A class of functions with low-valued Walsh spectrum

Discrete Applied Mathematics xxx (xxxx) xxx

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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam

A class of functions with low-valued Walsh spectrum✩ ∗

Fengwei Li a,b , , Yansheng Wu c , Qin Yue d a

School of Mathematics and Statistics, Zaozhuang University, Zaozhuang, 277160, PR China State Key Laboratory of Cryptology, P.O. Box 5159, Beijing, 100878, PR China c Department of Mathematics, Ewha Womans University, 52, Ewhayeodae-gil, Seodaemun-gu, Seoul, 03760, South Korea d Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, PR China b

article

info

a b s t r a c t

Article history: Received 21 November 2018 Received in revised form 19 October 2019 Accepted 24 October 2019 Available online xxxx Keywords: Walsh spectrum Gaussian sum

l(l−1)

Let l ≡ 3 (mod 4), l ̸ = 3, be a prime, N = l2 , f = 2 the multiplicative order of a prime p modulo N, and q = pf . In this paper, we investigate the Walsh spectrum of q−1

the monomial functions f (x) = Trq/p (x l2 ) in index two case. In special, we explicitly present the value distribution of the Walsh transform of f (x) if 1 + l = 4ph , where h is √ the class number of Q( −l). © 2019 Elsevier B.V. All rights reserved.

1. Introduction Walsh transform over finite fields is a basic tool in research of properties of cryptographic functions. The important information about cryptography can be obtained from the study of the Walsh transform. A long-standing problem about the Walsh transform is to find functions with a few Walsh transform values and determine its distribution. There are a few monomial functions with three-valued Walsh transform for special exponents [1,6,20,21,25]. The results on functions with at least four-valued Walsh transform were obtained in [5,7,9,19,22]. The research progress on a few Walsh transform values can be referred to literatures [8,13,18,28] and the references therein. Walsh transform is also closely related to Fourier transform, Gauss period, and the weight distribution of a cyclic code [10,14,15,23,27,30,31] et al.. Next let us introduce the definitions of the Walsh transform of a function over a finite field. Let Fq be a finite field with q elements, where q is a power of prime p. The trace function from Fq onto Fp is defined by Trq/p (x) = x + xp + · · · + xq/p , x ∈ Fq . Let f (x) be a function from Fq to Fp . The Walsh transform of f (x) is defined by

ˆ f (b) =

∑

f (x)+Trq/p (bx)

ζp

, b ∈ Fq ,

x∈Fq

where ζp is a complex primitive pth root of unity. The multiset {ˆ f (b) : b ∈ Fq } is called the Walsh spectrum of f (x) over Fq . If f (x) is a monomial function of the form Trq/p (axd ) with a ∈ Fq and d is a positive integer, then the Walsh transform of f (x) gives the cross correlation values of an m-sequence and their d-decimations [1,3,8,17,29]. This paper is organized as follows. In Section 2, we give several results about Gaussian sums in index two case. In l(l−1) Section 3, let l ≡ 3 (mod 4), l ̸ = 3, be a prime, N = l2 , f = 2 the multiplicative order of a prime p modulo N, ✩ The paper was supported by National Natural Science Foundation of China (Nos. 11601475, 61772015), the foundation of Science and Technology on Information Assurance Laboratory, China (No. KJ-17-010), and the foundation of innovative Science and technology for youth in universities of Shandong Province China (2019KJI001). ∗ Corresponding author at: School of Mathematics and Statistics, Zaozhuang University, Zaozhuang, 277160, PR China. E-mail addresses: [email protected] (F. Li), [email protected] (Y. Wu), [email protected] (Q. Yue). https://doi.org/10.1016/j.dam.2019.10.029 0166-218X/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.

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F. Li, Y. Wu and Q. Yue / Discrete Applied Mathematics xxx (xxxx) xxx q −1

and q = pf . We give the Walsh spectrum of an index two function f (x) = Trq/p (x l2 ) over Fq . In special, we explicitly √ present the value distribution of the Walsh transform of f (x) when 1 + l = 4ph , where h is the class number of Q( −l). In Section 4, we conclude this paper. 2. Preliminaries Let Trq/p be the trace function from Fq to Fp . An additive character of Fq is a nonzero function ψ from Fq to the set of complex numbers such that ψ (x + y) = ψ (x)ψ (y) for any pair (x, y) ∈ F2q . For each b ∈ Fq , the function Trq/p (bc)

ψb (c) = ζp

for all b ∈ Fq 2π

√ −1

defines an additive character of Fq , where ζp = e p denotes the pth primitive root of unity. When b = 1, the character ψ1 is called canonical additive character of Fq . It is well known that

∑

ψb (c) = 0 for b ̸= 0.

c ∈Fq

A multiplicative character of Fq is a nonzero function χ from Fq to the set of complex numbers such that χ (xy) = χ (x)χ (y) for all pairs (x, y) ∈ F∗q 2 . Let α be a fixed primitive element of Fq . For each j = 1, 2, . . . , q − 1, the function χj with

χj (α k ) = ζqjk−1 for k = 0, 1, . . . , q − 2 defines a multiplicative character with order

(2.1) q−1 gcd(q−1,j)

of Fq , where ζq−1 denotes the (q − 1)-th primitive root of unity.

Let q be odd and j = 2 in (2.1), we then get a multiplicative character denoted by η such that η(c) = 1 if c is the square of an element and η(c) = −1 otherwise. This η is called the quadratic character of Fq . Let χ be a multiplicative character with order k with k|(q − 1) and ψ an additive character of Fq . Then the Gaussian sum G(χ , ψ ) of order k is defined by q−1

G(χ , ψ ) =

∑

χ (x)ψ (x).

x∈F∗ q

Since G(χ , ψb ) = χ¯ (b)G(χ, ψ1 ), we just consider G(χ, ψ1 ), briefly denoted as G(χ ), in the sequel. Lemma 2.1 ([16]). Let ψ be a nontrivial additive character of Fq and χ a multiplicative character of Fq of order s = gcd(n, q−1). Then

∑

ψ (axn + b) = ψ (b)

x∈Fq

s−1 ∑

χ¯ j (a)G(χ j , ψ )

j=1

for any a, b ∈ Fq with a ̸ = 0. In general, the explicit determination of Gaussian sums is also a difficult problem. For future use, we state the quadratic Gaussian sums here. Lemma 2.2 ([16]). Suppose that q = pf and η is the quadratic multiplicative character of Fq , where p is an odd prime. Then G(η) =

√

(−1)f −1 √q, √ (−1)f −1 ( −1)f q,

{

if p ≡ 1 if p ≡ 3

(mod 4), (mod 4).

