Quantum Analysis of Trivium

Abstract

In this paper, we investigate the complexity measurements of the application of Grover’s search algorithm with respect to the stream cipher, Trivium. In the process, we first look into the problem of implementing the cipher using quantum gates for full 1152 rounds (which is probably the first, to the best of our knowledge). Equipped with the quantum implementation, we propose our key recovery algorithm using Grover’s search. The key size is taken small for the simulation to work, using IBM Qiskit, and we recover 3 and 4 unknown key qubits up to 1152 rounds of Trivium with a very high probability. We estimate the quantum cost of Trivium up to 1152 rounds and up to 90-qubit key stream generation. Based on this, we present the quantum complexity for Grover’s search on Trivium up to the full 1152 rounds. With the Toffoli decomposition on our Grover oracle, based on the $T$ + Clifford gate level, we present our benchmark cost of Grover’s search on Trivium up to the full 1152 rounds.