$p$-ary sequences with six-valued cross-correlation function: a new decimation of Niho type

p -ary sequences with six -v alued cross-correlation function: a new decimation of Niho typ e Y uh ua Sun 1 , Hui Li 1 , Zilong W ang 2 , and T ong jiang Y an 3 , 4 1 Key Lab ora tory of Computer Netw or ks and Info rmation Securit y , Xidian Univ ersit y , Xi’an 710071, Shaanxi, China 2 State Key Lab oratory of Inte grated Service Net w orks, Xidian Univ ersit y , Xi’an, 710071, Shaanxi, China 3 College of Mathematics and Computational Science , China Univ ersit y of P etroleum D ongying 257061 , Shandong, China 4 State Key Lab oratory of Informatio n Securit y (Institute of Soft w are, Chinese Academ y of Sciences ), Beijing 10 0049,China No v em b er 1 8, 2021 Abstract F or an odd prime p and n = 2 m , a n ew decimation d = ( p m − 1) 2 2 + 1 of Niho type of m -sequences is presented. Using generalized Niho’s Theorem, we sho w t hat the cross-correl ation function betw een a p -ary m -sequence of p erio d p n − 1 and its decimated sequ ence by the ab ov e d is at most six-val ued and we can easil y know th at the magnitude of the cross correlation is upp er b ound ed by 4 √ p n − 1. Index T erms. p - ary m -sequence, Niho t yp e, cross-correla tion. 1 In tro duction It is of great interest to find a decimatio n v alue d such that the cro ss-corr elation b etw een a p -ar y m - sequence { s t } of p erio d p n − 1 a nd its decimation { s dt } is low. F or the cas e gcd( d, p n − 1) = 1, the decimated sequence { s dt } is also an m -se q uence of p er io d p n − 1. Ba s ic r esults on the cro ss-corr elation betw een t wo m -sequences can b e found in [3], [5], [8] and [10]. F or the case gcd( d, p n − 1) 6 = 1, the reader is referre d to [7], [6], and [1]. Let p be a prime and n = 2 m . The cross cor relation functions for the type o f decimations d ≡ 1 (mo d p m − 1) w ere first studied by Niho [8] named as Niho t yp e of decimations. In [8], for p = 2 0 The w ork is supported b y the National N atural Science F ound ation of China (No.60772136 ), Shandong Pr o vincial Natural Science F oundation of China (No. ZR20 10FM017), the F unda menta l Researc h F unds for the Central Univ er- sities(No.K5051001001 2) and the open fund of State Key Lab oratory of In formation Security( Graduate Unive rsity o f Chinese Academ y of Sciences) (No.F1008001). 1 and d = s ( p m − 1) + 1 , Niho conv erted the problem of finding the v alues of cross-cor relation functions int o the problem of determining the n um b er of so lutions of a sy s tem o f equations . This result is ca lled Niho’s Theorem. In 2 006, Ro sendahl [9] genera lized Niho’s Theorem to nonbinary sequences . In 2007 , Helleseth et al. [4 ] proved that the cross correla tion function b etw een tw o m -sequences that differ by a decimation d of Niho t yp e is at le a st four-v alued. When d = 2 p m − 1 ≡ 1 (mo d p m − 1), the cr oss corr elation function b etw een a p -ary m -sequence { s t } of pe r io d p 2 m − 1 and its decima ted se quence { s dt } is four- v a lued, which was originally given by Niho [8] for the case p = 2 and by Helleseth [3] for the case p > 2. And when d = 3 p m − 2, the cros s c orrelatio n function b etw een t wo m -sequences that differ by d is at most six-v alued, especia lly , for p = 3, the cross correla tion function is at most fiv e-v alued [9]. In this note, w e study a new dec ima tion d = ( p m − 1) 2 2 + 1 of Niho t yp e. Employing ge neralized Niho’s Theorem, we show that the cro ss-cor r elation function b etw een a p - a ry m - s equence and its decima ted sequence by d is at mo st six -v alued. The r est o f this note is organize d as follows. Sectio n 2 pr esents some pre liminaries and definitions . Using generalized Niho’s Theorem, w e giv e an alternative pro of of a res ult b y Helleseth [3] where d = ( p m − 1)( p m +1) 2 + 1 in section 3. A new decimatio n d = ( p m − 1) 2 2 + 1 o f Niho t yp e is given in section 4. W e prove that the cross cor relation function be tw een a p -ary m -seq uence and its dec ima ted seq ue nc e by d takes at most six v alues. 