Let Z/N Z = {0, 1, . . . , N − 1} be the ring of integers modulo N and (Z/N Z)∗ a multiplicative group consisting of all invertible elements in Z/N Z. If ⟨p⟩ is a cyclic subgroup with a generator p of the group (Z/N Z)∗ such that [(Z/N Z)∗ : ⟨p⟩] = 2 and −1 ∈ / ⟨p⟩ ⊂ (Z/N Z)∗ , which is the so-called ‘‘quadratic residues’’ or ‘‘index 2’’ case, Gaussian sums are explicitly determined, see [26] and its references for details. We list some results [4,26] in the index 2 case below. l(l−1)

Lemma 2.3 ([26]). Let l ≡ 3 (mod 4) be a prime, l ̸ = 3, m a positive integer, N = lm , f = 2 the multiplicative order of a prime p modulo N, and q = pf . Suppose that χ is a primitive multiplicative character of order N over Fq . (1) For 1 ≤ i ≤ N − 1, let i = ult , 0 ≤ t ≤ m − 1 and gcd(u, l) = 1. Then G(χ ) = i

{

t

G(χ l )

if u ∈ ⟨p⟩ ⊂ Z∗N ,

G(χ )

if u ∈ / ⟨p⟩ ⊂ Z∗N .

lt

(2) For 0 ≤ t ≤ m − 1, t

G(χ l ) = p

f −hlt 2

(

√ a + b −l 2

)lt

,

Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.

F. Li, Y. Wu and Q. Yue / Discrete Applied Mathematics xxx (xxxx) xxx

3

√ where h is the ideal class number of Q( −l), a, b are integers given by { 2 a + lb2 = 4ph (2.2) l−1+2h a ≡ −2p 4 (mod l). √ √ √ √ Let O = Z[ −l] be all algebraic integers in Q( −l). Then pO = P1 P2 , where P1 = ⟨ a+b2 −l , p⟩ and P2 = ⟨ a−b2 −l , p⟩. In fact, the multiplicative character χ is correspondent to P2 (see [12]). 3. Explicit valuations of Walsh spectrum of f (x) = Trq/p (x

q−1 l2

)

l(l−1)

Let l ≡ 3 (mod 4), l ̸ = 3, be a prime, N = l2 , and f = 2 the multiplicative order of a prime p modulo N, i.e., f the smallest positive integer such that pf ≡ 1 (mod N). Let Fq be a finite field with q = pf elements. In this section, we shall give the value distributions of the Walsh transform of f (x) = Trq/p (x in F∗q and β = α

q −1 N

q −1 N

). We always denote α a primitive element

an Nth primitive root of unity in Fq . Let K = ⟨α N ⟩ denote the subgroup of Fq generated by α N . Then

⋃N −1

F∗q = ⟨α⟩ = i=0 α i K . (N ,q) Define Ci = α i ⟨α N ⟩ for i = 0, 1, . . . , N − 1. The cyclotomic numbers of order N are defined by (N ,q)

(i, j)N = |(1 + Ci

(N ,q)

) ∩ Cj

|

for all 0 ≤ i ≤ N − 1 and 0 ≤ j ≤ N − 1. The following lemma is proved in [24]. Lemma 3.1. If q ≡ 1 (mod 4), then (0, 0)2 =

q−5

(0, 1)2 =

q+1

, (0, 1)2 = (1, 0)2 = (1, 1)2 =

q−1

, (0, 0)2 = (1, 0)2 = (1, 1)2 =

q−3

4 If q ≡ 3 (mod 4), then

4 It is well known that

4

4

.

.

Z/l2 Z = {0} ∪ l(Z/l2 Z)∗ ∪ (Z/l2 Z)∗ . For convenience, denote S = S0 ∪ S1 ∪ S2 , where S = {i : 0 ≤ i ≤ l2 − 1}, S0 = {0}, S1 = l(Z/l2 Z)∗ = l(Z/lZ)∗ = {lu|gcd(u, l) = 1}, and S2 = (Z/l2 Z)∗ = {u|gcd(u, l) = 1}. Furthermore |S1 | = l − 1 and |S2 | = l(l − 1). (1) (0) Let γ be a primitive root of (Z/l2 Z)∗ , then γ is also a primitive root of (Z/lZ)∗ . Then (Z/lZ)∗ = H1 ∪ H1 , where (0) (1) (0) ∗ 2 H1 = ⟨γ ⟩ and H1 = γ H1 consist of all square elements and non-square elements of (Z/lZ) , respectively. Then (0) (1) (0) (1) (0) (Z/l2 Z)∗ = H2 ∪ H2 , where H2 = ⟨γ 2 ⟩ and H2 = γ H2 consist of all square elements and non-square elements of 2 ∗ (Z/l Z) , respectively. By [11], (0)

(0)

(1)

(1)

H2 = {a0 + a1 l|a0 ∈ H1 , a1 ∈ Z/lZ}, H2 = {a0 + a1 l|a0 ∈ H1 , a1 ∈ Z/lZ}. (0)

(1)

It is clear that |H2 | = |H2 | =

l(l−1) . 2 (0)

Hence (1)

(0)

(1)

S = S0 ∪ S1 ∪ S2 = S0 ∪ lH1 ∪ lH1 ∪ H2 ∪ H2 . Lemma 3.2. Suppose that l ≡ 3 (mod 4), l ̸ = 3, q = p cases: (1) If −l ̸ ≡ 1 (mod p), then for 0 ≤ i ≤ N − 1,

(3.1) l(l−1) 2

,β = α

q−1 N

∈ Fq , and ord(β ) = N = l2 . Then there are three

⎧ ⎪ if i = 0, ⎪ ⎪l(l − 1)/2, ⎪ ⎨ (0) lε ̸ = 0, if i ∈ lH1 , Trq/p (β i ) = ⎪ −l(1 + ε ) ̸= 0, if i ∈ lH1(1) , ⎪ ⎪ ⎪ ⎩ 0, otherewise, √ √ √ where ε = −1+2 −l and −l is an element in Fp such that ( −l)2 ≡ −l (mod p). Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.

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F. Li, Y. Wu and Q. Yue / Discrete Applied Mathematics xxx (xxxx) xxx

√ −l ≡ 1 (mod P1 ), then for 0 ≤ i ≤ N − 1,

(2) If −l ≡ 1 (mod p) and Trq/p (β i ) =

{

(1)

1, 0,

if i = 0 or i ∈ lH1 , otherwise.

√ −l ≡ −1 (mod P1 ), then for 0 ≤ i ≤ N − 1,

(3) If −l ≡ 1 (mod p) and Trq/p (β i ) =

{

√

where P1 = ⟨ a+b2

(0)

1, 0,

−l

if i = 0 or i ∈ lH1 , otherwise,

√ , p⟩ is a prime ideal of Q( −l) over p. 2

Proof. It is clear that xl − 1 = (x − 1)Φl (x)Φl2 (x), where Φl (x) and Φl2 (x) are cyclotomic polynomials. l(l−1) Since the order of p modulo l2 is 2 , there is an irreducible factorization over Fp : 2

(0)

(1)

(0)

(1)

xl − 1 = (x − 1)Φl (x)Φl Φl2 (x)Φl2 (x), (0)

where β l = ξl is a primitive lth root of unity in Fq , Φl (x) = (1)

(1)

and Φl2 (x) = Φl (xl ).

