Scalable method of searching for full-period Nonlinear Feedback Shift Registers with GPGPU. New List of Maximum Period NLFSRs.

This paper addresses the problem of efficient searching for Nonlinear Feedback Shift Registers (NLFSRs) with a guaranteed full period. The maximum possible period for an $n$-bit NLFSR is $2^n-1$ (all-zero state is omitted). %but omitting all-0 state makes the period $2^n-1$ in their longest cycle of states. A multi-stages hybrid algorithm which utilizes Graphics Processor Units (GPU) power was developed for processing data-parallel throughput computation.Usage of abovementioned algorithm allows to give an extended list of n-bit NLFSR with maximum period for 7 cryptographically applicable types of feedback functions

Using SAT solvers to finding short cycles in cryptographic algorithms

A desirable property of iterated cryptographic algorithms, such as stream ciphers or pseudo-random generators, is the lack of short cycles. Many of the previously mentioned algorithms are based on the use of linear feedback shift registers (LFSR) and nonlinear feedback shift registers (NLFSR) and their combination. It is currently known how to construct LFSR to generate a bit sequence with a maximum period, but there is no such knowledge in the case of NLFSR. The latter would be useful in cryptography application (to have a few taps and relatively low algebraic degree). In this article, we propose a simple method based on the generation of algebraic equations to describe iterated cryptographic algorithms and find their solutions using an SAT solver to exclude short cycles in algorithms such as stream ciphers or nonlinear feedback shift register (NLFSR). Thanks to the use of AIG graphs, it is also possible to fully automate our algorithm, and the results of its operation are comparable to the results obtained by manual generation of equations. We present also the results of experiments in which we successfully found short cycles in the NLFSRs used in KSG, Grain-80, Grain-128 and Grain-128a stream ciphers and also in stream ciphers Bivium and Trivium (without constants used in the initialization step)

Modified Alternating Step Generators

Irregular clocking of feedback shift registers is a popular technique to improve parameters of keystream generators in stream ciphers. Another technique is to implement nonlinear functions. We join these techniques and propose Modified Alternating Step Generators built with linear and nonlinear feedback shift registers. Adequate nonlinear Boolean functions are used as feedbacks or as filtering functions of shift registers in order to increase complexity of sequences produced by individual registers and the whole generator. We investigate basic parameters of proposed keystream generators, such as period, linear complexity and randomness

Searching for Nonlinear Feedback Shift Registers with Parallel Computing

Abstract. Nonlinear feedback shift registers (NLFSRs) are used to construct pseudorandom generators for stream ciphers. Their theory is not so complete as that of linear feedback shift registers (LFSRs). In general, it is not known how to construct all NLFSRs with maximum period. The direct method is to search for such registers with suitable properties. Advanced technology of parallel computing has been applied both in software and hardware to search for maximum period NLFSRs having a fairly simple algebraic normal form. Key words: nonlinear feedback shift registers, maximum period, quadratic m-sequences, parallel computing, FPGA implementation.