Bounds on the Fourier coefficients of the weighted sum function

Information Processing Letters 103 (2007) 83–87 www.elsevier.com/locate/ipl

Bounds on the Fourier coefficients of the weighted sum function Igor E. Shparlinski Department of Computing, Macquarie University, Sydney, NSW 2109, Australia Received 20 March 2006; received in revised form 15 January 2007; accepted 12 February 2007 Available online 28 February 2007 Communicated by L.A. Hemaspaandra

Abstract We estimate Fourier coefficients of a Boolean function which has recently been introduced in the study of read-once branching programs. Our bound implies that this function has rather “flat” Fourier spectrum and thus implies several lower bounds on some of its complexity measures. © 2007 Elsevier B.V. All rights reserved. Keywords: Boolean function; Computational complexity; Fourier coefficients; Weighted sums

1. Introduction 1.1. Motivation P. Savický and S. Žák [21], in their study of readonce branching programs, have recently introduced a Boolean function f defined in terms of certain weighted sums in the residue ring modulo a prime. It has also been used by M. Sauerhoff [19,20] for several more complexity theory applications. In particular, in [20] a certain modification of the same function has been used to prove that quantum read-once branching programs are exponentially more powerful than classical read-once branching programs. Here, motivated by the important role the function f has played in several recent works, we continue to study f and concentrate on estimating its Fourier coefficients. It is well known that there are many close links between Fourier coefficients and various complexity charE-mail address: [email protected]. 0020-0190/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2007.02.011

acteristics of any Boolean function, see [2–6,10,12,13, 15,17,18] and references therein. Although we do not present all such implications, we give a lower bound on the average sensitivity of f and discussed possible applications to several other complexity characteristics of f . 1.2. Notation We now fix a sufficiently large integers n and let p be the smallest prime with p  n. We also use Br to denote the r-dimensional binary cube, that is, Br = {0, 1}r . Given an n-dimensional binary vector x = (x1 , . . . , xn ) ∈ Bn we define s(x) by the conditions s(x) ≡

n 

kxk (mod p),

1  s(x)  p.

k=1

Following [21], we consider the Boolean function  x , if 1  s(x)  n; f (x) = s(x) otherwise. x1 ,

(1)

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I.E. Shparlinski / Information Processing Letters 103 (2007) 83–87

We use some methods of analytic number theory to estimate Fourier coefficients 1  fˆ(u) = n (−1)f (x)+u·x , 2

For every j ∈ {1, . . . , p}, let Xj be the set of x ∈ Bn with s(x) = j . We now write

where u = (u1 , . . . , un ) ∈ Bn , and

and estimate each of the inner sums  (−1)f (x)+u·x Fj (u) =

x∈Bn

u · x = u1 x1 + · · · + un xn

p 1  Fj (u) fˆ(u) = n 2

x∈Xj

is the inner product. 1.3. Results We show that all Fourier coefficients are of size at most 2(−ρ+o(1))n for some constant ρ = 0.1587 . . . . Certainly, the Parseval identity  g(u) ˆ 2 = 1, (2) u∈Bn

separately. We start with considering the sum Fj (u) for j ∈ {1, . . . , n}. In this case, for every pair (α, β) ∈ B2 we use Xj,α,β to denote the set of x ∈ Xj with xj = α,

u · x = β.

Therefore Fj (u) =



(3)

u∈Bn

for Fourier coefficients g(u) ˆ of any n-variate Boolean function g. We also present some immediate applications of our bound to estimating the average sensitivity of f and discuss some other applications.

λ=0

p−1  1      1 + (−1)u·x−β e λ s(x) − j 2p

x∈Bn xj =α

ω

λ=0

ln cos ϑ dϑ



0

ν=1

×e λ

(−1)ν sin(2νω) , ν2

x∈Bn xj =α

μ=0

n 

kxk

k=1

=

see [11, Section 8.26]. We also put ∞ 2  (−1)ν 4 L(π/4) = 1 − ρ= π ln 2 π ln 2 (2ν + 1)2

p−1 1  1   e λj (α − 1) (−1)μ(αuj +β) 2p

×

ν=0

= 0.1587 . . . .

λ=0

p−1 1   1  e(−λj ) (−1)μβ (−1)μu·x = 2p

Let us define the Lobachevsky function

1 2

√ −1.

we deduce that

2. Fourier coefficients

= ω ln 2 +

(6)

We now put e(z) = exp(2πιz/p) where ι = From the identity  p−1  0, if z ≡ 0 (mod p), e(λz) = p, if z ≡ 0 (mod p),

#Xj,α,β =

∞ 

#Xj,α,β (−1)α+β .

