Optimization of Additive Fibonacci Generators Based on Primitive Polynomials Over GF(p)

Abstract

This paper presents an approach to the modification of the additive Fibonacci generator by implementing it based on primitive polynomials over the field GF(p). The proposed modification addresses the inherent cryptographic instability of the classical additive Fibonacci generator and aims at its optimization. The modified additive Fibonacci generator (MAFG) introduced in this work achieves the maximum possible repetition period for the generated sequence. A critical drawback of generators constructed using primitive polynomials with odd values of p–namely, noncompliance with statistical requirements of the output sequence–has been eliminated. To address this issue, the original MAFG sequence with an odd p is combined via a logical XOR operation with a pseudorandom bit sequence exhibiting a uniform distribution of 0s and 1s. Statistical testing based on the NIST SP 800-22 methodology has demonstrated that this combination ensures maximum periodicity across all initial states and eliminates the occurrence of so-called “weak keys”. Extensive testing results confirm the high randomness quality of the output sequence and its suitability for cryptographic applications. Further optimization of such generators may focus on the selection and refinement of primitive polynomials for their implementation.

Keywords

• Combined generator
• field GF(p)
• information security
• primitive polynomials
• pseudorandom sequences
• statistical characteristics