∏

(1)

(0)

u∈H1

(x − ξlu ), Φl (x) =

∏

(0)

(1)

u∈H1

(0)

(x − ξlu ), Φl2 (x) = Φl (xl ),

By Lemma 2.2,

∑

ζlu =

−1 + 2

(0)

√ √ −l ∑ u −1 − −l , ζl = , 2

(1)

u∈H1

u∈H1

where ζl is a primitive lth root of unity in the complex number field. √ Let O = Z[ζl ] be the algebraic integer ring of Q(ζl ) and O the algebraic integer of Q( −l). Then there are prime ideal factorizations of the prime p in the integer rings O and O, respectively: pO = P1 P2 , pO = P1 P2 , P1 O = P1 , P2 O = P2 , where P1 and P2 are distinct prime ideals in O, P1 and P2 are distinct prime ideals in O. Let

ζl ≡ ξl = β l

(mod P1 ) and O/P1 = F

l−1 p 2

= Fp (ξl ),

where l−21 is the order of p modulo l. Suppose that −l ̸ ≡ 1 (mod p). Then

∑

ζlu =

−1 + 2

(0) u∈H1

and

∑

ζ = u l

√ −l

−1 −

ξlu = Trp(l−1)/2 (β lu ) = ε ̸= 0

(mod P1 ),

(0) u∈H1

√ −l

2

(1)

∑

≡

∑

≡

ξlu = Trp(l−1)/2 (β lu ) = −1 − ε ̸= 0

(mod P1 ).

(1)

u∈H1

u∈H1

Hence Trp(l−1)/2 /p (β i ) =

{

ε, −1 − ε,

(0)

if i = lu ∈ S1 , u ∈ H1 (1) if i = lu ∈ S1 , u ∈ H1 . (0)

(0)

(1)

(1)

Moreover, Trq/p (1) = l(l − 1)/2 ̸ = 0; Trq/p (β i ) = 0 for i ∈ (Z/l2 Z)∗ by Φl2 (x) = Φl (xl ) and Φl2 (x) = Φl (xl ).

√

Suppose that −l ≡ 1 (mod p) and

∑ (0) u∈H1

ζlu ≡

∑

ξlu = 0

(0) u∈H1

−l ≡ 1 (mod P1 ). Then

(mod P1 ),

∑

∑

ζlu ≡

(1) u∈H1

ξlu = −1

(mod P1 ).

(1) u∈H1

Hence Trp(l−1)/2 /p (β i ) =

{

0, −1,

(0)

if i = lu ∈ S1 , u ∈ H1 (1) if i = lu ∈ S1 , u ∈ H1 .

Moreover, Trq/p (1) = l(l − 1)/2 ≡ −l ≡ 1 (mod p) and Trq/p (β i ) = 0 for i ∈ (Z/l2 Z)∗ . Note that Trq/p (β i ) = Trq/p(l−1)/2 (Trp(l−1)/2 /p (β i )). Then the results follow. Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.

F. Li, Y. Wu and Q. Yue / Discrete Applied Mathematics xxx (xxxx) xxx

5

√ −l ≡ −1 (mod P1 ). The desired result follows from the similar way of the case

Suppose that −l ≡ 1 (mod p) and above. □ Suppose that f (x) = Trq/p (x

ˆ f (b) =

f (x)+Trq/p (bx)

∑

ζp

q −1 N

) is a function from Fq to Fp . Then the Walsh transform of f (x) can be described by q −1 Trq/p (x N +bx)

∑

=

x∈Fq

ζp

.

x∈Fq

If b = 0, then

ˆ f (0) =

q −1 Trq/p (x N )

∑

ζp

=

x∈Fq

ψ (x

q −1 N

).

x∈Fq q −1 N

If b ∈ F∗q and b

ˆ f (b) =

∑

q−1 Trq/p (x N +bx)

∑

q −1 N

= β k , where β = α

ζp

=1+

∑

and 0 ≤ k ≤ N − 1. then

q−1 q−1 − Trq/p (b N x N +x)

ζp

=1+

x∈F∗ q

x∈Fq

∑

q−1 Trq/p (β −k x N +x)

ζp

.

x∈F∗ q

In the following, we compute the valuations of ˆ f (b), b ∈ Fq . Lemma 3.3.

√ √ l − 1 f −hl a + b −l l a − b −l l l(l − 1) f −h ˆ f (0) = p 2 (( ) +( ))+ p 2 a, 2

2

2

√

2

where h is the ideal class number of Q( −l), and a, b are integers given by (2.2). Trq/p (x)

Proof. Let ψ (x) = ζp

ˆ f (0) =

∑

ψ (x

q −1 N

be a canonical additive character of Fq . By Lemma 2.1,

)=

x∈Fq

N −1 ∑

G(χ j ),

j=1

where χ is a multiplicative character of order N. (1) (0) (1) (0) By j ∈ lH1 ∪ lH1 ∪ H2 ∪ H2 and Lemma 2.3,

ˆ f (0) =

∑ (0) j∈lH1

=

G(χ lv ) +

∑

G(χ lv ) +

(1)

v∈H1

∑

∑

G(χ l ) +

(0) v∈H1

∑

G(χ l ) +

√

a + b −l

G(χ j )

(1)

j∈H2

∑

G(χ ) +

(0) j∈H2

∑

G(χ j ) +

(0)

∑

G(χ j )

(1) j∈H2

j∈H2

(1) v∈H1 f −hl 2

∑

G(χ j ) +

(0) j∈H2

∑

v∈H1

l−1

∑

G(χ j ) +

(1) j∈lH1

(0)

=

∑

G(χ j ) +

G(χ )

(1) j∈H2

√

a − b −l

l(l − 1)

f −h 2

a. □ 2 2 2 2 Let F∗q = ⟨α⟩. By the divisor algorithm, for an integer s with 0 ≤ s ≤ q − 2,

=

p

((

s = jN + i, 0 ≤ j ≤ For b ̸ = 0, β = α

ˆ f (b) = 1 +

q−1 N

q−1

, and b

q−2 ∑

N q−1 N

ψ (β −k α

)l + (

= β k, q−1 s N

+ αs ) = 1 +

= 1+

N

ψ (β i−k )

∑

i=0

x∈F∗ q

N −1

N −1

1 ∑ N

N −1 ∑

q −1 −1 N

ψ (β i−k )

i=0

N −1 1 ∑

i=0

p

− 1, 0 ≤ i ≤ N − 1.

s=0

= 1+

)l ) +

ψ (β i−k )

∑ j=0

ψ (α i x N ) = 1 +

∑

ψ (α i α Nj )

j=0 N −1 1 ∑

N

ψ (β i−k )(

∑ x∈Fq

i=0 N −1

χ j (α i )G(χ j ) = 1 +

1 ∑ N

ψ (α i xN ) − 1)

i=0

ψ (β i−k )

N −1 ∑

ζN−ij G(χ j ),

j=0

Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.

6

F. Li, Y. Wu and Q. Yue / Discrete Applied Mathematics xxx (xxxx) xxx 2π

√ −1

is the Nth primitive root of unity in the complex field, χ is a primitive multiplicative character of where ζN = e N order N over F∗q , χ (α ) = ζN , and G(χ 0 ) = −1. By Lemma 2.3, N −1 ∑

√ ζN−ij G(χ j ) = −1 + p

f −hl 2

((

√

a + b −l 2

j=0

)l

f −h 2

(

ζN−ij + (

a + b −l ∑ 2

a − b −l 2

(0)

j∈lH1

√ +p

∑

ζN−ij +

)l

ζN−ij )

(1)

j∈lH1

√

a − b −l ∑ 2

(0)

∑

ζN−ij ).

