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Analysis and design of post-quantum cryptographic algorithms : PKP-based signature scheme and ultra-short multivariate signatures

Abstract : The construction of large quantum computers would endanger most of the public-key cryptographic schemes in use today. Therefore, there is currently a large research effort to develop new post-quantum secure schemes. In particular, we are interested in post-quantum cryptographic schemes whose security relies on the hardness of solving some mathematical problems such as thePermuted Kernel Problem (PKP) and the Hidden Field Equations (HFE). This work investigates first the complexity of PKP. And after a thorough analysis of the State-of-theart attacks of PKP, we have been able to update some results that were not accurate, and to provide an explicit complexity formula which allows us to identify hard instances and secure sets of parameters of this problem. PKP was used in 1989 to develop the first Zero-Knowledge Identification Scheme (ZK-IDS) that has an efficient implementation on low-cost smart cards. In a second step, we optimize the PKP-based ZK-IDS and then we introduce PKP-DSS:a Digital Signature Scheme based on PKP. We construct PKP-DSS from the ZK-IDS based on PKP by using the traditional Fiat-Shamir (FS) transform that converts Identification schemes into Signature schemes. We develop a constant time implementation of PKP-DSS. It appears that our scheme is very competitive with other post-quantum FS signature schemes. Since that PKP is an NP-Complete problem and since there are no known quantum attacks for solving PKP significantly better than classical attacks, we believe that our scheme is post-quantum secure. On the other hand, we study multivariate public-key signature schemes that provide“ultra”-short signatures. We first analyze the most known attacks against multivariate signatures, and then define the minimal parameters that allow ultra-short signature. We also design some specific newmodes of operations in order to avoid particular attacks.Second, we provide various explicit examples of ultra-short signature schemes that are based on variants of HFE. We present parameters for several level of classical security: 80, 90, 100 bits in addition to 128, 192, and 256 bits; foreach level, we propose different choices of finite fields.
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https://tel.archives-ouvertes.fr/tel-03212069
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Submitted on : Thursday, April 29, 2021 - 12:12:08 PM
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Eliane Koussa. Analysis and design of post-quantum cryptographic algorithms : PKP-based signature scheme and ultra-short multivariate signatures. Cryptography and Security [cs.CR]. Université Paris-Saclay, 2020. English. ⟨NNT : 2020UPASG027⟩. ⟨tel-03212069⟩

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