Simulation of Shor algorithm for discrete logarithm problems with comprehensive pairs of modulo p and order q

• URL: http://arxiv.org/abs/2503.23939v2
• Date: Thu, 24 Apr 2025 12:51:36 GMT
• Title: Simulation of Shor algorithm for discrete logarithm problems with comprehensive pairs of modulo p and order q
• Authors: Kaito Kishi, Junpei Yamaguchi, Tetsuya Izu, Noboru Kunihiro,
• Abstract summary: We simulate quantum circuits that operate on general pairs of modulo $p and order $q.<n>We show how much the strength of safe-prime groups and Schnorr groups is the same if $p$ is equal.<n>In particular, it was experimentally and theoretically shown that when a carryer is used in the addition circuit, the cryptographic strength of a Schnorr group with $p=$48$ bits under Shor's algorithm is almost equivalent to that of a safe-prime group with $p=$48$ bits.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The discrete logarithm problem (DLP) over finite fields, commonly used in classical cryptography, has no known polynomial-time algorithm on classical computers. However, Shor has provided its polynomial-time algorithm on quantum computers. Nevertheless, there are only few examples simulating quantum circuits that operate on general pairs of modulo $p$ and order $q$. In this paper, we constructed such quantum circuits and solved DLPs for all 1,860 possible pairs of $p$ and $q$ up to 32 qubits using a quantum simulator with PRIMEHPC FX700. From this, we obtained and verified values of the success probabilities, which had previously been heuristically analyzed by Eker\r{a}. As a result, the detailed waveform shape of the success probability of Shor's algorithm for solving the DLP, known as a periodic function of order $q$, was clarified. Additionally, we generated 1,015 quantum circuits for larger pairs of $p$ and $q$, extrapolated the circuit sizes obtained, and compared them for $p=2048$ bits between safe-prime groups and Schnorr groups. While in classical cryptography, the cipher strength of safe-prime groups and Schnorr groups is the same if $p$ is equal, we quantitatively demonstrated how much the strength of the latter decreases to the bit length of $p$ in the former when using Shor's quantum algorithm. In particular, it was experimentally and theoretically shown that when a ripple carry adder is used in the addition circuit, the cryptographic strength of a Schnorr group with $p=2048$ bits under Shor's algorithm is almost equivalent to that of a safe-prime group with $p=1024$ bits.

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