Feedback with Carry Shift Registers

FCSRs have been used in the design of stream ciphers (such as the F-FCSR generator), in the cryptanalysis 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 ${\displaystyle q=q_r{N}^r+\cdots +q_1{N}^1-1}$. Associated with the output sequence is the N-adic number ${\displaystyle a=a_0+a_{1}N+a_2{N}^2+\dots }$ The fundamental theorem of FCSRs says that there is an integer ${\displaystyle u}$ so that ${\displaystyle a=u/q}$, a rational number. The output sequence is strictly periodic if and only if ${\displaystyle u}$ is between ${\displaystyle -q}$ and ${\displaystyle 0}$. It is possible to express u as a simple quadratic polynomial involving the initial state and the q_i.

There is also an exponential representation of FCSRs: if ${\displaystyle g}$ is the inverse of ${\displaystyle N(\mathrm{mod}q)}$, and the output sequence is strictly periodic, then ${\displaystyle a_i=(A{g}_{i}modq)modN}$, where ${\displaystyle A}$ 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. In this case, the period is ${\displaystyle q-1}$. 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. They 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 ${\displaystyle N=2}$; 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 ${\displaystyle 2L}$ 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. A general reference on the subject of LFSRs, FCSRs, and AFSRs is the book.