(1)

j∈H2

j∈H2

Hence

√ √ f −hl p 2 a + b −l l (0) − I0 a − b −l l (1) ˆ + (( ) I1 + ( ) I1 ) f (b) = 1 + N N 2 2 √ √ f −h a − b −l (1) p 2 a + b −l (0) ( I2 + I2 ), + N

2

(3.2)

2

where I0 =

N −1 ∑

ψ (β i−k ) =

i=0

(0)

I1 =

N −1 ∑

ψ (β i ),

i=0

N −1 ∑

∑

ψ (β i−k )

ζN−ij , I1(1) =

(0)

i=0 (0)

∑

ψ (β i−k )

∑

ζN−ij ,

(1)

i=0

j∈lH1

j∈lH1

N −1

N −1

I2 =

N −1 ∑

∑

ψ (β i−k )

ζN−ij , I2(1) =

(0)

i=0

∑

ψ (β i−k )

ζN−ij .

(1)

i=0

j∈H2

∑ j∈H2

In the following, we shall compute the values of ˆ f (b), b ̸ = 0, in two cases: −l ̸ ≡ 1 (mod p) and −l ≡ 1 (mod p). Lemma 3.4. Suppose that −l ̸ ≡ 1 (mod p). Let b ̸ = 0 and b I0 = ζpl(l−1)/2 +

l−1 2

ζplε +

l−1 2

q −1 N

= β k , 0 ≤ k ≤ N − 1. Then

ζp−l(1+ε) + l(l − 1).

(1) If k = 0, then l−1

(1)

(0)

I1 = I1 = (0)

I2 = (1)

I2 =

2

l(l − 1) 2 l(l − 1) 2

ζpl(l−1)/2 + (

l−1 2

)2 ζplε + (

√

ζpl(l−1)/2 +

l(l − 1) −1 −

ζpl(l−1)/2 +

l(l − 1) −1 +

2

2

2

−l

l−1 2

)2 ζp−l(1+ε) −

2

2

ζplε +

l(l − 1) −1 +

ζplε +

l(l − 1) −1 −

√ −l

l(l − 1)

2

2

2

√ −l √ −l

2

. ζp−l(1+ε) . ζp−l(1+ε) .

(0)

(2) If k ∈ lH1 , then (0)

l−1

(1)

I1 = I1 =

2

ζpl(l−1)/2 + (

√ (0)

I2 = l (1)

I2 = l

−1 −

−l

2

ζpl(l−1)/2 +

√

−1 +

−l

2

ζpl(l−1)/2 +

l−1 2

l2 + l 4 l2 + l 4

)2 ζplε + (

l−1 2

)2 ζp−l(1+ε) −

√

ζp−l(1+ε) + lζplε (

1+

ζp−l(1+ε) + lζplε (

1−

−l

)2 .

−l

)2 .

2

l(l − 1) 2

.

√

2

(1)

(3) If k ∈ lH1 , then (0)

l−1

(1)

I1 = I1 =

2

ζpl(l−1)/2 + (

√ (0)

I2 = l

−1 + 2

−l

ζpl(l−1)/2 +

l−1 2

l2 + l 4

)2 ζplε + (

l−1 2

)2 ζp−l(1+ε) −

ζplε + lζp−l(1+ε) (

√ 1−

−l 2

l(l − 1) 2

.

)2 .

Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.

F. Li, Y. Wu and Q. Yue / Discrete Applied Mathematics xxx (xxxx) xxx

√ −l

−1 −

(1)

I2 = l

2

l2 + l

ζpl(l−1)/2 +

4

ζplε + lζp−l(1+ε) (

√ −l

1+

2

7

)2 .

(0)

(4) If k ∈ H2 , then

√

(0)

I1 = l

1+ 2

(l − 1)(−1 −

+

4

√

(1)

I1 = l (0)

1−

√

√ −l −l

2

−l)

−1 −

(ζplε + ζp−l(1+ε) ) +

−1 +

(ζplε + ζp−l(1+ε) ) +

−1 +

(ζplε + ζp−l(1+ε) ) +

−1 −

2

√

(l − 1)(−1 +

+

(ζplε + ζp−l(1+ε) ) +

−l)

4

−l

ζpl(l−1)/2 .

−l

ζpl(l−1)/2 .

−l

ζpl(l−1)/2 .

−l

ζpl(l−1)/2 .

√

2

(1)

I2 = I2 = 0. (1)

(5) If k ∈ H2 , then 1−

−l 2

(l − 1)(−1 +

+

4

√

(1)

I1 = l

1+

−l 2

√

√

√

(0)

I1 = l

−l) √

(l − 1)(−1 −

+

2

−l)

4

√

2

(1)

(0)

I2 = I2 = 0. Proof. Note that I0 =

N −1 ∑

ψ (β i−k ) =

N −1 ∑

i=0

ψ (β i ) .

i=0

By Lemma 3.2, I0 = ζpl(l−1)/2 +

l−1 2

ζplε +

l−1 2

ζp−l(1+ε) + l(l − 1).

(1) Suppose that k = 0. Then

∑

(0)

I1 =

ψ (β i )

l−1 2

i∈l(Z/l2 Z) (0)

If j ∈ lH1

∑

(0)

j∈lH1

∑

+

i∈(Z/l2 Z)∗

(0)

(1)

(0)

ζN−ij =

2

ζN−ij .

j∈lH1

and i ∈ H2 , then −ij ∈ lH1 √ −1+ −l

∑

ψ (β i )

by l ≡ 3 (mod 4) and

∑

(0)

j∈lH1

ζN−ij =

√ −1− −l 2

l−1

=

2 l−1

=

2

(1)

(1)

and i ∈ H2 , then

. Hence

√

(0) I1

(0)

. If j ∈ lH1

l(l−1)/2 p

ζ

+(

ζpl(l−1)/2 + (

l−1 2 l−1 2

)2 plε

ζ +(

l−1

)2 ζplε + (

l−1

2 2

ζ

)2 p−l(1+ε)

l(l − 1) −1 − + ( 2 2 l(l − 1)

)2 ζp−l(1+ε) −

2

√ −l

+

−1 + 2

−l

)

.

(0)

Similarly, I1 = I1 . ∑ −ij (0) (0) (1) (1) If i ∈ (Z/l2 Z)∗ , then by Φl2 (x) = Φl (xl ) and Φl2 (x) = Φl (xl ), = 0. Hence (0) ζN j∈H 2

(0) I2

∑

=

ψ (β )

i∈l(Z/l2 Z)∗ (0)

∑

i

−ij

ζN .

(0) j∈H2

(0)

−ij

(0)

−i j

(1)

If j ∈ H2 and i ∈ lH1 , and i = i0 l, j = j0 + j1 l, where i0 , j0 ∈ H1 , j1 ∈ Z/lZ, then ζN = ζl 0 0 and −i0 j0 ∈ H1 by l ≡ 3 ∑ ∑ −i0 j0 −ij (mod 4). Note that j1 ∈ {0, 1, . . . , l − 1}. Then the valuation of is exactly l times valuation of . (0) ζN (0) ζl j∈ H j ∈H √

Hence

∑ (0)

I2

ζN−ij = l −1−2

I2 =

−l

(0)

(1) H2

l(l − 1) 2

(1)

; if j ∈ H2 and i ∈ lH1 , then

2

Similarly, if j ∈ (1)

2

√

ζN−ij = l −1+2 −l ; Hence √ √ l(l − 1) l(l−1)/2 l(l − 1) −1 − −l lε l(l − 1) −1 + −l −l(1+ε) = ζp + ζp + ζp .

