Group key management based on semigroup actions

In this work we provide a suite of protocols for group key management based on general semigroup actions. Construction of the key is made in a distributed and collaborative way. Examples are provided that may in some cases enhance the security level and communication overheads of previous existing protocols. Security against passive attacks is considered and depends on the hardness of the semigroup action problem in any particular scenario.Comment: accepted for publication in Journal of algebra and its application

Про нові потоковi алгоритми створення чутливих дайджестiв електронних документів

Для прийняття обґрунтованих планових рішень у суспільно-економічній сфері спеціалісти повинні користуватися перевіреними документами. До засобів перевірки документів належать криптографічно стабільні алгоритми компресії великого файлу в дайджест визначеного розміру, чутливий до будь-якої зміни символів на вході. Пропонуються нові швидкі алгоритми компресії, криптографічна стабільність яких пов’язується зі складними алгебраїчними проблемами, такими як дослідження систем алгебраїчних рівнянь великої степені та задача розкладу нелінійного відображення простору за твірними. Запропоновані алгоритми створення чутливих до змін дайджестів документів будуть використані для виявлення кібератак та аудиту усіх файлів системи після зареєстрованого втручання.Specialists must use well checked documents to elaborate well founded,decisions and plans in the socio-economic field. Check tools include cryptographically stable algorithms for compressing a large file into a digest of a specified size, sensitive to any change in the characters on the input. New fast compression algorithms are proposed, whose cryptographic stability is associated with complex algebraic problems, such as the study of systems of algebraic equations of large power and the problem of the expansion of nonlinear mapping of space by generators. The proposed algorithms for creation of change-sensitive digests will be used to detect cyberattacks and audit all system files after a registered intervention

On semigroups of multiplicative Cremona transformations and new solutions of Post Quantum Cryptography.

Noncommutative cryptography is based on the applications of algebraic structures like noncommutative groups, semigroups and noncommutative rings. Its intersection with Multivariate cryptography contains studies of cryptographic applications of subsemigroups and subgroups of affine Cremona semigroups defined over finite commutative ring K. We consider special semigroups of transformations of the variety (K*)^n, K=F_q or K=Z_m defined via multiplications of variables. Efficiently computed homomorphisms between such subsemigroups can be used in Post Quantum protocols schemes and their inverse versions when correspondents elaborate mutually inverse transformations of (K*)n. The security of these schemes is based on a complexity of decomposition problem for element of the semigroup into product of given generators. So the proposed algorithms are strong candidates for their usage in postquantum technologies

On effective computations in special subsemigroups of polynomial transformations and protocol based multivariate cryptosystems

Large semigroups and groups of transformations of finite affine space of dimension n with the option of computability of the composition of n arbitrarily chosen elements in polynomial time are described in the paper. Constructions of such families are given together with effectively computed homomorphisms between members of the family. These algebraic platforms allow us to define protocols for several generators of subsemigroup of affine Cremona semigroups with several outputs. Security of these protocols rests on the complexity of the word decomposition problem, It allows to introduce algebraic protocols expanded to cryptosystems of El Gamal type which are not a public key system. In particular symbiotic combination of these protocol of Noncommutative cryptography with one time pad encryption is given. Some of these nonclassical multivariate cryptosystems are implemented with platforms of cubical transformations

On Extremal Expanding Algebraic Graphs and post-quantum secure delivery of passwords, encryption maps and tools for multivariate digital signatures.

Expanding graphs are known due to their remarkable applications to Computer Science. We are looking for their applications to Post Quantum Cryptography. One of them is postquantum analog of Diffie-Hellman protocol in the area of intersection of Noncommutative and Multivariate Cryptographies .This graph based protocol allows correspondents to elaborate collision cubic transformations of affine space Kn defined over finite commutative ring K. Security of this protocol rests on the complexity of decomposition problem of nonlinear polynomial map into given generators. We show that expanding graphs allow to use such output as a ‘’seed’’ for secure construction of infinite sequence of cubic transformation of affine spaces of increasing dimension. Correspondents can use the sequence of maps for extracting passwords for one time pads in alphabet K and other symmetric or asymmetric algorithms. We show that cubic polynomial maps of affine spaces of prescribed dimension can be used for transition of quadratic public keys of Multivariate Cryptography into the shadow of private areas

On short digital signatures with Eulerian transformations

Let n stands for the length of digital signatures with quadratic multivariate public rule in n variables. We construct postquantum secure procedure to sign O(n^t), t ≥1 digital documents with the signature of size n in time O(n^{3+t}). It allows to sign O(n^t), t 1 documents of size n in time O(n^{t+3}), t>1. The multivariate encryption map has linear degree O(n) and density O(n^4). We discuss the idea of public key with Eulerian transformations which allows to sign O(n^t), t≥0 documents in time O(n^{t+2}). The idea of delivery and usage of several Eulerian and quadratic transformations is also discussed

On semigroups of multivariate transformations constructed in terms of time dependent linguistic graphs and solutions of Post Quantum Multivariate Cryptography.

