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Feedback with Carry Shift Registers

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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 is an integer, then an N-ary FCSR of length is a finite state device with a state consisting of a vector of elements in and an integer .[1][2][3][4] The state change operation is determined by a set of coefficients and is defined as follows: compute . Express s as with in . Then the new state is . By iterating the state change an FCSR generates an infinite, eventually periodic sequence of numbers in .

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[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 . Associated with the output sequence is the N-adic number The fundamental theorem of FCSRs says that there is an integer so that , a rational number. The output sequence is strictly periodic if and only if is between and . 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 is the inverse of , and the output sequence is strictly periodic, then , where 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 . In this case the output sequence is called an l-sequence (for "long sequence").[1]

l-sequences have many excellent statistical properties[1][3] 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 ;[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 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.[4]

References

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  1. ^ a b c d e f Klapper, Andrew; Goresky, Mark (March 1997). "Feedback Shift Registers, 2-Adic Span, and Combiners With Memory" (PDF). Journal of Cryptology. 10 (2): 111–147. CiteSeerX 10.1.1.37.5637. doi:10.1007/s001459900024. S2CID 206885831.
  2. ^ a b Couture, Raymond; L’Ecuyer, Pierre (April 1994). "On the lattice structure of certain linear congruential sequences related to AWC/SWB generators" (PDF). Mathematics of Computation. 62 (206): 799–808. doi:10.2307/2153540. JSTOR 2153540.
  3. ^ a b Goresky, Mark; Klapper, Andrew (October 2003). "Efficient Multiply-with-Carry Random Number Generators with Maximal Period" (PDF). ACM Transactions on Modeling and Computer Simulation. 13 (4): 310–321. CiteSeerX 10.1.1.22.9007. doi:10.1145/945511.945514. S2CID 13334372.
  4. ^ a b Goresky, Mark; Klapper, Andrew (2012). Algebraic Shift Register Sequences. Cambridge University Press. ISBN 978-1-107-01499-2. Table of contents.
  5. ^ Marsaglia, George; Zaman, Arif (August 1991). "A new class of random number generators" (pdf). Annals of Applied Probability. 1 (3): 462–480. doi:10.1214/aoap/1177005878.
  6. ^ Schneier, Bruce (1996). Applied Cryptography. New York: John Wiley & Sons.
  7. ^ Mahler, Kurt (January 1940). "On a geometrical representation of p-adic numbers" (PDF). Annals of Mathematics. 2. 41 (1): 8–56. doi:10.2307/1968818. JSTOR 1968818. MR 0001772.
  8. ^ de Weger, B. M. M. (September 1986). "Approximation lattices of p–adic numbers" (PDF). Journal of Number Theory. 24 (1): 70–88. doi:10.1016/0022-314X(86)90059-4.
  9. ^ Klapper, Andrew; Xu, Jinzhong (March 2004). "Register Synthesis for Algebraic Feedback Shift Registers Based on Non-Primes" (PDF). Designs, Codes and Cryptography. 31 (3): 227–250. doi:10.1023/B:DESI.0000015886.71135.e1. S2CID 13138878.
  10. ^ Klapper, Andrew; Xu, Jinzhong (17 September 1999). "Algebraic Feedback Shift Registers" (PDF). Theoretical Computer Science. 226 (1–2): 61–93. CiteSeerX 10.1.1.36.8645. doi:10.1016/S0304-3975(99)00066-3.