Finite Non-commutative Algebras Research Articles

A new type of digital signature algorithms with a hidden group

The known designs of digital signature schemes with a hidden group, which use finite non-commutative algebras as algebraic support, are based on the computational complexity of the so-called hidden discrete logarithm problem. A similar design, used to develop a signature algorithm based on the difficulty of solving a system of many quadratic equations in many variables, is introduced. The significant advantage of the proposed method compared with multivariate-cryptography signature algorithms is that the said system of equations, which occurs as the result of performing the exponentiation operations in the hidden group, has a random look and is specified in a finite field of a higher order. This provides the ability to develop post-quantum signature schemes with significantly smaller public-key sizes at a given level of security.

Post-quantum signature algorithms on noncommutative algebras, using difficulty of solving systems of quadratic equations

A recently proposed new concept for constructing algebraic signature schemes with a hidden group is used to develop two new post-quantum signature algorithms on four-dimensional and six-dimensional finite noncommutative associative algebras. As in the case of the post-quantum signature schemes of multivariate cryptography, the security of the introduced algorithms is based on the computational difficulty of solving systems of many quadratic equations (44 and 42) with many unknowns (40 and 36). The signature represents a pair of a natural number e and a vector S. The latter enters three times in the verification equation used, providing resistance to the forging signature attacks by using the value S as a fitting parameter. A public key is generated in the form of a set of vectors, each of which is calculated as the product of triples of secret vectors. With a special choice of these triples, a signer has the possibility of calculating a signature that satisfies the verification equation. The developed post-quantum signature algorithms are practical, having sufficiently small signatures (97 and 109 bytes), public keys (387 and 291 bytes), and secret keys (315 and 451 bytes). A significant difference from the public-key algorithms of multivariate cryptography is that in the developed signature schemes, the system of quadratic equations is derived from the formulas for generating the public-key elements in the form of a set of vectors of m-dimensional finite noncommutative algebra with an associative vector multiplication operation. The formulas define the system of n quadratic vector equations, which reduces to the system of m quadratic equations over a finite field.

A New Method for Developing Signature Algorithms on Finite Non-commutative Algebras

A new method for developing signature schemes on finite non-commutative associative algebras is introduced. A signature algorithm is developed on a 4-dimensional algebra defined over the ground field $GF(p)$. The public key element and one of the signature elements represent vectors calculated using exponentiation operations in a hidden commutative group. Decomposition of the algebra into commutative subalgebras is taken into account while designing the algorithm. The method extends the class of algebraic digital signature schemes and opens up the possibility of developing a number of practical post-quantum digital signature algorithms, the main merit of which is comparatively small size of the public key, secret key, and signature.

Structure of a finite non-commutative algebra set by a sparse multiplication table

Four-dimensional finite non-commutative associative algebras represent practical interest as algebraic support of post-quantum digital signature algorithms, especially algebras with two sided global unit, set by sparse basis vectors multiplication tables. A new algebra of the latter type, set over the field GF(p), is proposed and its structure is investigated. The studied algebra is described as a set of p2 + p + 1 commutative subalgebras of three different types. All subalgebras intersect strictly in the subset of scalar vectors. Formulas are derived for the number of subalgebras of each type.