2 Preliminaries W e will use the following notation in the re s t of this note. Let p b e an o dd prime, GF( p n ) the finite field with p n elements and GF ( p n ) ∗ = GF( p n ) \{ 0 } . The trace function T r n m from the field GF ( p n ) o n to the subfield GF( p m ) is defined a s T r n m ( x ) = x + x p m + x p 2 m + · · · + x p ( l − 1) m , where l = n m is an in teger. W e may assume that a p -ar y m -sequence { s t } of p er io d p n − 1 is given b y s t = T r n 1 ( α t ) , where α is a primitive element of the finite field GF( p n ) and T r n 1 is the trace function from GF( p n ) onto GF( p ). The perio dic cros s c o rrelatio n function C d ( τ ) betw een { s t } and its decimated s equence { s dt } is defined a s C d ( τ ) = Σ p n − 2 t =0 ω s t + τ − s dt , 2 where ω = e 2 π √ − 1 p and 0 ≤ τ ≤ p n − 2. W e will a lwa ys assume that n = 2 m is even in this note unless otherwise sp ecified. 3 An alternativ e pro of of a kno wn result F or p = 2, Niho [8] pr esented Niho’s Theorem ab out decimations of Niho t yp e of m - sequences. Rosendahl [9] generalized this result as follows. Lemma 1 (gener alize d Niho’s The or em) [9] L et p , n , and m b e define d as in se ction 2. Assum e that d ≡ 1 (mo d p m − 1) , and denote s = d − 1 p m − 1 . Then when y = α τ runs thr ough the nonzer o elements of the field GF( p n ) , C d ( τ ) assum es exactly the values − 1 + ( N ( y ) − 1) · p m , wher e N ( y ) is the numb er of c ommon solutions of x 2 s − 1 + y p m x s + y x s − 1 + 1 = 0 , x p m +1 = 1 . In 1976 , Helleseth [3] pr ov ed the following result. Here, using the ge neralized Niho’s Theo rem, we give a simple pro of. Theorem 1 [3] L et the symb ols b e define d as in se ction 2, p b e an o dd prime and d = p n − 1 2 + 1 . Then C d ( τ ) ∈ {− 1 − p m , − 1 , − 1 + p m , − 1 + p m − 1 2 p m , − 1 + p m +1 2 p m } . Pro of of Theorem 1. Since d = p n − 1 2 + 1 = p m +1 2 ( p m − 1) + 1, we get s = d − 1 p m − 1 = p m +1 2 . By Lemma 1, we hav e C d ( τ ) = − 1 + ( N ( y ) − 1 ) · p m , where y = α τ , 0 ≤ τ ≤ p n − 2, and N ( y ) is the num ber of co mmo n solutions of x ( p m +1) − 1 + y p m x p m +1 2 + y x p m +1 2 − 1 + 1 = 0 , (1) x p m +1 = 1 . (2) Note that Eq. (2) implies x p m +1 2 = 1 (3) 3 or x p m +1 2 = − 1 . (4) Substituting (3) and (4) in to (1) re sp e c tiv ely , we get C d ( τ ) = − 1 + ( N 1 ( y ) + N − 1 ( y ) − 1 ) · p m , where N 1 ( y ) is the num b er of the co mmon solutions of (3 . 1 . 1) ( ( y p m + 1) x + ( y + 1) = 0 , x p m +1 2 = 1 , and N − 1 ( y ) is the n umber o f solutio ns of (3 . 1 . 2) ( ( y p m − 1) x + ( y − 1) = 0 , x p m +1 2 = − 1 . Obviously , for y 6 = ± 1, 0 ≤ N 1 ( y ) + N − 1 ( y ) ≤ 2 . Let y = 1. First, it is straig h tforward to g et N − 1 (1) = p m +1 2 . Second, we see that x = − 1 is the only so lutio n of (3.1.1) for p m + 1 ≡ 0 mod 4 and N 1 (1) = 0 for p m + 1 ≡ 2 mod 4. Hence, we ha ve N 1 (1) + N − 1 (1) = ( 1 + p m +1 2 , if p m + 1 ≡ 0 mo d 4 , p m +1 2 , if p m + 1 ≡ 2 mo d 4 . Similarly , for y = − 1, we hav e N 1 ( − 1) + N − 1 ( − 1) = ( p m +1 2 , if p m + 1 ≡ 0 mo d 4 , 1 + p m +1 2 , if p m + 1 ≡ 2 mod 4 . The r esult follows.  In Theorem 1, the v alue s of Niho t yp e decimatio n d is equal to p m +1 2 corres p o nding to Lemma 1 . Motiv ated by the ab ov e pr o of, we take s as the v alue p m − 1 2 , a new decimation o f Niho type will be presented, and cross correla tion v alues will b e deter mined in the following section. 4 A n ew decimation of Niho t yp e In this section, w e give a new decimation d of Niho’s t yp e and we show that the cro ss cor relation function b etw een a p -ary m - sequence and its decimated sequence by d is at most six-v alued. 