Quantum Analysis

Authors: Kento Oonishi and Noboru Kunihiro Abstract: Shor’s algorithm is a quantum algorithm that can efficiently solve the problems of integer factorization and discrete logarithms, which potentially break currently used public-key cryptosystems such as RSA and elliptic curve cryptography. However, a significant challenge in solving these problems is the noise inherent in quantum computers. Previous research advanced fault-tolerant techniques such as surface codes to mitigate this noise in quantum computers. Analyzing the security of currently used public-key cryptosystems on fault-tolerant quantum computers is essential. In this paper, we estimate the computational cost required to solve integer factorization using Shor’s algorithm on surface codes. We evaluated computational resources required for surface codes and magic state distillation and derived the computational cost of Shor’s algorithm based on these evaluations. Our computational complexity analysis indicates that there is no significant difference in the order of complexity. However, it underscores the paramount importance of optimizing the coefficient part for future efficiency improvements. ...

Authors: Kaito Kishi, Junpei Yamaguchi, Tetsuya Izu, and Noboru Kunihiro 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 article, we constructed such quantum circuits and solved DLPs for all 1860 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 Ekerå (2019). 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. In addition, we generated 1015 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 basic 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. ...