The Hidden Subgroup Problem and Post-quantum Group-Based Cryptography

Horan, Kelsey; Kahrobaei, Delaram

doi:10.1007/978-3-319-96418-8_26

Cited by 11 publications

(5 citation statements)

References 42 publications

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“…It is hoped this work may encourage machine learning and hyper-heuristic approaches in cryptology. This may have an impact upon post-quantum cryptography, with such problems as the hidden subgroup problem ripe for the attack [28]. If the approach is effective, then it confirms a given problem combined with a platform group is breakable, whereas if it is not effective then this may provide further evidence to validate as "quantum-safe" proposed cryptographic structures.…”

Section: Discussionmentioning

confidence: 99%

Evolution of Group-Theoretic Cryptology Attacks using Hyper-heuristics

Craven,

Woodward

2020

Preprint

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In previous work, we developed a single Evolutionary Algorithm (EA) to solve random instances of the Anshel-Anshel-Goldfeld (AAG) key exchange protocol over polycyclic groups. The EA consisted of six simple heuristics which manipulated strings. The present work extends this by exploring the use of hyper-heuristics in group-theoretic cryptology for the first time. Hyper-heuristics are a way to generate new algorithms from existing algorithm components (in this case the simple heuristics), with the EAs being one example of the type of algorithm which can be generated by our hyper-heuristic framework. We take as a starting point the above EA and allow hyper-heuristics to build on it by making small tweaks to it. This adaptation is through a process of taking the EA and injecting chains of heuristics built from the simple heuristics.We demonstrate we can create novel heuristic chains, which when placed in the EA create algorithms which out-perform the existing EA. The new algorithms solve a markedly greater number of random AAG instances than the EA for harder instances. This suggests the approach could be applied to many of the same kinds of problems, providing a framework for the solution of cryptology problems over groups. The contribution of this paper is thus a framework to automatically build algorithms to attack cryptology problems.

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Section: Discussionmentioning

confidence: 99%

Evolution of Group-Theoretic Cryptology Attacks using Hyper-heuristics

Craven,

Woodward

2020

Preprint

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“…where φ n means the automorphism φ composed with itself n times. 1 The difficulty of DLP and the hidden subgroup problem in various groups is beyond the scope of this report: for a survey of the state of the hidden subgroup problem in various platform groups, see [4].…”

Section: The Semidirect Productmentioning

confidence: 99%

Semidirect Product Key Exchange: the State of Play

Battarbee¹,

Kahrobaei²,

Shahandashti³

2022

Preprint

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In this report we survey the various proposals of the key exchange protocol known as semidirect product key exchange (SDPKE). We discuss the various platforms proposed and give an overview of the main cryptanalytic ideas relevant to each scheme. Introduction 1.MotivationFew fields possess a text as foundational as New Directions in Cryptography [1], which presents a key agreement mechanism today known as the Diffie-Hellman Key Exchange (DHKE). The protocol remains relevant in modern cryptographic applications and works as follows:1. Suppose Alice and Bob wish to establish a shared secret key K. They agree on a public, finite group G and a generator g ∈ G.

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“…Compared to previous pair schemes, our scheme has a larger pair advantage in terms of efciency, since all messages are encoded as low-dimensional matrices, and the scaling rate in terms of storage and computational overhead is linear compared to plaintext implementations. Horan K. et al [33] mentioned that the CSP problem is in a general linear group GL d (R) (where R represents the real number feld); if d > 4, CSP can be proved to be antiquantum secure, so when we encode the message M as a matrix, it is necessary to keep its dimension greater than 4. Specifcally, we assume that G is a general noncommutative semigroup, a ∈ G − 1 and b ∈ G, and the function F a (M) can be regarded as a pair of preprocessing for the message M. For any message M originating from the real domain R, we can encode b |M| as a 6-dimensional upper triangular matrix, denoted by M ∈ R 6×6 .…”

Section: Theorem Specifcally Assuming That There Is An Adversarymentioning

confidence: 99%

New Constructions of Existential Unforgeable Aggregate Signature Scheme from CSP

Zou

Huang

et al. 2022

Security and Communication Networks

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In future, hundreds of years of mathematical problems that the security of public key cryptography algorithms rely on may be defeated by quantum algorithms. How can a digital signature scheme gracefully balance security and efficiency? This study uses the conjugate search problem and the left self-distributive system to combine and uses the RSA-like algorithm as the underlying structure to propose a new aggregated signature scheme. We, through the EUF game, under the random metaphor model, prove that the security of the scheme satisfies the adaptation unforgeability under selective message attack, the scheme can be finally reduced to the discrete logarithm problem or large prime number decomposition problem. In addition, we can achieve antiquantum attack and exhaustive attack by performing matrix calculations on the message, defining and changing the structure of the matrix by encoding, and setting thresholds for the matrix dimension and the length of the private key. In terms of efficiency, the message signature implementation is linear compared with the expansion rate in terms of storage and computing overhead, and the generation and verification of the final signature pair have nothing to do with the number of users. In addition, the length of the signature is fixed and the size is only the length of a single group, which effectively reduces the generation of public and private key pairs and saves a lot of storage space. The storage space and computational complexity are also effectively improved compared with other solutions.

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The Hidden Subgroup Problem and Post-quantum Group-Based Cryptography

Cited by 11 publications

References 42 publications

Evolution of Group-Theoretic Cryptology Attacks using Hyper-heuristics

Evolution of Group-Theoretic Cryptology Attacks using Hyper-heuristics

Semidirect Product Key Exchange: the State of Play

New Constructions of Existential Unforgeable Aggregate Signature Scheme from CSP

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