(0) j∈H2

2

and i ∈

(0) lH1 ,

ζpl(l−1)/2 +

then

2

(1)

j∈H2

l(l − 1) −1 + 2

2

−ij

∑

2

ζN = l

√ −l

√ −1+ −l 2

0

1

∑

(0) j∈H2

2

. Hence

√ ζplε +

l(l − 1) −1 − 2

2

−l

ζp−l(1+ε) .

Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.

8

F. Li, Y. Wu and Q. Yue / Discrete Applied Mathematics xxx (xxxx) xxx (0)

(2) Suppose that k ∈ lH1 . Then (0)

I1

∑

=

ψ (β i−k )

l−1 2

i∈l(Z/l2 Z)

∑

=

ψ (β i )

l−1

=

2

(1)

ζpl(l−1)/2 + (

∑

+

2

2

ζN−ij

(0)

j∈lH1

∑

ψ (β i )

i∈(Z/l2 Z)∗

l−1

∑

ψ (β i−k )

i∈(Z/l2 Z)∗

l−1

i∈l(Z/l2 Z)

∑

+

ζN−ij

(0)

j∈lH1

)2 ζplε + (

l−1 2

)2 ζp−l(1+ε) −

l(l − 1) 2

.

(0)

Similarly, I1 = I1 . (0) (0) By k ∈ lH1 , there is a unique i ∈ lH1 such that i = k; there are (1, 1)2 = there are (1, 0)2 = i∈

(1) lH1

l−3 4

elements i ∈

such that i − k ∈

(0) I2

∑

=

ψ (β

(0) lH1 ;

i−k

l−3 4

∑ (0)

(1) lH1

elements i ∈

such that i − k ∈

(0)

+l

2

−l

elements

Thus

2

√

4

ζplε l

j∈H2

−1 −

l+1

lε p l

−l

−1 +

√ −l

2

√

−l

−1 −

l−3

ζN−ij

(0)

i∈lH1

ζpl(l−1)/2 +

∑

ψ (β i−k )

(1)

j∈H2

√

−1 −

∑

ζN−ij +

(0)

i∈lH1

l(l − 1)

∑

ψ (β i−k )

−l(1+ε )

2

4

∑

=

(1) lH1 .

l+1 4

ζN

4

(1)

(0)

On the other hand, there are (0, 1)2 =

√

l−3

ζp l + ζ + ζ 2 4 2 4 p √ √ −1 − −l l(l−1)/2 l2 + l −l(1+ε) 1 + −l 2 =l ζp + ζp + lζplε ( ) ,

and I2

(0)

elements i ∈ lH1 such that i − k ∈ lH1 ;

−ij

∑

ζN0 +

j∈H2

+

l−3 4

(0) j∈H2

= ψ (β 0−k )

l−3

(1) lH1 .

such that i − k ∈

there are (0, 0)2 =

∑

)

i∈l(Z/l2 Z)

= ζp−l(1+ε)

(0) lH1

∑

ψ (β i−k )

i∈l(Z/l2 Z)

−1 +

l

−l

2

2

ζN−ij

(1) j∈H2

= ψ (β 0−k )

∑

∑

ζN0 +

(1)

∑

ψ (β i−k )

(0)

j∈H2

= ζp−l(1+ε)

−l(1+ε )

l(l − 1)

+l

2

−1 +

−l

2

ζpl(l−1)/2 +

l−3 4

ζN−ij

(1)

i∈lH1

j∈H2

√

∑

ψ (β i−k )

(1)

(1)

i∈lH1

∑

ζN−ij +

j∈H2

√

ζplε l

−1 +

−l

2

√ √ √ −1 + −l l + 1 lε −1 − −l l − 3 −l(1+ε) −l − −l ζp−l(1+ε) l + ζp l + ζp l 4 2 4 2 4 2 √ √ −1 + −l l(l−1)/2 l2 + l −l(1+ε) 1 − − l =l ζp + ζp + lζplε ( )2 .

+

l−3

2

(3) Suppose that k ∈ (0)

(1)

I1 = I1 =

l−1 2

4

(1) lH1 .

Then

ζpl(l−1)/2 + (

(1)

By k ∈ lH1 , there are (0, 0)2 = such that i − k ∈ (1)

(1) lH1 .

2

l−1 2 l−3 4

)2 ζplε + (

l−1 2

(0)

=

∑

(0)

i∈l(Z/l2 Z)

= ψ (β 0−k )

∑

(0)

l−3 4

(1) lH1

l+1 4

such that i = k; there are (1, 0)2 =

(1)

(1)

(0)

elements i ∈ lH1 l−3 4

elements

elements i ∈ lH1 such that i − k ∈ lHl . Thus

ζN−ij

(0)

ζN0 +

(0)

l(l − 1) 2

.

j∈H2

∑

+

∑

ψ (β i−k )

(0)

j∈H2

= ζplε

2

(0)

On the other hand, there is a unique i ∈

ψ (β i−k )

l(l − 1)

elements i ∈ lH1 such that i − k ∈ lH1 ; there are (0, 1)2 =

i ∈ lH1 such that i − k ∈ lHl ; there are (1, 1)2 = I2

)2 ζp−l(1+ε) −

4

ζplε l

ζN−ij +

(0)

i∈lH1

l−3

∑

√

−1 − 2

+

ψ (β i−k )

(1)

j∈H2

−l

∑

4

ζp−l(1+ε) l

ζN−ij

(0)

i∈lH1

l+1

∑ j∈H2

−1 −

√ −l

2

Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.

F. Li, Y. Wu and Q. Yue / Discrete Applied Mathematics xxx (xxxx) xxx

+l =l

√ −1 + −l 2

−1 +

√ −l

2

(1)

and I2

∑

=

l−3

ζpl(l−1)/2 +

l2 + l

ζpl(l−1)/2 +

4

∑

ψ (β i−k )

i∈l(Z/l2 Z)

l−3 4

l(l − 1) 2√

−1 −

(0)

−1 −

l−3 4

√ −l

1−

2

ζpl(l−1)/2

−l

ζpl(l−1)/2

2

(1)

(0)

∑

=

ψ (β i−k )

elements i ∈

l−1 2

i∈l(Z/l2 Z)

=

2 l−1

+

=l (1)

I1

ζ

+ζ

∑

+

l(l−1) 2

2

∑

+

∑

ψ (β i−k )

−1 −

−l +

l−1

+

2

i∈l(Z/l2 Z)

ζ

lε p

=

2 l−1

+

=l

+ζ

l(l−1)/2 p

−l(1+ε ) −1

ζ 2 √p 1 − −l 2

+

+

√ 2 −l

∑

ψ (β i−k )

∑

=

ψ (β i−k )

i∈l(Z/l2 Z)

=

l(l − 1) 2 l(l − 1) 2

+l

√ −l

−1 + +

√ l(l − 1) −1 −

(ζplε + ζp−l(1+ε) ) +

−l

2 √ 2 −1 + −l l(l−1)/2 2

ζp

.

