# 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 $N >1$ is an integer, then an N-ary FCSR of length $r$ is a finite state device with a state $(a;z) = (a_0,a_1,\dots,a_{r-1};z)$ consisting of a vector of elements $a_i$ in $\{0,1,\dots,N-1\}=S$ and an integer $z$.[1][2][3][4] The state change operation is determined by a set of coefficients $q_1,\dots,q_n$ and is defined as follows: compute $s = q_r a_0+q_{r-1} a_1+\dots+q_1 a_{r-1} + z$. Express s as $s = a_r + N z'$ with $a_r$ in $S$. Then the new state is $(a_1,a_2,\dots,a_r; z')$. 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[1]), and in generating pseudorandom numbers for quasi-Monte Carlo (under the name Multiply With Carry (MWC) generator - invented by Couture and L'Ecuyer,[2]) generalizing work of Marsaglia and Zaman.[5]

FCSRs are analyzed using number theory. Associated with the FCSR is a connection integer $q = q_r N^r + \dots + q_1 N^1 - 1$. Associated with the output sequence is the N-adic number $a = a_0 + a_1 N + a_2N^2+\dots$ 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.[1]

There is also an exponential representation of FCSRs: if $g$ is the inverse of $N \mod q$, and the output sequence is strictly periodic, then $a_i = (A g_i \mod q) \mod N$, 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. In this case, the period is $q-1$. In this case the output sequence is called an l-sequence (for "long sequence").[1]

l-sequences have many excellent statistical properties[1][4] that make them candidates for use in applications,[6] 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[7] and De Weger's[8] lattice based analysis of N-adic numbers when $N=2$;[1] by a variant of the Euclidean algorithm when N is prime; and in general by Xu's adaptation of the Berlekamp-Massey algorithm.[9] 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.[10] A general reference on the subject of LFSRs, FCSRs, and AFSRs is the book.[11]

## References

1. A. Klapper and M. Goresky, Feedback Shift Registers, 2-Adic Span, and Combiners With Memory, in Journal of Cryptology vol. 10, pp. 111-147, 1997, [1]
2. ^ a b R. Couture and P. L’Ecuyer, On the lattice structure of certain linear congruential sequences related to AWC/SWB generators, Math. Comp. vol. 62, pp. 799–808, 1994, [2],
3. ^ M. Goresky and A. Klapper, Algebraic Shift Register Sequences, 2009, [3]
4. ^ a b M. Goresky and A. Klapper, Efficient Multiply-with-Carry Random Number Generators with Optimal Distribution Properties, ACM Transactions on Modeling and Computer Simulation, vol 13, pp 310-321, 2003, [4]
5. ^ G. Marsaglia and A. Zaman, A new class of random number generators, Annals of Applied Probability, vol 1, pp. 462–480, 1991
6. ^ B. Schneier, Applied Cryptography. John Wiley & Sons, New York, 1996
7. ^ K. Mahler, On a geometrical representation of p–adic numbers, Ann. of Math., vol. 41, pp. 8–56, 1940
8. ^ B. M. M. de Weger, Approximation lattices of p–adic numbers, J. Num. Th., vol 24, pp. 70–88, 1986
9. ^ A. Klapper and J. Xu, Register Synthesis for Algebraic Feedback Shift Registers Based on Non-Primes, Designs, Codes, and Cryptography vo. 31, pp. 227-250", 2004
10. ^ A. Klapper and J. Xu, Algebraic Feedback Shift Registers, Theoretical Computer Science, vol. 226, pp. 61-93, 1999, [5]
11. ^ M. Goresky and A. Klapper, Algebraic Shift Register Sequences, Cambridge University Press, 2012