Алгоритмы ЭЦП на конечных некоммутативных алгебрах над полями характеристики два

Abstract Purpose of work is the increase in performance and decrease in the hardware implementation cost of the of post-quantum algebraic signature algorithms based on the computational difficulty of solving systems of many quadratic equations with many unknowns. Research method is i) the development of post-quantum signature algorithms on finite non-commutative associative algebras defined over finite fields of characteristic two, which have high performance and small sizes of signature and public and secret keys; ii) using the concept of constructing algebraic signature algorithms with a hidden commutative group, characterized by the use of a power-type vector verification equation with multiple occurrences of the signature S as a factor; iii) the choice of the degree of extension z of the field GF(2z) in which the order of the hidden group is divisible only by prime divisors of at least 24 bits. Results of the study are the formulated main provisions for the implementation of post-quantum signature algorithms with a hidden group, the security of which is based on the computational difficulty of solving systems of many quadratic equations with many unknowns, when using the finite non-commutative algebras given over the GF(2z) fields as algebraic support. The values of the extension degree z are established for which the order of the hidden commutative group is divisible only by prime divisors of a sufficiently large size. A new post-quantum signature algorithm with relatively high performance and small sizes of the signature and public and secret keys have been developed. Using an informal security index in the form of a product of the binary logarithm of the order of the field and the number of unknowns, the developed and known post-quantum algorithms for a given level of security are compared. Practical relevance. The main provisions for constructing signature algorithms with a hidden group are formulated for the case of using finite non-commutative algebras with computationally efficient operations of multiplication and exponentiation, providing prerequisites for improving performance and reducing the hardware implementation cost of post-quantum signature algorithms.

Алгебраические алгоритмы ЭЦП, основанные на трудности решения систем уравнений

Purpose of work is the development of post-quantum digital signature algorithms with comparatively small sizes of the public and secret keys and the signature. Research method is the use of a new concept for constructing signature algorithms on finite non-commutative associative algebras, which is distinguished by the multiple occurrences of the signature S in the power verification equation. A public key is generated in the form of a set of vectors every of which is calculated as the product of triples of secret vectors. With a special choice of these triples, it is possible to calculate a signature that satisfies the verification equation. Results of the study are two developed algebraic post-quantum digital signature algorithms of a new type, security of which is based on the computational difficulty of solving systems of many quadratic equations with many unknowns. The difference from the public-key algorithms of multivariate cryptography is that the system of quadratic equations is derived from the formulas for generating the public-key elements in the form of a set of vectors of m-dimensional finite non-commutative algebra with an associative vector multiplication operation. The said formulas define the system of n quadratic vector equations, which reduces to the system of mn quadratic equations over a finite field. Thanks to the “natural” mechanism for the occurrence of the specified system, it is set above the field, the order of which has a large size (97 and 129 bits). The used procedures for generating the public key and signature include the exponentiation operations to the degree of a large size (96 and 128 bits), which are performed over the elements of the secret (hidden) commutative group contained in the algebra. The signature is formed in the form of two elements: a randomizing natural number e and a “fitting” vector S. The signature authentication equation includes a multiple occurrence of the S element and every entry of the vector S is associated with the formation of a product that is exponentiated to a degree dependent on the value of the e element. A significant reduction in the size of public and secret keys and signatures has been achieved, as well as an increase in performance compared to foreign analogues, considered currently as basic algorithms for the adoption of post-quantum digital signature standards.

Digital Signature Scheme with Hidden Group Possessing Two-Dimensional Cyclicity

A method is proposed for constructing digital signature schemes based on the hidden discrete logarithm problem, which meet ageneral criterion of post-quantum resistance. The method provides a relatively small size of the public key and signature. Based on the method, a practical digital signature scheme has been developed, in which the exponentiation operation in a hidden group with two-dimensional cyclicity is the basic cryptographic primitive. The algebraic support of a cryptoscheme is a four-dimensional finite non-commutative algebra with associative multiplication operation. By specifying algebra using abasis vector multiplication table with half of empty cells, the performance of signature generation and authentication procedures is improved. A public key is a triple of four-dimensional vectors calculated as images of elements of a hidden group which are mapped using two types of masking operations: 1) mutually commutative with the exponentiation operation and 2) not having this property.

Cryptosystems based on RS and BCH codes over finite noncommutative algebras

The purpose of this paper is to introduce new cryptosystems based on linear Reed-Solomon (RC) and Bose-Chaudhuri-Hocquenghem (BCH) codes over finite Cayley-Dickson and finite Clifford algebras with fast code and encode procedures based on fast Fourier- Clifford-Galois transforms

MAXIMUM NUMBER OF SOLUTIONS OF xq = x IN A FINITE NONCOMMUTATIVE ALGEBRA

We obtain a bound on the number of solutions of xq = x in a finite noncommutative algebra over a field with q elements. Furthermore, we completely characterize those rings for which this maximum number is attained.