ζN−ij = 0. If i ∈ l(Z/l2 Z), then i − k ∈ (Z/l2 Z)∗ . Hence

∑

∑ ∑

−ij

ζN +

(0)

(1)

i∈lH1

√

2

,

ζN−ij

(3.3)

i∈l(Z/l2 Z) j∈H (0)

j∈H2

−1 −

ζp

2

∑ ∑ (0)

ζ

lε p

2

∑

ζN−ij =

(0)

i∈lH1

=

∑

(0)

j∈H2

j∈H2

+

l−1

2 √ 2 l(l − 3) −1 + −l

∑

2

(1)

+

4

(0)

(0)

+

−l

2 √ 2 −1 − −l l(l−1)/2

ζN−ij

j∈lH1

2 √ 2 (l − 1)(−1 + −l)

By k ∈ H2 . If i ∈ (Z/l2 Z)∗ , then I2

−l

√ l(l − 1) −1 +

(ζplε + ζp−l(1+ε) ) +

i∈(Z/l2 Z)∗

−1 +

such that such that

−l +

2

√ l(l − 1)

− l elements i ∈ − ∈ /l2 Z)∗ . Then

√

−1 −

2 √ 2 l(l − 3) −1 − −l

∑

+

(0)

(0) H2

elements i ∈ H2

(0)

4 l−1

l−1 2

(1) l(l−1) lH1 ; there are 2 (1) H2 such that i k (Z

ζN−ij

j∈lH1

2 √ 2 (l − 1)(−1 − −l)

ψ (β i−k )

j∈H2

√

such that i = k; there are

elements i ∈

i∈(Z/l2 Z)∗

√ 2 − 1 − −l −l(1+ε )

2 √p 1 + −l

=

l(l−1)/2 p

ζN−ij

such that i − k ∈

√ l(l − 1)

∑ (1)

i∈lH1

(0)

(0) H2

i − k ∈ (Z/l Z) . On the other hand, there are I1

ψ (β i−k )

(1)

j∈H2

√ −1 + −l

(0)

l−1 2

are

−l

2

)2 .

∑

ζN−ij +

(4) Suppose that k ∈ H2 . Then there is a unique i ∈ H2 i−k ∈

−1 +

l + 1 −l(1+ε) −1 + −l l + ζp 2 2 √4 √ l − 3 lε −1 − −l l − 3 −l(1+ε) −1 − −l + l ζ l + ζ 4 p 2 4 p 2 √ 2 l + l lε 1 + − l + ζp + lζp−l(1+ε) ( )2 . 4 2

ζplε l

−l

∑

ψ (β i−k )

i∈lH1

+

2√

(0) lH1 ; there 2 ∗

ζp−l(1+ε) l

ζN−ij

∑

ζN0 +

(1)

=l

ζplε + lζp−l(1+ε) (

+

(1)

j∈H2

+l

2

9

√

j∈H2

∑

= ψ (β 0−k ) = ζplε

ζplε l

4

√ −1 + −l

−l l − 1 2

+l

ζN−ij

(0)

j∈H2

√

−1 + 2

−l l − 1 2

= 0.

(1)

Similarly, I2 = 0. (1) (1) (5) Suppose that k ∈ H2 . Then there is a unique i ∈ H2 such that i = k; there are i−k ∈

(0) lH1 ;

there are

l−1 2

elements i ∈

(1) H2

such that i − k ∈

(1) lH1 ;

there are

l(l−1) 2

l−1 2

(1)

such that

(1) H2

such that

elements i ∈ H2

− l elements i ∈

Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.

10

F. Li, Y. Wu and Q. Yue / Discrete Applied Mathematics xxx (xxxx) xxx

i − k ∈ (Z/l2 Z)∗ . On the other hand there are (0)

I1

∑

=

ψ (β i−k )

l−1 2

i∈l(Z/l2 Z)

= +

2 √ l(l − 3) −1 + −l 2√

=l (1)

I1

+ ζp

−l 2

∑

=

+

2

i∈l(Z/l2 Z)

= +

2 l−1

=l

+ζ

l(l−1)/2 p

ζ

−1 −

2 √ 1 + −l 2

+

+

∑

+

√ −l

(ζ

4

+

l−1 2

ζplε

−1 +

−l

2

√

∑

−1 +

l−1

ζ

ζpl(l−1)/2 ,

ζN−ij √

−l(1+ε ) −1

2

−l

2

(1) j∈lH1

2 2 √p l(l − 3) −1 − −l

2 2√ (l − 1)(−1 − −l)

√ −l

2

+ ζp−l(1+ε) ) +

ψ (β i−k )

+

−1 +

2 lε p

i∈(Z/l2 Z)∗

−1 −

−l

√

l−1

ζ −l(1+ε) 2 2√ p l(l − 1) −1 − −l

+

√ lε p

(0)

4 l−1

ζN−ij

j∈lH1

2 2√ (l − 1)(−1 + −l)

ψ (β i−k )

l(l − 1)

+

∑

ψ (β i−k )

i∈(Z/l2 Z)∗

√ −l l(l−1)/2 −1 +

l(l − 1)

1−

∑

+

(0)

elements i ∈ H2 such that i − k ∈ (Z/l2 Z)∗ . Hence

l(l−1) 2

−

−l

2 √ l(l − 1) −1 + −l

+

2

(ζplε + ζp−l(1+ε) ) +

2 √

−1 − 2

−l

ζpl(l−1)/2 .

(1)

(0)

Similarly, I2 = I2 = 0. This completes the proof. □ Suppose that −l ≡ 1 (mod p). Similar to the proof of Lemma 3.4, the following result follows and the proof is omitted here.

√ Lemma 3.5. Suppose that −l≡1 (mod p) and I0 =

l+1 2

−l ≡ 1 (mod P1 ). Let b ̸= 0 and b

q−1 N

= β k , 0 ≤ k ≤ N − 1. Then

(ζp − 1) + l2 .

(1) If k = 0, then l2 − 1

(1)

(0)

I1 = I1 =

4

√ (0)

(ζp − 1), I2 =

l(l − 1) 1 + 2

−l 2

(ζp − 1),

√ (1) I2

=

l(l − 1) 1 − 2

−l 2

(ζp − 1).

(0)

(2) If k ∈ lH1 , then (0)

l2 − 1

(1)

I1 = I1 =

4

√ (0)

(ζp − 1), I2 = l(

1+

−l 2

)2 (1 − ζp ),

√ (1) I2

= l(

1−

−l 2

)2 (1 − ζp ).

(1)

(3) If k ∈ lH1 , then (0)

(1)

I1 = I1 =

l2 − 1 4

(0)

(1)

(ζp − 1), I2 = I2 =

l2 + l 4

(1 − ζp ),

(0)

(4) If k ∈ H2 , then

√ (0)

I1 =

(l + 1)(1 +

−l)

4

(1)

(1 − ζp ), I1 =

(l + 1)(1 −

√ −l)

4

(0)

(1)

(0)

(1)

(1 − ζp ), I2 = I2 = 0.

