Semi-primitive roots and the discrete logarithm module $2^k$

TL;DR

This paper explores the relationship between semi-primitive roots modulo powers of two and discrete logarithm algorithms, extending previous results to all such roots and proposing a generalized algorithm with potential cryptographic applications.

Contribution

It generalizes the discrete logarithm algorithm for modulo 2^k to any semi-primitive root, broadening its applicability and efficiency.

Findings

Extended the algorithm to all semi-primitive roots modulo 2^k

Provided a generalized conversion formula for bases

Potential applications in cryptography and hardware arithmetic

Abstract

We establish a connection between semi-primitive roots of the multiplicative group of integers modulo $2^{k}$ where $k\geq 3$, and the logarithmic base in the algorithm introduced by Fit-Florea and Matula (2004) for computing the discrete logarithm modulo $2^{k}$. Fit-Florea and Matula used properties of the semi-primitive root 3 modulo $2^{k}$ to obtain their results and provided a conversion formula for other possible bases. We show that their results can be extended to any semi-primitive root modulo $2^{k}$ and also present a generalized version of their algorithm to find the discrete logarithm modulo $2^{k}$. Various applications in cryptography, symbolic computation, and others can potentially benefit from higher precision hardware integer arithmetic. The algorithm is suitable for hardware support of applications where fast arithmetic computation is desirable.