[Paper Review] Pseudo-random Sequences Generated by Cellular Automata

Bruno Martin, Patrick Solé|ArXiv.org|Jul 24, 2008

Cellular Automata and Applications14 references13 citations

TL;DR

This paper proposes a hardware-efficient method for generating cryptographically strong pseudo-random sequences using cellular automata (CA) and hybrid CA (HCA), leveraging the synthesis of Linear Feedback Shift Registers (LFSR) via CA to enable fast, reconfigurable FPGA implementations. The key contribution is a systematic CA-based realization of LFSR-like sequences with high statistical quality, suitable for VLSI and stream cipher applications.

ABSTRACT

Generation of pseudo random sequences by cellular automata, as well as by hybrid cellular automata is surveyed. An application to the fast evaluation and FPGA implementation of some classes of boolean functions is sketched out.

Motivation & Objective

• To develop a hardware-efficient method for generating cryptographically strong pseudo-random sequences using cellular automata (CA).
• To extend the synthesis of Linear Feedback Shift Registers (LFSR) to hybrid CA (HCA) for improved implementation flexibility.
• To enable fast, reconfigurable evaluation of boolean functions related to cyclic codes using CA-based architectures.
• To provide a VLSI- and FPGA-friendly realization of LFSR-like sequences via CA, supporting dynamic reconfiguration.
• To demonstrate the feasibility of using elementary and hybrid CA rules for generating sequences with good statistical properties.

Proposed method

• The paper uses elementary CA with 3-cell neighborhoods and Boolean transition functions to generate pseudo-random sequences, focusing on rules like 30 and 86.
• It applies a synthesis algorithm to realize LFSRs using HCA, based on irreducible polynomials over 𝔽₂ and their corresponding characteristic polynomials.
• The method constructs HCA transition matrices as block-diagonal matrices from irreducible polynomials, enabling concatenation of primitive CA machines.
• It leverages the Mattson-Solomon polynomial theory to compute minimal polynomials of field elements, crucial for connection polynomial derivation in cyclic codes.
• The approach uses genetic algorithms to select optimal CA rules (e.g., 30, 86, 101) for HCA, aiming for statistical randomness and resistance to correlation attacks.
• The framework maps boolean functions of the form f(x) = Tr(ax + bx^s) to sequences of period 2^n, enabling efficient hardware evaluation on FPGAs.

Experimental results

Research questions

• RQ1 Can cellular automata be effectively used to generate pseudo-random sequences of cryptographic quality?
• RQ2 How can Linear Feedback Shift Registers (LFSRs) be synthesized using hybrid cellular automata (HCA)?
• RQ3 Which elementary CA rules produce sequences with good statistical randomness and low correlation immunity?
• RQ4 Can CA-based architectures enable fast, reconfigurable evaluation of boolean functions related to cyclic codes?
• RQ5 What is the relationship between the minimal polynomials of field elements and the connection polynomials of cyclic codes in this CA framework?

Key findings

• Elementary CA rules such as 30, 86, and 101 were found to generate sequences with good statistical properties, though not correlation-immune, as confirmed by statistical testing.
• The synthesis of LFSRs via HCA is possible using irreducible polynomials over 𝔽₂, with transition matrices constructed as block-diagonal combinations of component CA matrices.
• The method enables the construction of long-period sequences by concatenating primitive HCA machines, achieving scalability in hardware design.
• The use of Mattson-Solomon polynomials allows efficient computation of minimal polynomials, which are essential for deriving connection polynomials of cyclic codes.
• Boolean functions of the form f(x) = Tr(ax + bx^s), including those from Gold, Kasami, Welch, and Niho exponents, can be efficiently evaluated using the proposed CA architecture.
• The framework supports FPGA implementation with dynamic reconfiguration, enabling high-speed generation of pseudo-random sequences through hardware acceleration.