(1)

(5) If k ∈ H2 , then

√ (0) I1

=

(l + 1)(1 − 4

−l)

(1 − ζp ),

(1) I1

=

(l + 1)(1 + 4

√ −l)

(1 − ζp ), I2 = I2 = 0.

Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.

F. Li, Y. Wu and Q. Yue / Discrete Applied Mathematics xxx (xxxx) xxx

√ q −1 −l ≡ −1 (mod P1 ). Let b ̸= 0 and b N = β k , 0 ≤ k ≤ N − 1. Then

Lemma 3.6. Suppose that −l≡1 (mod p) and I0 =

l+1 2

11

(ζp − 1) + l2 .

(1) If k = 0, then (0) I1

=

(1)

I2 =

(1) I1

l2 − 1

=

4

√ (ζp − 1),

(0) I2

l(l − 1) 1 −

=

2

√

l(l − 1) 1 + 2

−l 2

−l 2

(ζp − 1),

(ζp − 1).

(0)

(2) If k ∈ lH1 , then (0)

l2 − 1

(1)

I1 = I1 =

4

(0)

(1)

(ζp − 1), I2 = I2 =

l2 + l 4

(1 − ζp ).

(1)

(3) If k ∈ lH1 , then (0) I1

(1) I2

=

(1) I1

= l(

l2 − 1

=

1+

4

√ −l 2

(ζp − 1),

(0) I2

= l(

√ −l

1−

2

)2 (1 − ζp ),

)2 (1 − ζp ).

(0)

(4) If k ∈ H2 , then (0) I1

=

√

(l + 1)(1 +

−l)

4

(1 − ζp ),

(1) I1

=

(l + 1)(1 −

√ −l)

4

(0)

(1)

(0)

(1)

(1 − ζp ), I2 = I2 = 0.

(1)

(5) If k ∈ H2 , then (0)

I1 =

√

(l + 1)(1 −

−l)

4

(1)

(1 − ζp ), I1 =

(l + 1)(1 +

√ −l)

4

(1 − ζp ), I2 = I2 = 0.

By (3.2) and Lemma 3.4, we have the following theorem. l(l−1)

Theorem 3.7. Suppose that l ≡ 3 (mod 4), l ̸ = 3, be a prime. Let N = l2 be an integer and f = 2 be the smallest positive integer such that pf ≡ 1 (mod N). Let q = pf for gcd(p, f ) = 1. Suppose that −l̸ ≡1 (mod p). Then the Walsh spectrum of f (x) = Trq/p (x

q −1 N

) is given as follows: Value Frequency l−1

(A + B + lp

2 1

1+

N 1

1+ 1+

+

l−1 2l

p

1+ 1

+ p

1 N

N

1 N

l

1+ 1

+ p

1 N

(l−1)(q−1) 2l

times,

(−∆1 + A∆6 + B∆5 )

occurs

(l−1)(q−1) 2l

times,

occurs

q−1 l2

occurs

(l−1)(q−1) 2l2

times,

occurs

(l−1)(q−1) 2l2

times,

l−1

∆2 −

2 l−1 2l

∆3 −

2 l−1 2l

f −h 2

∆4 −

l−1 2l

times,

(A + B))∆1

(A + B + 2)

l−1 2

(A + B))∆1

(A + B + 2)

l−1

(− 1 +

l

occurs 1 time, occurs

(− 1 +

f −h 2

a)

(−∆1 + A∆5 + B∆6 )

(− 1 +

f −h 2

f −h 2

(A + B))∆1

(A + B + 2)

where l(l−1) 2

∆ 1 = ζp

+

l−1 2

ζplε +

l−1 2

ζp−l(1+ε) ,

Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.

12

F. Li, Y. Wu and Q. Yue / Discrete Applied Mathematics xxx (xxxx) xxx

l(l−1) 2

∆2 = aζp

−a + bl

+

∆3 =

−a + bl

∆4 =

−a − bl

2 2

2 l(l−1) 2

ζp

+

l(l−1) 2

ζp

+

∆5

2

f −hl 2

l(l−1) 2

ζp

√ −1 − −l

∆6 = A=p

−l

2

4 a(l + 1) 4

+

), B=p

f −hl 2

+

a(l + 1) 4

a − al + 2bl

ζplε +

4

√ −l)

4

l(l−1) 2

−l l

ζp−l(1+ε) ,

ζplε +

(l − 1)(−1 +

ζp

√

( a+b2

2

a − al − 2bl

√ −1 +

−a − bl

ζplε +

lε p

(ζ

−l)

4 √

ζp−l(1+ε) ,

+ ζp

−l(1+ε )

)+

l(1 −

√ −l)

2

√

(l − 1)(−1 −

( a−b2

ζp−l(1+ε) ,

(ζplε + ζp−l(1+ε) ) +

,

√

l(1 +

−l)

2

,

√

) , h is the ideal class number of Q( −l), and a, b are integers given by (2.2).

−l l

By (3.2) and Lemmas 3.5 and 3.6, we obtain Theorem 3.8. l(l−1)

Theorem 3.8. Suppose that l ≡ 3 (mod 4), l ̸ = 3, be a prime. Let N = l2 be an integer and f = 2 be the smallest positive integer such that pf ≡ 1 (mod N). Let q = pf for gcd(p, f ) = 1. Suppose that −l≡1 (mod p). Then the Walsh spectrum of f (x) = Trq/p (x

q −1 N

) is given as follows: Value Frequency l−1 l+1 2N l+1

(1 − ζp )(1 +

(A + B + lp

√2 1 + −l 2√

A+

f −h 2

√ 1 − −l 2√

a)

occurs 1 time,

B)

occurs

(l−1)(q−1) 2l

times,

occurs

(l−1)(q−1) 2l

times,

occurs

q−1 l2

occurs

(l−1)(q−1) 2l2

times,

occurs

(l−1)(q−1) 2l2

times,

−l 1 + −l A+ B) 2 2 1 l2 − 1 (l − 1)(a + δ bl) f −h (1 − ζp )(l + 1 − (A + B) − lp 2 ) 2N 2 2 l2 − 1 a − al + 2δ bl f −h 1 (1 − ζp )(l + 1 − (A + B) + lp 2 ) 2N 2 2 l2 − 1 a(l + 1) f −h 1 (1 − ζp )(l + 1 − (A + B) + lp 2 ) 2N 2 2 2N

where A = p

δ=

f −hl 2

(1 − ζp )(1 +

√

( a+b2

{ −1, 1,

−l l

), B=p

f −hl 2

√

1−

√

( a−b2

times,

√

) , h is the ideal class number of Q( −l), a, b are integers given by (2.2), and

−l l

if √−l ≡ 1 (mod P1 ), if −l ≡ −1 (mod P1 ).

√

Let l is a prime congruent to 3 modulo 4, l ̸ = 3. Suppose that 1 + l = 4ph , where h is the ideal class number of Q( −l). l−1+2h Then in (2.2), a, b take ±1. By a2 + lb2 = 4ph , [(Z/lZ)∗ : ⟨p⟩] = 2, and a ≡ −2p 4 (mod l),

{ a=

1, −1,

if l ≡ 3 if l ≡ 7

(mod 8), (mod 8).