4 Theorem 2 L et the symb ols b e define d as in s e ction 2. L et d = ( p m − 1) 2 2 + 1 . Then C d ( τ ) ∈ {− 1 + ( j − 1) · p m | 0 ≤ j ≤ 5 } is at most six-value d. Pro of of Theorem 2. since d = ( p m − 1) 2 2 + 1 = p m − 1 2 · ( p m − 1) + 1 ≡ 1 mo d ( p m − 1) , w e kno w that the v alue s corresp onding to that in Lemma 1 is p m − 1 2 . By the s a me argument a s in Theorem 1 , w e g et C d ( τ ) = − 1 + ( N 1 ( y ) + N − 1 ( y ) − 1 ) · p m , where N 1 ( y ) is the num b er of solutions of (4 . 1 . 1) ( x 3 + y p m x 2 + y x + 1 = 0 , x p m +1 2 = 1 , and N − 1 ( y ) is the n umber o f solutio ns of (4 . 1 . 2) ( x 3 − y p m x 2 − y x + 1 = 0 , x p m +1 2 = − 1 . By the basic algebraic theor em, we know that 0 ≤ N 1 ( y ) ≤ 3 and 0 ≤ N − 1 ( y ) ≤ 3, i.e., 0 ≤ N 1 ( y ) + N − 1 ( y ) ≤ 6. F urther, we will pr ov e 0 ≤ N 1 ( y ) + N − 1 ( y ) ≤ 5, i.e., we will prov e N 1 ( y ) + N − 1 ( y ) 6 = 6. Suppo se that N 1 ( y ) + N − 1 ( y ) = 6. Then N 1 ( y ) = 3 and N − 1 ( y ) = 3, i.e., bo th (4.1.1) and (4.1 .2 ) hav e three solutions. Now, for i = 1 , 2 , 3 , let x i and x ∗ i be the solutions of (4.1.1) and (4 .1.2), resp ectively . Since x i satisfies x p m +1 2 = 1 a nd x ∗ i satisfies x p m +1 2 = − 1, we kno w that there exists some even in teger j i satisfying x i = α j i ( p m − 1) and that ther e exists some o dd integer j ∗ i satisfying x ∗ i = α j ∗ i ( p m − 1) , wher e i = 1 , 2 , 3 . Simultaneously , since x i and x ∗ i satisfy the fir st equations of (4.1 .1)and (4.1.2) resp ectively , we ha ve 3 Y i =1 x i = α ( p m − 1) 3 P i =1 j i = − 1 , 3 Y i =1 x ∗ i = α ( p m − 1) 3 P i =1 j ∗ i = − 1 . By multiplying the above tw o eq uations, we get 3 Y i =1 x i 3 Y i =1 x ∗ i = α ( p m − 1)( 3 P i =1 j i + 3 P i =1 j ∗ i ) = 1 , and induce p m + 1 | 3 P i =1 j i + 3 P i =1 j ∗ i . This contradicts to the fact that p m + 1 is ev en but 3 P i =1 j i + 3 P i =1 j ∗ i is o dd. Ther efore, we get N 1 ( y ) + N − 1 ( y ) 6 = 6, i.e., 0 ≤ N 1 ( y ) + N − 1 ( y ) ≤ 5. The result follows.  5 Remark 1 The de cimate d se quenc e { s dt } in The or em 2 is n ot ne c essarily a n m -se quenc e. In fact, d = p m − 1 2 ( p m − 1) + 1 ≡ p m − 1 2 ( − 2) + 1 ≡ 3 mo d ( p m + 1) . F or p ≡ − 1 mo d 3 , m o dd, we know t hat gcd( d, p n − 1) = 3 , { s dt } is not an m -se quenc e. F or t he other c ase, gcd( d, p n − 1) = 1 , and { s dt } is an m -se quenc e. Remark 2 The or etic al ly, the nu mb er of t he values of C d ( τ ) c an not b e r e duc e d to less than 6. F ol lowing is an ex ample whose cr oss c orr elatio n funct ion b etwe en an m - se quenc e and its de cimate d se quenc e by d has exactly six values. Example 1 L et p = 3 , n = 6 , m = 3 and d = ( p m − 1) 2 2 + 1 = 339 . The p olynomial f ( x ) = x 6 + x 5 + 2 is primitive over GF(3 ) . L et α b e a r o ot of f ( x ) , then s t = T r 6 1 ( α t ) , s dt = T r 6 1 ( α 339 t ) . Computer exp eriment gives the fol lowing cr oss c orr elation values: − 1 − 3 3 o ccurs 246 times , − 1 o ccurs 284 times , − 1 + 3 3 o ccurs 144 times , − 1 + 2 · 3 3 o ccurs 42 times , − 1 + 3 · 3 3 o ccurs 3 times , − 1 + 4 · 3 3 o ccurs 9 times . Conclusion In this note, us ing generalize d Niho’s Theor em, we g ive an alterna tive pr o of of a known res ult. By changing the form o f the known decimation facto r , we g ive a new dec imation d = ( p m − 1) 2 2 + 1 of Niho t yp e. W e prove tha t the cross correlatio n function b etw een a p -ary m -seq uenc e of per io d p n − 1 and its decimated sequence by the v a lue d is at mo st six-v alued, and w e can ea sily see that the mag nitude of the cr oss corr elation v alues is upp er b o unded by 4 √ p n − 1. References [1] S.-T. Cho i, J.-S. No, H. 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