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Lecture notes in electrical engineering

An Optimization of Shor's Algorithm Based on Quantum Parallelism

Shuaishuai Zhu, Ming Yuan

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.1007/978-981-95-3320-6_53

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Quantum computers leverage the principles of quantum mechanics to solve complex problems beyond the capabilities of classical computers, such as prime factorization, which poses a significant threat to cryptographic algorithms like RSA and ECC. Although current quantum resources are insufficient for effective decryption, ongoing research focuses on optimizing Shor's algorithm to overcome these limitations. Shor's algorithm, a quantum algorithm designed for efficient integer factorization, exploits quantum parallelism and entanglement to perform prime factorization exponentially faster than classical algorithms. This paper reviews recent trends in Shor algorithm optimization, particularly in enhancing the efficiency of ripple-carry operators and the Karatsuba algorithm in multiplication and proposes a multiplication method that applies the Strassen algorithm to constant adders
The present study explores the groundbreaking possibilities of Shor's algorithm in cracking RSA encryption, hence enhancing the security consequences of this extensively employed cryptographic protocol. Due to Shor's algorithm's exponential speed advantage over classical algorithms, RSA encryption—a popular public key cryptosystem—is vulnerable to quantum attacks. To keep its security, RSA depends on the difficulty of factoring huge semi-primes. RSA's security presumptions are called into question by Shor's algorithm's quick factorization on a quantum computer, which has prompted research into post-quantum cryptography solutions to guarantee secure communication in the quantum age. Even with parallel computation, the speed and storage capacity of classical computing are constrained. A key idea in quantum computing is quantum parallelism, which allows quantum systems to carry out several computations at once. In contrast to classical portions. The scope of this paper is to understand the fundamentals of Quantum Computing, the current state of Shor’s algorithm implementation, and efficiency, particularly its application in compromising public key cryptosystems like RSA. The project aims to achieve how factorization is done through classical methods and further, how is this done through Shor’s algorithm and QFT techniques.
An algorithm of Ekerå and Gärtner, inspired by Regev’s factoring algorithm, computes discrete logarithms in multiplicative groups of prime fields. Asymptotically, it has the same qubit and gate complexity as Shor’s algorithm but splits the computation into d independent, parallel runs. The optimal value of this parameter is d∼n where n is the bit size of the cryptographic group. We propose an extension of this algorithm to hyperelliptic curves. For curves of genus g∼n, which are not used in cryptography, we prove unconditionally that the optimal number of parallel runs d∼n can still be achieved. For genus-two curves, we propose a heuristic algorithm. While its runtime speedup is difficult to quantify in general, we show that it can be expected to use up to d=8 parallel runs for specific curves relevant to cryptography, including GLV-friendly and pairing-friendly curves.