(In)security of Stream Ciphers Against Quantum Annealing Attacks on the Example of the Grain 128 and Grain 128a Ciphers

Abstract

The security level of a cipher is a key parameter. While general-purpose quantum computers significantly threaten modern symmetric ciphers, other quantum approaches like quantum annealing have been less concerning. However, this paper argues that a quantum annealer specifically designed to attack Grain 128 and Grain 128a ciphers could soon be technologically feasible. Such an annealer would require 5,751 (6,761) qubits and 77,496 (94,865) couplers, with a qubit connectivity of 225 (245). This work also shows that modern stream ciphers like Grain 128 and Grain 128a may be vulnerable to quantum annealing attacks. Although the exact complexity of quantum annealing is unknown, heuristic estimates suggest that for many problems with $N$ variables, a $\sqrt{N}$ exponential advantage over simulated annealing may hold. We detail how to transform algebraic attacks on Grain ciphers into the QUBO problem, making our attack potentially more efficient than classical brute-force methods. We demonstrate that applying our attack to rescaled Grain cipher versions, Grain $l$ and Grain $la$, overtakes brute-force and Grover’s attacks for sufficiently large $l$, assuming quantum annealing’s exponential benefit over simulated annealing.