## FANDOM

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In sequence design, a Feedback with Carry Shift Register (or FCSR) is the arithmetic or with carry analog of a Linear feedback shift register (LFSR). If N >1 is an integer, then an N-ary FCSR of length r is a finite state device with a state (a;z) = (a0,a1,...,ar-1;z) consisting of a vector of elements ai in {0,1,...,N-1}=S and an integer z. The state change operation is determined by a set of coefficients q1,...,qn and is defined as follows: compute s = qra0+qr-1a1+..+q1ar-1 + z. Express s as s = ar+Nz' with ar in S. Then the new state is (a1,a2,...,ar;z'). By iterating the state change an FCSR generates an infinite, eventually period sequence of numbers in S.

FCSRs have been used in the design of stream ciphers (such as the F-FCSR generator), in the cryptanalyis of the summation combiner stream cipher (the reason Goresky and Klapper invented them), and in generating pseudorandom numbers for quasi-Monte Carlo (under the name Multiply With Carry (MWC) generator - invented by Couture and L'Ecuyer, generalizing work of Marsaglia and Zaman.

FCSRs are analyzed using number theory. Associated with the FCSR is a connection integer q = qrNr + ... + q1N - 1. Associated with the output sequence is the N-adic number a = a0 + a1N + a2N2+.... The fundamental theorem of FCSRs says that there is an integer u so that a = u/q, a rational number. The output sequence is strictly periodic if and only if u is between -q and 0. It is possible to express u as a simple quadratic polynomial involving the initial state and the qi.

There is also an exponential representation of FCSRs: if g is the inverse of N modulo q, and the output sequence is strictly periodic, then ai = (Agi mod q) mod N, where A is an integer. It follows that the period is at most the order of N in the multiplicative group of units modulo q. This is maximized when q is prime and N is a primitive element modulo q. Then the period is q-1. In this case the output sequence is called an l-sequence (for "long sequence").

l-sequences have many excellent statistical properties that make them candidates for use in applications, including near uniform distribution of sub-blocks, ideal arithmetic autocorrelations, and the arithmetic shift and add property. The are the with-carry analog of m-sequences or maximum length sequences.

There are efficient algorithms for FCSR synthesis. This is the problem: given a prefix of a sequence, construct a minimal length FCSR that outputs the sequence. This can be solved with a variant of Mahler and De Weger's lattice based analysis of N-adic numbers when N=2; by a variant of the Euclidean algorithm when N is prime; and in general by Xu's adaptation of the Berlekamp-Massey algorithm. If L is the size of the smallest FCSR that outputs the sequence (called the N-adic complexity of the sequence), then all these algorithms require a prefix of length about 2L to be successful and have quadratic time complexity. It follows that, as with LFSRs and linear complexity, any stream cipher whose N-adic complexity is low should not be used for cryptography.

FCSRs and LFSRs are special cases of a very general algebraic construction of sequence generators called Algebraic Feedback Shift Registers (AFSRs) in which the integers are replaced by an arbitrary ring R and N is replaced by an arbitrary non-unit in R.

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