Space Kn Research Articles

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 continue to investigate inverse protocols of Non-commutative cryptography defined in terms of subsemigroups of Affine Cremona Semigroups over finite fields or arithmetic rings Zm and homomorphic images of these semigroups as possible instruments of Post Quantum Cryptography. This approach allows to construct cryptosystem which are not public keys, when protocol finish correspondents have mutually inverse transformations on affine space Kn or variety (K*)n where K is the field or 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 usage combinations of two cryptosystems with cipherspaces(K*)n and Kn to form a new cryptosystem with the plainspace(K*)n, ciphertextKn and nonbijective highly nonlinear encryption map.
 
 

On security of GIS systems with N-tier architecture and family of graph based ciphers

Discovery of q-regular tree description in terms of an infinite system of quadratic equations over finite field Fq had an impact on Computer Science, theory of graph based cryptographic algorithms in particular. It stimulates the development of new graph based stream ciphers. It turns out that such encryption instruments can be efficiently used in GIS protection systems which use N-Tier Architecture. We observe known encryption algorithms based on the approximations of regular tree, their modifications defined over arithmetical rings and implementations of these ciphers. Additionally new more secure graph based ciphers suitable for GIS protection will be presented.Algorithms are constructed using vertices of bipartite regular graphs D(n,K) defined by a finite commutative ring K with a unit and a non-trivial multiplicative group K*. The partition of such graphs are n-dimensional affine spaces over the ring K. A walk of even length determines the transformation of the transition from the initial to the last vertex from one of the partitions of the graph. Therefore, the affine space Kn can be considered as a space of plaintexts, and walking on the graph is a password that defines the encryption transformation.With certain restrictions on the password the effect when different passwords with K*)2s, s <[(n+5)/2]/2 correspond to different ciphertexts of the selected plaintext with Kn can be achieved.In 2005, such an algorithm in the case of the finite field F127 was implemented for the GIS protection. Since then, the properties of encryption algorithms using D(n, K) graphs (execution speed, mixing properties, degree and density of the polynomial encryption transform) have been thoroughly investigated.The complexity of linearization attacks was evaluated and modifications of these algorithms with the resistance to linearization attacks were found. It turned out that together with D(n, K) graphs, other algebraic graphs with similar properties, such as A(n, K) graphs, can be effectively used.The article considers several solutions to the problem of protecting the geological information system from possible cyberattacks using stream ciphers based on graphs. They have significant advantages compared to the implemented earlier systems.

On computations with double Schubert automaton and stable maps of multivariate cryptography

The families of bijective transformations Gn of affine space Kn over general commutative ring K of increasing order with the property of stability will be constructed. Stability means that maximal degree of elements of cyclic subgroup generated by the transformation of degreed is bounded by d. In the case K = Fq these transformations of Kncan be of an exponential order. We introduce large groups formed by quadratic transformations and numerical encryption algorithm protected by secure protocol of Noncommutative Cryptography. The construction of transformations is presented in terms of walks on Double Schubert Graphs.

Polynomial Realizations of Finite-Dimensional Lie Algebras

It is proved that any finite-dimensional Lie algebra over an algebraically closed field K of characteristic 0 can be embedded (realized) as a transitive Lie subalgebra of the Lie algebra of polynomial vector fields on the space Kn. The same is also proved for arbitrary real Lie algebras and in some other cases.

Vanishing Ideals Over Odd Cycles

Abstract Let K be a finite field. Let X * be a subset of the a ne space Kn , which is parameterized by odd cycles. In this paper we give an explicit Gröbner basis for the vanishing ideal, I(X *), of X *. We give an explicit formula for the regularity of I(X *) and finally if X * is parameterized by an odd cycle of length k, we show that the Hilbert function of the vanishing ideal of X * can be written as linear combination of Hilbert functions of degenerate torus.

On desynchronised multivariate algorithms of El Gamal type for stable semigroups of affine Cremona group

Families of stable cyclic groups of nonlinear polynomial transformations of affine spaces Kn over general commutative ring K of increasing with n order can be used in the key exchange protocols and related to them El Gamal multivariate cryptosystems. To use high degree of noncommutativity of affine Cremona group correspondents have to modify multivariate El Gamal algorithm via the usage of conjugations for two polynomials of kind gk and g−1 given by key holder (Alice) or giving them as elements of different transformation groups. The idea of hidden tame homomorphism and comlexity of decomposition of polynomial transwormation into word of elements of Cremona semigroup can be used. We suggest usage of new explicit constructions of infinite families of large stable subsemigroups of affine Cremona group of bounded degree as instruments of multivariate key exchange protocols. Recent results on generation of families of stable transformations of small degree and density via technique of symbolic walks on algebraic graphs are observed. Some of them used for the implementation of schemes as above with feasible computational complexity. We consider an example of a new implemented quadratic multivariate cryptosystem based on the above mentioned ideas.

