Breaking SHA-1 through QUBO models
Shunsuke Tsukiyama, Koji Nakano, Yasuaki Ito, Takumi Kato, Yuya Kawamata · 2024
A Quadratic Unconstrained Binary Optimization (QUBO) model is defined by a quadratic objective function comprising binary variables, each taking values of 0 or 1. The objective of the QUBO problem is to determine the optimal assignment of these binary variables to minimize the QUBO model. QUBO models are instrumental in solving various combinatorial optimization problems, leading to extensive research on solver development, as many combinatorial optimization problems can be reformulated as QUBO problems. Notably, D-Wave Systems, a company specializing in quantum technologies, has developed quantum annealers which are programmable QUBO solvers that utilize quantum mechanics. SHA-1 is a cryptographic hash function that generates a 160-bit hash value from a message of an arbitrary length. This hash value makes it difficult to compute the original message within a practical timeframe. A pre-image attack, which aims to compute the original message from a hash value, is important in cryptographic hash functions. Previous research has shown that a QUBO can be designed to simulate SHA-1, facilitating the execution of a pre-image attack. In this paper, we present a novel QUBO model with fewer bits, linear terms, and quadratic terms than previous QUBO models for simulating hash values and pre-image calculations in SHA-1. While the previous QUBO model requires 34,016 bits, 32,584 linear terms, and 168,188 quadratic terms, our new model requires only 32,096 bits, 26,600 linear terms, and 157,308 quadratic terms. This approach can efficiently execute pre-image attacks using a quantum annealer, providing valuable insights for evaluating the robustness of cryptographic security systems.