Feedback with Carry Shift Registers

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 formula_1 is an integer, then an "N"-ary FCSR of length formula_2 is a finite state device with a state formula_3 consisting of a vector of elements formula_4 in formula_5 and an integer formula_6. The state change operation is determined by a set of coefficients formula_7 and is defined as follows: compute formula_8. Express s as formula_9 with formula_10 in formula_11. Then the new state is formula_12. By iterating the state change an FCSR generates an infinite, eventually periodic sequence of numbers in formula_11.
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 formula_14. Associated with the output sequence is the N-adic number formula_15 The fundamental theorem of FCSRs says that there is an integer formula_16 so that formula_17, a rational number. The output sequence is strictly periodic if and only if formula_16 is between formula_19 and formula_20. 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 formula_21 is the inverse of formula_22, and the output sequence is strictly periodic, then formula_23, where formula_24 is an integer. It follows that the period is at most the order of in the multiplicative group of units modulo . This is maximized when is prime and is a primitive element modulo . In this case, the period is formula_25. 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 formula_26; by a variant of the Euclidean algorithm when 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 formula_27 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 and is replaced by an arbitrary non-unit in . A general reference on the subject of LFSRs, FCSRs, and AFSRs is the book.