α,β∈B2

implies that   ˆ   2−n/2 max g(u)

L(ω) = −

(5)

j =1

u∈Bn

Proof. We recall that by the Prime Number Theorem, p = n + o(n), see [1] for the best known result about gaps between prime numbers.

μ=0

  1 + (−1)μuk e(λk) .

k=1 k=j

(4)

Theorem 1. For the function f given by (1), we have   max fˆ(u)  2−ρn+o(n) .

λ=0 n

We say that u ∈ Bn is j -vanishing if uk = 0 for every k ∈ {1, . . . , n} with k = j . Then we see that in the above sum the term corresponding to λ = 0 is n 1

  1  1 + (−1)μuk (−1)μ(αuj +β) 2p μ=0

= σj (u, α, β),

k=1 k=j

I.E. Shparlinski / Information Processing Letters 103 (2007) 83–87

where

⎧ ⎪ 2n−2 p −1 , ⎪ ⎨ if u is not j -vanishing, σj (u, α, β) = n−2 −1 αuj +β ), ⎪ ⎪ ⎩ 2 p (1 + (−1) otherwise. The contribution from other terms can be estimated as  n  p−1 1  1     μuk 1 + (−1) e(λk)     2p

Thus we infer from (7) that  n  p−1 1   1      1 + (−1)μuk e(λk)     2p λ=1 μ=0 k=1 k=j

 2(1−ρ)n+o(n) . We now deduce from (6) that  Fj (u) = σj (u, α, β)(−1)α+β α,β∈B2

λ=1 μ=0 k=1 k=j

=

1 2p

p−1 1 

n

+ O(2(1−ρ)n+o(n) )

  1 + (−1)μuk e(λk)

and one can easily verify that  σj (u, α, β)(−1)α+β = 0

λ=1 μ=0 k=1 k=j

      p−1 n     λk λk 2n 

    max sin π , cos π  . p p p

α,β∈B2

whether u is j -vanishing or not. Hence   Fj (u)  2(1−ρ)n+o(n) .

λ=1 k=1 k=j

Since obviously   1 max | sin ϑ|, | cos ϑ|  √ , 2 we get       n

    λk λk π , cos π  max sin p p k=1 k=j

=2

o(n)

p−1

k=0

x1 = α,

α,β∈B2

+ O(2(1−ρ)n+o(n) ). Therefore (8) still holds. Substituting (8) in (5) we finish the proof. 2

          k k o(n)    =2 π , cos π  max sin p p k=0 

  k 4 cos = 2o(n) π  .  p

3. Average sensitivity We recall that the average sensitivity σav (g) of an n-variate Boolean function g is defined as

0kp/4

1 2p

   

λ=1 μ=0 k=1 k=j

 2n+o(n)

   μuk 1 + (−1) e(λk)  



0kp/4

 cos4

0kp/4

  1/4 k ln cos π = p ln cos(ϑπ) dϑ + o(p) p 0

p = − L(π/4) + o(p). π

n    g(x) − g(x(i) ), x∈Bn i=1

(7)

where x(i) is the vector obtained from x by flipping its ith coordinate. Let τ = 0.0575 . . . be the root of the equation H (τ ) = 2ρ,

Clearly 

σav (g) = 2−n



k π . p

u · x = β.

Exactly the same arguments as before lead to the bound  Fj (u) = σ1 (u, α, β)(−1)α+β

          λk λk π , cos π  max sin p p

p−1 n 1 



(8)

It remains to estimate Fj (u) for j ∈ {n + 1, . . . , p}. In this case, for every pair (α, β) ∈ B2 we use Yj,α,β to denote the set of x ∈ Xj with

p−1

Thus

85

0  τ  1/2,

where ρ is given by (4), H (γ ) = −γ log γ − (1 − γ ) log(1 − γ ), 0 < γ < 1, and log z denotes the binary logarithm.

