On the distinctness of modular reductions of maximal length sequences modulo odd prime powers

Abstract

Abstract. We discuss the distinctness problem of the reductions modulo M of maximal length sequences modulo powers of an odd prime p, where the integer M has a prime factor different from p. For any two different maximal length sequences generated by the same polynomial, we prove that their reductions modulo M are distinct. In other words, the reduction modulo M of a maximal length sequence is proved to contain all the information of the original sequence.

Periods of termwise exclusive ors of maximal length FCSR sequences

“…Primitive sequences over integer residue rings have been extensively studied, see [10][11][12] for the case of Z/(2 e ) and see [13][14][15] for the case of Z/(p e ) with p an odd prime number. In this paper, however, we only concern with the very simple case-primitive sequences of order 1 over Z/(p e ) with p an odd prime number, i.e., s = {A · ξ t mod p e } ∞ t=0 , where ξ is a primitive root modulo p e and gcd( A, p) = 1.…”

“…It has already been shown in [15] that the "mod 2" operation does not change the period of the primitive sequences over Z/(p e ) with odd prime number power p e . Here we give the conclusion on terms of 1st-order sequences.…”

“…(See [15].) Let p e be an odd prime number power and a be a primitive sequence of order 1 over Z/(p e ).…”

On the distinctness of primitive sequences over Z/(p e q) modulo 2

“…Note that we can derive 31 sequences totally from the 2-adic expansion of a = a 0 +a 1 ·2+· · ·+a 30 ·2 30 , called the 2-adic coordinate sequences of a. The essential rationality for the application of primitive sequences over Z/(2 31 − 1) is that they are pairwise distinct modulo 2 [3,Theorem 4.2] i.e. a = b iff [a] mod 2 = [b] mod 2 , where a and b are two primitive sequences over Z/(2 31 − 1) generated by a primitive polynomial.…”

“…The case that N is an odd prime power integer has been completely solved in [3]. Besides, several results for square-free N can be found in [4][5][6][7][8].…”