## 1 Introduction  
Since the mid-1950s, RNS arithmetic has attracted significant attention from researchers and practitioners across various domains, including number-theoretic methods, computer technology, digital signal processing, cryptography, and communications [1, 2, 3, 4, 5, 6, 7, 8, 9]. RNS arithmetic is particularly valuable in parallel processing systems, where it enables fast and accurate computations. The inherent parallelism of modular computational structures provides several key advantages over positional numeral systems, especially for operations involving large numbers. These advantages include the following:  
* performance time independence of parallel modular operations from the moduli number and, consequently, the length of the residue code;
* high adaptability of modular arithmetic algorithms to tabular calculations and pipeline processing;
* flexibility in using lookup table techniques for reconfiguring modular computational structures, among others. A well-known application of modular arithmetic is in cryptographic systems, where multiplicative operations over large moduli form the foundation for various information security mechanisms [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. These operations are extensively utilized in electronic signature schemes, public key cryptosystems based on RSA and Rabin's method, and other cryptographic tasks. Therefore, advancing computational technologies that enhance modular multiplication and exponentiation efficiency over large-number ranges remains a crucial research direction. One of the most promising approaches for achieving high-performance modular arithmetic is using RNS-based modular computational structures. As the degree of modularity in computational processes increases, the efficiency of these structures improves significantly. This effect is particularly evident when handling large-number arithmetic operations. A notable example is the Montgomery multiplication (MM) algorithm implemented using RNS [10; 11; 12; 13; 22]. Montgomery multiplication replaces general division with an integer division operation, which is well suited for RNS configurations [24]. This property makes MM in RNS highly efficient. However, a principal computational bottleneck in modular arithmetic is the residue code extension required in MM implementations. Optimizing these operations is a key challenge in developing high-performance cryptographic hardware and software for large-number computations. One promising solution is to use minimally redundant RNS, which allows for more efficient execution of fundamental non-modular operations such as base extension and reverse conversion based on the Chinese Remainder Theorem (CRT) [5; 6; 25; 26; 27; 28]. This paper aims to take fundamental advantages of minimally redundant RNS arithmetic to optimize multiplication algorithms based on [PERSON]'s scheme. By refining the base extension and conversion processes, we enhance the overall efficiency of MM in large-number cryptographic applications. This paper provides a rigorous theoretical framework for optimizing modular multiplication in minimally redundant Residue Number Systems (RNSs), offering new insights into the computational efficiency of cryptographic operations. Using minimally redundant arithmetic, the study improves the performance of modular computations, which are fundamental in public-key cryptography, secure computing, and high-performance arithmetic. Rather than focusing on software or hardware implementation, this work contributes by presenting formal algorithmic descriptions, mathematical proofs, and correctness guarantees. These theoretical foundations serve as a basis for future advances in secure computing, number-theoretic algorithms, and parallel processing architectures. For clarity and reproducibility, the paper provides detailed algorithmic methodologies and mathematical derivations, making it accessible to researchers developing efficient cryptographic protocols, modular arithmetic techniques, and computational number theory applications.