Linear complexity problems of level sequences of Euler quotients and their related binary sequences

Abstract

The Euler quotient modulo an odd-prime power p r (r > 1) can be uniquely decomposed as a p-adic number of the formwhere 0 ≤ a j (u) < p for 0 ≤ j ≤ r − 1 and we set all a j (u) = 0 if gcd(u, p) > 1. We firstly study certain arithmetic properties of the level sequences (a j (u)) u≥0 over F p via introducing a new quotient. Then we determine the exact values of linear complexity of (a j (u)) u≥0 and values of k-error linear complexity for binary sequences defined by (a j (u)) u≥0 .

On the Linear Complexity of a Class of Periodic Sequences Derived from Euler Quotients

“…Firstly, the distribution of vectors of consecutive Euler-Fermat quotients modulo a composite was described in [9]. Trace representations of sequences deduced from Euler quotients modulo a prime power were studied in [10]. Furthermore, the -error linear complexity of binary sequences deduced from Euler quotients modulo a prime power was determined [11].…”

Linear Complexity of a Family of Binary pq 2-Periodic Sequences From Euler Quotients

“…Chen and Winterhof generalized the Fermat quotient to the polynomial quotient in [7]. Then the k-error linear complexity was determined for binary sequences derived from the polynomial quotient modulo a prime [5] or its power [22], respectively. In [23], a series of optimal families of perfect polyphase sequences were derived from the array structure of Fermat-quotient sequences.…”

On the k-error linear complexity of 2p2-periodic binary sequences

“…We give an example of generalized cyclotomic binary sequences (applying Theorem 3) in Appendix C.4.1 and found that such sequences are not "good" pseudorandom sequences. In recent years, the pseudorandom sequences derived from Fermat quotients have attracted extensive attention [7][8][9], these sequences are of high linear complexity, and the problem of their k-error linear complexity deserves further study. We also give another numerical example (applying Theorem 4) in Appendix C.4.2.…”