Candidate for practical post-quantum signature scheme

Abstract

A new criterion of post-quantum security is used to design a practical signature scheme based on the computational complexity of the hidden discrete logarithm problem. A 4-dimensional finite non-commutative associative algebra is applied as algebraic support of the cryptoscheme. The criterion is formulated as computational intractability of the task of constructing a periodic function containing a period depending on the discrete logarithm value. To meet the criterion, the hidden commutative group possessing the 2-dimensional cyclicity is exploited in the developed signature scheme. The public-key elements are computed depending on two vectors that are generators of two different cyclic groups contained in the hidden group. When computing the public key two types of masking operations are used: i) possessing the property of mutual commutativity with the exponentiation operation and ii) being free of such property. The signature represents two integers and one vector S used as a multiplier in the verification equation. To prevent attacks using the value S as a fitting element the signature verification equation is doubled.

A novel method for developing post-quantum cryptoschemes and a practical signature algorithm

“…As a substantiation of resistance to quantum attacks, it should be noted that the proposed signature scheme satisfies the general criterion of post-quantum security used to develop HDLP-based signature schemes described in the papers [12][13][14]. The mentioned criterion is formulated as follows [12]: "Based on the public parameters of the signature scheme, the construction of a periodic function containing a period with the length depending on the discrete logarithm value should be a computationally intractable task."…”

“…As a substantiation of resistance to quantum attacks, it should be noted that the proposed signature scheme satisfies the general criterion of post-quantum security used to develop HDLP-based signature schemes described in the papers [12][13][14]. The mentioned criterion is formulated as follows [12]: "Based on the public parameters of the signature scheme, the construction of a periodic function containing a period with the length depending on the discrete logarithm value should be a computationally intractable task." The fulfillment of this criterion in the developed signature scheme is ensured by the fact that the elements of the first (second) public key form the basis of a primary group of the order q 3 in the first (second) algebra used as an algebraic carrier, therefore, all possible products Y i 1 Z j 1 U k 1 in the first FAA and Y i 2 Z j 2 U k 2 the second FAA for i, j, k 5 0, 1, 2 .…”

“…The technique of doubling the verification equation when designing a signature scheme was previously used in [12,14], but in the proposed method it is extended to the case of using two different algebras as a single algebraic carrier of the signature scheme. At the same time, it has a new purpose, which is to provide binding of public key elements to a fixed hidden group in each of the used algebras.…”

“…The finalists in the category of post-quantum digital signatures were Falcon [17] and Crystals-Dilithium [16], and Rainbow [18]. It is interesting to compare the proposed signature scheme with the finalists, with other HDLP-based signatures [12,14], and with 2048-bit RSA signature algorithm [19]. Table 3 presents a rough comparison which uses the published results of comparing the performance of the finalists with each other and with the algorithm RSA-2048.…”

“…One of attractive approaches to the development of post-quantum signature algorithm relates to exploiting computational difficulty of the so-called hidden discrete logarithm problem (HDLP) defined usually in non-commutative finite associative algebras (FAAs). Different forms of the HDLP were used to develop signature algorithms on non-commutative FAAs [7,12,13]. For the first time, a HDLP-based signature algorithm on a commutative FAA was proposed in [14].…”