While b can be determined up to√ sign. In Theorem 3.8, suppose that √ −l ≡ 1 (mod P1 ) for some p. If l ≡ 3 (mod 8), then take b = −1; if l ≡ 7 (mod 8), then take b = 1. Suppose that −l ≡ −1 (mod P1 ) for some p. If l ≡ 3 (mod 8), then take b = 1; if l ≡ 7 (mod 8), then take b = −1. Without loss of generality, we give Corollary 3.9 based on one of above conditions. Corollary 3.9. The notations are as Theorem 3.8. Suppose that 1 + l = 4ph and (mod 8), then the Walsh spectrum of function f (x) = Trq/p (x

q−1 N

√ −l ≡ 1 (mod P1 ) for some prime p. If l ≡ 3

) is given as follows:

Value Frequency l−1 l+1 2N

(1 − ζp )(1 +

√2 1 + −l 2

(A + B + lp

f −h 2

)

occurs 1 time,

√ A+

1−

−l 2

B)

occurs

(l−1)(q−1) 2l

times,

Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.

F. Li, Y. Wu and Q. Yue / Discrete Applied Mathematics xxx (xxxx) xxx

√

√ 1 + −l

1 − −l (1 − ζp )(1 + A+ B) 2N 2 2 f − h l+1 (1 − ζp )(2 − (l − 1)(A + B + lp 2 )) 4N f −h l+1 (1 − ζp )(2 − (l − 1)(A + B) + lp 2 ) 4N

l+1

where A = p

f −hl 2

(

√ 1− −l l 2

), B=p

f −hl 2

(

occurs

(l−1)(q−1) 2l

occurs

q−1 l2

occurs

(l−1)(q−1) l2

times,

times, times,

√

√ 1+ −l l 2

13

) , and h is the ideal class number of Q( −l). φ (72 )

Example 3.10. Let p = 2 and l = 7. It is straightforward to check that ord72 (2) = 21 = 2 = f . The class number h of √ Q( −7) is 1 (see [2, P.514]). Therefore we indeed have 1 + l = 4ph in this case. From Corollary 3.9, the Walsh spectrum of f (x) = Tr221 /2 (x

221 −1 7

) over F221 is five-valued. φ (1072 )

Example 3.11. Let p = 3 and l = 107. It is straightforward to check that ord1072 (3) = 5671 = = f . Let q = 35671 . 2 √ h The class number h of Q( −107) is 3 (see [2, P.514]). Therefore we also have 1 + l = 4p in this case. From Corollary 3.9, q−1

the Walsh spectrum of f (x) = Trq/3 (x 1072 ) over Fq is five-valued. 4. Concluding remarks q−1

In this paper, we investigated the Walsh spectrum of f (x) = Trq/p (x N ) over Fq in index two case, where l ≡ 3 (mod 4), l(l−1) l ̸ = 3, is a prime, N = l2 , the order of a prime p modulo N is f = 2 , and q = pf . In particular, a class of monomial h functions √ with five-valued Walsh spectrum are presented if l has the form l + 1 = 4p , where h is the ideal class number of Q( −l). q−1 In fact, the Walsh spectrum of f (x) = Trq/p (x N ) for the general case N = lm and m > 2 can also be settled by the same method presented in this paper. But the process is more complex and the Walsh spectrum may have many values. Acknowledgments The authors are very grateful to the reviewers and the Editor for their valuable suggestions that much improved the quality of this paper. References [1] A. Canteaut, P. Charpin, H. Dobbertin, Binary m-sequences with three-valued crosscorrelation: A proof of Welch’s conjecture, IEEE Trans. Inform. Theory 46 (1) (2000) 4–8. [2] H. Cohen, A Course in Computational Algebraic Number Theory, in: GTM, vol. 138, Springer, 1996. [3] T.W. Cusick, H. Dobbertin, Some new three-valued crosscorrelation functions for binary m-sequences, IEEE Trans. Inform. Theory 42 (4) (1996) 1238–1240. [4] C. Ding, J. Yang, Hamming weights in irreducible cyclic codes, Discrete Math. 313 (2013) 434–446. [5] K. Feng, J. Luo, Values distributions of exponential sums from perfect nonlinear functions and their applications, IEEE Trans. Inform. Theory 53 (9) (2007) 3035–3041. [6] T. Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discrete Math. 16 (3) (1976) 209–232. [7] T. Helleseth, L. Hu, A. Kholosha, X. Zeng, N. Li, W. Jiang, Period-different m-sequences with at most four-valued cross correlation, IEEE Trans. Inform. Theory 55 (7) (2009) 3305–3311. [8] T. Helleseth, A. Kholosha, Crosscorrelation of m-sequences, exponential sums, bent functions and Jacobsthal sums, Cryptogr. Commun. 3 (4) (2011) 281–291. [9] T. Helleseth, P. Rosendahl, New pairs of m-sequences with 4-level cross-correlation, Finite Fields Appl. 11 (4) (2005) 674–683. [10] Z. Heng, Q. Yue, Several classes of cyclic codes with either optimal three weights or a few weights, IEEE Trans. Inform. Theory 62 (2016) 4501–4513. [11] L. Hu, Q. Yue, M. Wang, The linear complexity of whiteman’s generalize cyclotomic sequences of period pm+1 qn+1 , IEEE Trans. Inform. Theory 58 (2012) 5534–5543. [12] P. Langevin, Caluls de certaines sommes de Gauss, J. Number theory 32 (1997) 59–64. [13] N. Li, T. Helleseth, A. Kholosha, X. Tang, On the Walsh transform of a class of functions from Niho exponents, IEEE Trans. Inform. Theory 59 (7) (2013) 4662–4667. [14] C. Li, Q. Yue, The Walsh transform of a class of monomial functions and cyclic codes, Cryptogr. Commun. 7 (2) (2015) 217–228. [15] C. Li, Q. Yue, F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl. 28 (2014) 94–114. [16] R. Lidl, H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 2008. [17] J. Luo, Binary sequences with three-valued cross correlations of different lengths, IEEE Trans. Inform. Theory 62 (12) (2016) 7532–7537. [18] J. Luo, K. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inform. Theory 54 (12) (2008) 5332–5344. [19] J. Luo, Y. Tang, H. Wang, Cyclic codes and sequences: The generalized Kasami case, IEEE Trans. Inform. Theory 56 (5) (2010) 2130–2142. [20] G.J. Ness, T. Helleseth, Cross correlation of m-sequences of different lengths, IEEE Trans. Inform. Theory 52 (4) (2006a) 1637–1648. [21] G.J. Ness, T. Helleseth, A new three-valued cross correlation between m-sequences of different lengths, IEEE Trans. Inform. Theory 52 (10) (2006b) 4695–4701. [22] G.J. Ness, T. Helleseth, A new family of four-valued cross correlation between m-sequences of different lengths, IEEE Trans. Inform. Theory 53 (11) (2007) 4308–4313.

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Please cite this article as: F. Li, Y. Wu and Q. Yue, A class of functions with low-valued Walsh spectrum, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.10.029.