2-Adic shift registers

Periodicity and Distribution Properties of Combined FCSR Sequences

“…These sequences have been studied since the time of Gauss [3], [2] (p. 163). The reverse of this sequence is known as an FCSR sequence [4], [5] since it is the output sequence of a feedback with carry shift register with connection integer q, with cell contents taken from Z/(2), and with initial loading that depends on h, cf. [5].…”

“…It follows that the possible variation in N e is one half the possible variation in Y 4 . By the definition of Y 4 we have Y 4 ≤ min(Q, R) and…”

“…The security of the summation combiner was called into question following the introduction of feedback-with-carry shift registers, or FCSRs [4], [5], and the associated rational approximation algorithm [5]. This is because the 2-adic complexity of the output of the summation combiner is no more than the sum of the 2-adic complexities of the constituent sequences.…”

“…Recall that a binary -sequence is a maximal length FCSR sequence [4] of 0's and 1's. Such a sequence is obtained whenever the connection integer q ≥ 3 is a prime number such that 2 is a primitive root modulo q.…”

“…Such a sequence is obtained whenever the connection integer q ≥ 3 is a prime number such that 2 is a primitive root modulo q. The period of such an -sequence is q − 1 and it is known to have a number of desirable statistical properties, one of which is that the number of occurrences of any given block f = (f 0 , f 1 , · · · , f k−1 ) of size k differs at most by one, as f ranges over all 2 k possibilities [4].…”