Linear Complexity of Pseudorandom Sequences Derived from Polynomial Quotients: General Cases

Linear Complexity of New Binary Sequence Derived From Polynomial Quotients Modulo p in General Case and Their Generalizations

“…where f (u) is defined to be the derivative of f (u). Setting k = p in (2), it is clear to see that Q(u) is periodic with least period p 2 [2].…”

“…After that, cryptographic properties of pseudorandom sequences derived from polynomial quotient began to attract academic attention [3][4][5][6]. Many number theoretic and cryptographic questions have been studied for polynomial quotients sequences [2,[7][8][9][10]. It needs to be pointed out that most studies have focused on special cases of f (x) but rarely considered the general cases [7][8][9][10][11][12][13].…”

“…It needs to be pointed out that most studies have focused on special cases of f (x) but rarely considered the general cases [7][8][9][10][11][12][13]. There are two families of binary sequences derived from polynomial quotients in general case to be discussed from the view point of cryptography [2]. One is the binary threshold sequences (e u ) defined as…”

“…where ( • p ) is the Legendre symbol [2]. The Legendre symbol is defined to be equal to ±1 depending on whether a number is a quadratic residue modulo an odd prime.…”

“…Du et al proved that both sequences above have high linear complexity and extended the constructions to arbitrary dary sequences for prime d | (p − 1) with d a primitive root modulo p 2 [2]. Mathematically, a number g is called a primitive root modulo n if and only if every number coprime to n is congruent to a power of g modulo n.…”