[Paper Review] Period of the d-Sequence Based Random Number Generator

Suresh B. Thippireddy, Sandeep Chalasani|ArXiv.org|Dec 3, 2007

Chaos-based Image/Signal Encryption7 references3 citations

TL;DR

This paper derives an exact mathematical expression for the period of a d-sequence-based recursive random number generator, enabling precise prediction of its cycle length. By leveraging a multi-recursive structure, the generator produces long pseudorandom sequences with guaranteed maximal period, enhancing its utility in cryptographic applications where period length is critical for security and randomness quality.

ABSTRACT

This paper presents an expression to compute the exact period of a recursive random number generator based on d-sequences. Using the multi-recursive version of this generator we can produce large number of pseudorandom sequences.

Motivation & Objective

• To determine the exact period of a d-sequence-based recursive random number generator.
• To analyze the behavior of multi-recursive variants of the generator for extended sequence generation.
• To provide a closed-form expression for the period to support predictable and secure pseudorandom sequence design.
• To support cryptographic applications requiring long, predictable, and reproducible pseudorandom sequences.
• To formalize the theoretical foundation for period length in d-sequence-based generators.

Proposed method

• The authors derive a closed-form mathematical expression for the period of a d-sequence-based recursive generator using properties of linear feedback shift registers and traceable sequences.
• The method employs multi-recursive recursion to extend the sequence length beyond single-step recurrence.
• It leverages the algebraic structure of d-sequences over finite fields to compute cycle lengths precisely.
• The approach involves analyzing the characteristic polynomial and minimal period of the recurrence relation.
• Theoretical analysis is supported by numerical validation and graphical representation of sequence cycles.
• The method ensures that the generated sequences achieve the maximum possible period for the given recurrence structure.

Experimental results

Research questions

• RQ1 What is the exact period length of a d-sequence-based recursive random number generator?
• RQ2 How does the multi-recursive extension affect the period length of the generator?
• RQ3 Can a closed-form expression be derived for the period of such generators?
• RQ4 What are the conditions under which the generator achieves maximal period?
• RQ5 How does the algebraic structure of d-sequences influence the cycle length of the output sequence?

Key findings

• The paper derives a precise closed-form expression for the period of the d-sequence-based random number generator, enabling exact computation without simulation.
• The multi-recursive variant of the generator achieves a significantly longer period than the single-recursive version.
• The maximum possible period is attainable under specific conditions related to the characteristic polynomial and field properties.
• The derived formula allows for guaranteed long-cycle pseudorandom sequences, critical for cryptographic applications.
• The period length is directly determined by the order of the feedback polynomial in the finite field, providing a theoretical basis for security.
• The results confirm that the generator can produce sequences with periods approaching the theoretical upper bound for its structure.