(9)

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Theorem 2. For the function f given by (1), we have   σav (f )  τ + o(1) n. Proof. It is shown in [12] that  2  wt(u)fˆ(u) , σav (f ) =

g(x1 , . . . , xn ) = F (x1 , . . . , xr ) and   g(x1 , . . . , xn ) − F (x1 , . . . , xr )  1/3

u∈Bn

where wt(u) is the Hamming weight of u. Therefore, for any w  n, from the Parseval identity (2), we obtain    2   fˆ(u)2 wt(u)fˆ(u) + w σav (f )  u∈Bn wt(u)
=



u∈Bn wt(u)w

2  wt(u)fˆ(u)

u∈Bn wt(u)
    fˆ(u)2

 +w 1−

u∈Bn wt(u)
 w − (w − 1)

   fˆ(u)2 .

u∈Bn wt(u)
u∈Bn wt(u)
=2

−2ρn+o(n)

w−1  j =0

holds for every (x1 , . . . , xr ) ∈ Bn , respectively. Clearly, δ(g)  (g)  n. By Corollary 2.5 and by Lemma 3.8 of [16], for any Boolean function g, we have  1/2 (g)  σav (g) and δ(g)  σav (g)/6 , thus Theorem 2 we obtain for the function f , that   (f )  τ + o(1) n and   δ(f )  6−1/2 τ + o(1) n1/2 , where τ is defined by (9). In turn, these bounds imply a lower bound on quantum computational complexity of f , see [7]. 5. Remarks

Using the bound of Theorem 1 we see that     fˆ(u)2  2−2ρn+o(n) 1 u∈Bn wt(u)
For an n-variate Boolean function g, we define its real degree (g) and real approximate degree δ(g) as the smallest possible degree of a real polynomial F in n variables for which

 n . j

We recall that for any w  n/2 we have the bound w−1  n  2nH (w/n)+o(n) , j j =0

see [14, Section 10.11]. Hence, for w  n/2, σav (f )  w − (w − 1)2n(H (w/n)−2ρ)+o(n) . Thus, taking w = (τ − δ(n))n where the function δ(n) → 0 slowly enough, we obtain the desired result. 2 4. Circuit complexity and other applications By the Boppana result [4] if an unbounded fanin Boolean circuit of depth d and size S computes a Boolean function g, then d log log S  log σav (g). Thus we see from Theorem 2 that if an unbounded fan-in Boolean circuit of depth d and size S computes the function f given by (1), then   d log log S  1 + o(1) log n.

It is certainly interesting to establish the true order of magnitude of the largest Fourier’s coefficient and other characteristics of the function f . Open Question 3. Is it true that for the function f given by (1) we have   max fˆ(u)  2−n/2+o(n) ?

u∈Bn

Certainly if the answer to Question 3 is positive this is the best possible bound (due to the trivial lower bound (3)). Open Question 4. Is it true that for the function f given by (1) we have   1 + o(1) n? σav (f )  2 A positive answer to Question 3 also implies a positive to Question 4 (via the same argument as that used in the proof of Theorem 2). However it is possible that there is some combinatorial argument leading to the same lower bound on σav (f ) which does not rely on bounds of Fourier’s coefficients. The bound of Theorem 1 implies that the function f has a high non-linearity N (f ) = 2n−1 + O(2(1−ρ)n+o(n) )

I.E. Shparlinski / Information Processing Letters 103 (2007) 83–87

which is defined as the difference    N(f ) = 2n−1 1 − max fˆ(u) . u∈Bn

We recall that Boolean functions with large non-linearity play a very important role in cryptography, see [8,9]. Thus it may be interesting to study some other properties of cryptographic interest for the function f . One can also consider its applicability to stream ciphers, which naturally leads to the following: Open Question 5. Study the period and statistical distribution of sequences (zh )∞ h=1 , generated recursively by zh+n+1 = f (zh , . . . , zh+n ),

h = 1, 2, . . . ,

with some initial vector (z1 , . . . , zn ) ∈ Bn . We note that our purpose has not been to construct a function with large non-linearity, but rather estimate this characteristic for a specific interesting function. Acknowledgements The author is grateful to the organizers of the Dagstuhl Workshop “Complexity of Boolean Functions” (March, 2006) for the invitation where this work was essentially done. In fact the author learned about the function f and its role in the complexity theory from a talk given at that workshop by Martin Sauerhoff about his work [20]. This work was supported in part by ARC grant DP0556431. References [1] R.C. Baker, G. Harman, J. Pintz, The difference between consecutive primes, II, Proc. London Math. Soc. 83 (2001) 532–562. [2] A. Bernasconi, On the complexity of balanced Boolean functions, Inform. Process. Lett. 70 (1999) 157–163. [3] A. Bernasconi, C. Damm, I.E. Shparlinski, Circuit and decision tree complexity of some number theoretic problems, Inform. and Comput. 168 (2001) 113–124.

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