Time dependent linguistic graphs over abelian group H are introduced. In the case $H=K*$ such bipartite graph with point set $P=H^n$ can be used for generation of Eulerian transformation of $(K*)^n$, i.e. the endomorphism of $K[x_1, x_2,… , x_n]$ sending each variable to a monomial term. Subsemigroups of such endomorphisms together with their special homomorphic images are used as platforms of cryptographic protocols of noncommutative cryptography. The security of these protocol is evaluated via complexity of hard problem of decomposition of Eulerian transformation into the product of known generators of the semigroup. Nowadays the problem is intractable one in the Postquantum setting. The symbiotic combination of such protocols with special graph based stream ciphers working with plaintext space of kind $K^m$ where $m=n^t$ for arbitrarily chosen parameter $t$ is proposed. This way we obtained a cryptosystem with encryption/decryption procedure of complexity $O(m^{1+2/t})$

On inverse protocols of Post Quantum Cryptography based on pairs of noncommutative multivariate platforms used in tandem

Non-commutative cryptography studies cryptographic primitives and systems which are based on algebraic structures like groups, semigroups and noncommutative rings. We con-tinue to investigate inverse protocols of Non-commutative cryptography defined in terms of subsemigroups of Affine Cremona Semigroups over finite fields or arithmetic rings $Z_m$ and homomorphic images of these semigroups as possible instruments of Post Quantum Cryptography. This approach allows to construct cryptosystems which are not public keys, as outputs of the protocol correspondents receive mutually inverse transformations on affine space $K^n$ or variety $(K^*)^n$, where $K$ is a field or an arithmetic ring. The security of such inverse protocol rests on the complexity of word problem to decompose element of Affine Cremona Semigroup given in its standard form into composition of given generators. We discuss the idea of the usage of combinations of two cryptosystems with cipherspaces $(K^*)^n$ and $K^n$ to form a new cryptosystem with the plainspace $(K^*)^n$, ciphertext $K^n$ and nonbijective highly nonlinear encryption map

On Noncommutative Cryptography and homomorphism of stable cubical multivariate transformation groups of infinite dimensional affine spaces

Noncommutative cryptography is based on applications of algebraic structures like noncommutative groups, semigroups and non-commutative rings. Its inter-section with Multivariate cryptography contains studies of cryptographic applications of subsemigroups and subgroups of affine Cremona semigroups defined overfinite commutative rings. Efficiently computed homomorphisms between stable subsemigroups of affine Cremona semigroups can be used in tame homomorphisms protocols schemes and their inverse versions. The implementation scheme with the sequence of subgroups of affine Cremona group, which defines projective limit was already suggested. We present the implementation of other scheme which uses two projective limits which define two different infinite groups and the homomorphism between them. The security of corresponding algorithm is based on a complexity of decomposition problem for an element of affine Cremona semigroup into product of given generators. These algorithms may be used in postquantum technologies

On Multivariate Algorithms of Digital Signatures on Secure El Gamal Type Mode.

The intersection of Non-commutative and Multivariate cryptography contains studies of cryptographic applications of subsemigroups and subgroups of affine Cremona semigroups defined over finite commutative ring K with the unit. We consider special subsemigroups (platforms) in a semigroup of all endomorphisms of K[x_1, x_2, …, x_n]. Efficiently computed homomorphisms between such platforms can be used in Post Quantum key exchange protocols when correspondents elaborate common transformation of (K*)^n. The security of these schemes is based on a complexity of decomposition problem for an element of a semigroup into a product of given generators. We suggest three such protocols (with a group and with two semigroups as platforms) for their usage with multivariate digital signatures systems. The usage of protocols allows to convert public maps of these systems into private mode, i.e. one correspondent uses the collision map for safe transfer of selected multivariate rule to his/her partner. The ‘’ privatisation’’ of former publicly given map allows the usage of digital signature system for which some of cryptanalytic instruments were found ( estimation of different attacks on rainbow oil and vinegar system, cryptanalytic studies LUOV) with the essentially smaller size of hashed messages. Transition of basic multivariate map to safe El Gamal type mode does not allow the usage of cryptanalytic algorithms for already broken Imai - Matsumoto cryptosystem or Original Oil and Vinegar signature schemes proposed by J.Patarin. So even broken digital signatures schemes can be used in the combination with protocol execution during some restricted ‘’trust interval’’ of polynomial size. Minimal trust interval can be chosen as a dimension n of the space of hashed messages, i. e. transported safely multivariate map has to be used at most n times. Before the end of this interval correspondents have to start the session of multivariate protocol with modified multivariate map. The security of such algorithms rests not on properties of quadratic multivariate maps but on the security of the protocol for the map delivery and corresponding NP hard problem