = C2 × C2 × C2, etc.