Layered cellular automata for pseudorandom number generation

The proposed Layered Cellular Automata (L-LCA), which comprises of a main CA with L additional layers of memory registers, has simple local interconnections and high operating speed. The time-varying L-LCA transformation at each clock can be reduced to a single transformation in the set formed by the transformation matrix of a maximum length Cellular Automata (CA), and the entire transformation sequence for a single period can be obtained. The analysis for the period characteristics of state sequences is simplified by analyzing representative transformation sequences determined by the phase difference between the initial states for each layer. The L-LCA model can be extended by adding more layers of memory or through the use of a larger main CA based on widely available maximum length CA. Several L-LCA (L=1,2,3,4) with 10- to 48-bit main CA are subjected to the DIEHARD test suite and better results are obtained over other CA designs reported in the literature. The experiments are repeated using the well-known nonlinear functions and in place of the linear function used in the L-LCA. Linear complexity is significantly increased when or is used

Efficient linear feedback shift registers with maximal period

We introduce and analyze an efficient family of linear feedback shift registers (LFSR's) with maximal period. This family is word-oriented and is suitable for implementation in software, thus provides a solution to a recent challenge posed in FSE '94. The classical theory of LFSR's is extended to provide efficient algorithms for generation of irreducible and primitive LFSR's of this new type

Duality of Channel Encoding and Decoding - Part I: Rate-1 Binary Convolutional Codes

In this paper, we revisit the forward, backward and bidirectional Bahl-Cocke-Jelinek-Raviv (BCJR) soft-input soft-output (SISO) maximum a posteriori probability (MAP) decoding process of rate-1 binary convolutional codes. From this we establish some interesting explicit relationships between encoding and decoding of rate-1 convolutional codes. We observe that the forward and backward BCJR SISO MAP decoders can be simply represented by their dual SISO channel encoders using shift registers in the complex number field. Similarly, the bidirectional MAP decoding can be implemented by linearly combining the shift register contents of the dual SISO encoders of the respective forward and backward decoders. The dual encoder structures for various recursive and non-recursive rate-1 convolutional codes are derived.Comment: 32 pages, 20 figures, to appear in ET

Permutation and sampling with maximum length CA for pseudorandom number generation

In this paper, we study the effect of dynamic permutation and sampling on the randomness quality of sequences generated by cellular automata (CA). Dynamic permutation and sampling have not been explored in previous CA work and a suitable implementation is shown using a two CA model. Three different schemes that incorporate these two operations are suggested - Weighted Permutation Vector Sampling with Controlled Multiplexing, Weighted Permutation Vector Sampling with Irregular Decimation and Permutation Programmed CA Sampling. The experiment results show that the resulting sequences have varying degrees of improvement in DIEHARD results and linear complexity compared to the CA

On the Optimal Space Complexity of Consensus for Anonymous Processes

The optimal space complexity of consensus in shared memory is a decades-old open problem. For a system of $n$ processes, no algorithm is known that uses a sublinear number of registers. However, the best known lower bound due to Fich, Herlihy, and Shavit requires $\Omega(\sqrt{n})$ registers. The special symmetric case of the problem where processes are anonymous (run the same algorithm) has also attracted attention. Even in this case, the best lower and upper bounds are still $\Omega(\sqrt{n})$ and $O(n)$. Moreover, Fich, Herlihy, and Shavit first proved their lower bound for anonymous processes, and then extended it to the general case. As such, resolving the anonymous case might be a significant step towards understanding and solving the general problem. In this work, we show that in a system of anonymous processes, any consensus algorithm satisfying nondeterministic solo termination has to use $\Omega(n)$ read-write registers in some execution. This implies an $\Omega(n)$ lower bound on the space complexity of deterministic obstruction-free and randomized wait-free consensus, matching the upper bound and closing the symmetric case of the open problem

Revisiting LFSMs

Linear Finite State Machines (LFSMs) are particular primitives widely used in information theory, coding theory and cryptography. Among those linear automata, a particular case of study is Linear Feedback Shift Registers (LFSRs) used in many cryptographic applications such as design of stream ciphers or pseudo-random generation. LFSRs could be seen as particular LFSMs without inputs. In this paper, we first recall the description of LFSMs using traditional matrices representation. Then, we introduce a new matrices representation with polynomial fractional coefficients. This new representation leads to sparse representations and implementations. As direct applications, we focus our work on the Windmill LFSRs case, used for example in the E0 stream cipher and on other general applications that use this new representation. In a second part, a new design criterion called diffusion delay for LFSRs is introduced and well compared with existing related notions. This criterion represents the diffusion capacity of an LFSR. Thus, using the matrices representation, we present a new algorithm to randomly pick LFSRs with good properties (including the new one) and sparse descriptions dedicated to hardware and software designs. We present some examples of LFSRs generated using our algorithm to show the relevance of our approach.Comment: Submitted to IEEE-I