Hidden Shift Quantum Cryptanalysis and Implications

Dan Boneh, Richard J. Lipton

Kahrobaei, Delaram, Flores, Ramón, Marialaura Noce +2 more

Xavier Bonnetain

In this thesis, we study the security of cryptographic systems against an adversary who has access to a quantum computer. In quantum computing, we studied the hidden period and hidden shift problems, which are among the few known problems that have some applications in cryptogaphy and for which the best known quantum algorithm is more than polynomially faster than the best known classical algorithm. We proposed some improvements, new tradeoffs between classical and quantum time and memory, and extended their scope of applications to cases where only a classical oracle is available. In cryptanalysis, in symmetric cryptography, we proposed some attacks against symmetric constructions based on hidden shifts, and generalized many attacks using hidden periods to cases where the construction is only accessible classically. We proposed a quantum cryptanalysis of the different versions of the authenticated cipher AEZ and some quantum versions of multiple slide attacks, which are a classical family of cryptanalyses. This rewriting of attacks in the formalism of hidden periods has allowed us to propose a new classical attack against multiple variants of the cipher MiMC. In asymmetric cryptography, we proposed a concrete and asymptotic quantum security analysis of some isogeny-based key exchanges. Finally, we studied quantum security in some cases where these hidden structure problems do not apply, with in particular the first quantum security analysis of AES, the most used symmetric cipher to date.

Evangelos Mavridis

Tehzeeb Ali Gows Basha

Modern public key cryptosystems rely on two assumptions regarding computational difficulty: the integer factorization problem (RSA) and the discrete logarithm problem (elliptic curve cryptography).These problems have stood the test of time for four decades of cryptanalysis.Yet, their algebraic and periodic nature makes them vulnerable to quantum algorithms, most notably Shor's algorithm (1994).This paper presents a detailed comparison between classical and quantum-resistant cryptographic systems with an emphasis on lattice-based systems.The Learning With Errors (LWE) problem and its variants (Ring-LWE, Module-LWE) do not have the same periodicity that makes problems like integer factorization and discrete logarithms susceptible to quantum algorithms.Furthermore, LWE problems have reductions from worst-case lattice problems and have mathematical proofs of quantum resistance.Due to the vulnerability of ECDSA algorithms for cryptocurrencies, it is estimated that current cryptocurrencies will no longer be secure once quantum computer technology matures within the next 10-30 years, potentially compromising holdings estimated at between $260 billion and $390 billion as of August 2026.The implications of cryptanalysis on public key cryptography systems motivate mathematical recommendations regarding the security of current cryptosystems and the need for a shift to post-quantum cryptographic standards.

Marta Conde Pena, Raúl Durán Dı́az, Jean‐Charles Faugère +2 more

At STOC 2012, Aaronson and Christiano proposed a noisy and a noiseless version of the first public‐key quantum money scheme endowed with a security proof. This paper addresses the so‐called noisy hidden subspaces problem , on which the noisy version of their scheme is based. The first contribution of this work is a non‐quantum cryptanalysis of the above‐mentioned noisy quantum money scheme extended to prime fields , with , that runs in randomised polynomial time. This finding is supported with experimental results showing that, in practice, the algorithm presented is efficient and succeeds with overwhelming probability. The second contribution is a non‐quantum randomised polynomial‐time cryptanalysis of the noisy quantum money scheme over succeeding with a certain probability for values of the noise lying within a certain range. This result disproves a conjecture made by Aaronson and Christiano about the non‐existence of an algorithm that solves the noisy hidden subspaces problem over and succeeds with such probability.