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# Nonlinear-feedback shift register

A nonlinear-feedback shift register (NLFSR) is a shift register whose input bit is a non-linear function of its previous state.

For an n-bit shift register r its next state is defined as:

${\displaystyle r_{i+1}(b_{0},b_{1},b_{2},\ldots ,b_{n-1})=r_{i}(b_{1},b_{2},\ldots ,f(b_{0},b_{1},b_{2},\ldots ,b_{n-1}))}$,

where f is the non-linear feedback function.[1]

## Applications

Nonlinear-feedback shift registers are components in modern stream ciphers, especially in RFID and smartcard applications. NLFSRs are known to be more resistant to cryptanalytic attacks than Linear Feedback Shift Registers (LFSRs).

## Generating

It is known how to generate an n-bit NLFSR of maximal length 2n, generating a De Bruijn sequence, by extending a maximal-length LFSR with n stages;[2] but the construction of other large NLFSRs with guaranteed long periods remains an open problem.[3] Using bruteforce methods, a list of maximum-period n-bit NLFSRs for n ≤ 25 has been made as well as for n=27.[4][1]

New methods suggest usage of evolutionary algorithms in order to introduce non-linearity.[5] In these works, an evolutionary algorithm learns how to apply different operations on strings from LFSR to enhance their quality to meet the criteria of a fitness function, here the NIST protocol,[6] effectively.

## See also

NLFSR-based ciphers:

## References

1. ^ a b Rachwalik, Tomasz; Szmidt, Janusz; Wicik, Robert; Zabłocki, Janusz (3 June 2012). Generation of Nonlinear Feedback Shift Registers with special-purpose hardware (PDF). Military Communication Institute (Warsaw). p. 1. Retrieved 3 May 2017.
2. ^ C.G. Günther, "Alternating Step Generator Controlled by de Bruijn Sequence", Advances in Cryptology — EUROCRYPT '87, doi:10.1007/3-540-39118-5_2
3. ^
4. ^ E. Dubrova, "A List of Maximum Period NLFSRs", Cryptology ePrint Archive, Report 2012/166, March 2012, [eprint.iacr.org].
5. ^ A. Poorghanad, A. Sadr, A. Kashanipour" Generating High Quality Pseudo Random Number Using Evolutionary Methods", IEEE Congress on Computational Intelligence and Security, vol. 9, pp. 331–335, May 2008 [1]
6. ^ NIST." A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications". NIST, Special Publication April 2010