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Tight Bounds for Simon's Algorithm.

Xavier Bonnetain

Published Jan 1, 2020• 1 citation
OpenAlex

Abstract

Simon’s algorithm is the first example of a quantum algorithm exponentially faster than any classical algorithm, and has many applications in cryptanalysis. While these quantum attacks are often extremely efficient, they are generally missing some precise cost estimate. This article aims at resolving this issue by computing precise query costs for the different use cases of Simon’s algorithm in cryptanalysis.

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Simon’s algorithm is the first example of a quantum algorithm exponentially faster than any classical algorithm, and has many applications in cryptanalysis. While these quantum attacks are often extremely efficient, they are generally missing some precise cost estimate. This article aims at resolving this issue by computing precise query costs for the different use cases of Simon’s algorithm in cryptanalysis.
This article presents an algorithm for fast factorization on quantum computer. Quantum mechanic introduces to information processing new effects that can be used to solve efficiently problems classical known as hard. Simon algorithm was one of the first proofs that quantum computer has some advantages over classical model of computation. Shor generalized ideas introduce by Simon and show that real world problems are in range of quantum computation. Article presents also general description of quantum Fourier transform (QFT), which is without doubt the most powerful tool used in algorithms described in this paper. Generalization of algorithms for Simon problem and fast factorization gives new class of problems for which fast quantum algorithms can be construct.
Limited by today's physical devices, quantum circuits with a long depth are usually noisy and difficult to realize in practice. The novel computing architecture of distributed quantum computing is expected to reduce the noise and depth of quantum circuits. In this paper, we study Simon's problem in distributed scenarios and design a distributed quantum algorithm to solve the problem. The algorithm proposed by us has the advantage of exponential acceleration compared with classical distributed computing and has the advantage of square acceleration compared with the best distributed quantum algorithm proposed before in query complexity. In particular, the previous distributed quantum algorithm for Simon's problem cannot be extended to the case of more than two computing nodes (i.e., two subproblems), but our distributed quantum algorithm can be extended to the case of multiple computing nodes (i.e., multiple subproblems).
With the development of quantum technology, quantum computing has an increasingly significant impact on cryptanalysis. Several quantum algorithms, such as Simon’s algorithm, Grover’s algorithm, the Bernstein–Vazirani algorithm, Shor’s algorithm, and the Grover-meets-Simon algorithm, have been proposed successively. However, almost all cryptanalysis is based on the quantum chosen-plaintext attack (qCPA) model. This paper focuses on a powerful cryptanalytic model, quantum related-key attack (qRKA), and proposes a strategy of qRKAs against symmetric ciphers using Simon’s algorithm. We construct a periodic function to efficiently recover the secret key of symmetric ciphers if the attacked symmetric ciphers satisfy Simon’s promise, and present the complexity analysis on specific symmetric ciphers. Then, we apply qRKA to the Even–Mansour cipher and SoEM construction, recover their secret keys, and show their complexity comparison in the distinct attack models. This work is of great significance for the qRKA cryptanalysis of existing provably secure cryptographic schemes and the design of future quantum secure cryptographic schemes.
“Simon's Algorithm: Collision Finding” explains Simon's Algorithm, focusing on how it helps in finding collisions. It starts with an overview of how the algorithm works. A literature review is included, covering both Simon's algorithm and its uses in cryptography. The section then introduces Simon's problem, comparing how classical and quantum methods solve it, and explains how to build a quantum circuit for a blackbox function. Different examples of blackbox functions (oracles) are shown, such as 1-to-1 and 2-to-1 mappings, with detailed implementations of Simon's algorithm for both 4-bit and 8-bit cases. The section also looks at a suggested attack method on the Grain-128a cipher, explaining how the key and initialization vector (IV) are set up, how the keystream is generated, and how the authenticated tag is created. It provides a quantum circuit design for the Simon attack on Grain-4a, along with steps for carrying out the attack. The section also presents experimental results from several test cases and quantum resource estimates. It wraps up with a discussion of the results and thoughts on future improvements and developments in using Simon's algorithm for cryptographic attacks.
We present new connections between quantum information and the field of classical cryptography. In particular, we provide examples where Simon’s algorithm can be used to show insecurity of commonly used cryptographic symmetric-key primitives. Specifically, these examples consist of a quantum distinguisher for the 3-round Feistel network and a forgery attack on CBC-MAC which forges a tag for a chosen-prefix message querying only other messages (of the same length). We assume that an adversary has quantum-oracle access to the respective classical primitives. Similar results have been achieved recently in independent work by Kaplan et al. [KLLNP16]. Our findings shed new light on the post-quantum security of cryptographic schemes and underline that classical security proofs of cryptographic constructions need to be revisited in light of quantum attackers.
The first quantum algorithm to offer an exponential speedup (in the query complexity setting) over classical algorithms was Simon's algorithm for identifying a hidden exclusive-or mask. Here we observe how part of Simon's algorithm can be interpreted as a Clebsch-Gordan transform. Inspired by this we show how Clebsch-Gordan transforms can be used to efficiently find a hidden involution on the group G^n where G is the dihedral group of order eight (the group of symmetries of a square.) This problem previously admitted an efficient quantum algorithm but a connection to Clebsch-Gordan transforms had not been made. Our results provide further evidence for the usefulness of Clebsch-Gordan transform in quantum algorithm design.