Metrizable quotients of Cp-spaces

The famous Rosenthal–Lacey theorem asserts that for each infinite compact set K the Banach space C(K) admits a quotient which is either an isomorphic copy of c or ℓ2. What is the case when the uniform topology of C(K) is replaced by the pointwise topology? Is it true that Cp(X) always has an infinite-dimensional separable (or better metrizable) quotient? In this paper we prove that for a Tychonoff space X the function space Cp(X) has an infinite-dimensional metrizable quotient if X either contains an infinite discrete C⁎-embedded subspace or else X has a sequence (Kn)n∈N of infinite compact subsets such that for every n the space Kn contains two disjoint topological copies of Kn+1. Applying the latter result, we show that under ◊ there exists a zero-dimensional Efimov space K whose function space Cp(K) has an infinite-dimensional metrizable quotient. These two theorems essentially improve earlier results of Ka̧kol and Śliwa on infinite-dimensional separable quotients of Cp-spaces.

On the Families of Stable Multivariate Transformations of Large Order and Their Cryptographical Applications

Abstract Families of stable cyclic groups of nonlinear polynomial transformations of affine spaces Kn over general commutative ring K of with n increasing order can be used in the key exchange protocols and El Gamal multivariate cryptosystems related to them. We suggest to use high degree of noncommutativity of affine Cremona group and modify multivariate El Gamal algorithm via conjugations of two polynomials of kind gk and g−1 given by key holder (Alice) or giving them as elements of different transformation groups. Recent results on the existence of families of stable transformations of prescribed degree and density and exponential order over finite fields can be used for the implementation of schemes as above with feasible computational complexity.

Finite and infinite dimensional Lie group structures on Riordan groups

We introduce a Frechet Lie group structure on the Riordan group. We give a description of the corresponding Lie algebra as a vector space of infinite lower triangular matrices. We describe a natural linear action induced on the Frechet space KN by any element in the Lie algebra. We relate this to a certain family of bivariate linear partial differential equations. We obtain the solutions of such equations using one-parameter groups in the Riordan group. We show how a particular semidirect product decomposition in the Riordan group is reflected in the Lie algebra. We study the stabilizer of a formal power series under the action induced by Riordan matrices. We get stabilizers in the fraction field K((x)) using bi-infinite representations. We provide some examples. The main tool to get our results is the paper [18] where the Riordan group was described using inverse sequences of groups of finite matrices.

Radius of approximate controllability of linear retarded systems under structured perturbations

In this paper we shall deal with the problem of calculation of the radius of approximate controllability in the Banach state space Kn×L2([−hk,0],Kn) for linear retarded systems of the form ẋ(t)=A0x(t)+A1x(t−h1)+⋯+Akx(t−hk)+Bu(t). By using multi-valued linear operators we are able to derive computable formulas for this radius when the system’s coefficient matrices are subjected to structured perturbations. Some examples are provided to illustrate the obtained results.

Explicit constructions of extremal graphs and new multivariate cryptosystems

New multivariate cryptosystems are introduced. Sequences f(n) of bijective polynomial transformations of bijective multivariate transformations of affine spaces Kn, n = 2, 3, ... , where K is a finite commutative ring with special properties, are used for the constructions of cryptosystems. On axiomatic level, the concept of a family of multivariate maps with invertible decomposition is proposed. Such decomposition is used as private key in a public key infrastructure. Requirements of polynomiality of degree and density allow to estimate the complexity of encryption procedure for a public user. The concepts of stable family and family of increasing order are motivated by studies of discrete logarithm problem in Cremona group. Statement on the existence of families of multivariate maps of polynomial degree and polynomial density with the invertible decomposition is formulated. We observe known explicit constructions of special families of multivariate maps. They correspond to explicit constructions of families of nonlinear algebraic graphs of increasing girth which appeared in Extremal Graph Theory. The families are generated by pseudorandom walks on graphs. This fact ensures the existence of invertible decomposition; a certain girth property guarantees the increase of order for the family of multivariate maps, good expansion properties of families of graphs lead to good mixing properties of graph based private key algorithms. We describe the general schemes of cryptographic applications of such families (public key infrastructure, symbolic Diffie—Hellman protocol, functional versions of El Gamal algorithm).

Polynomial-time algorithms for quadratic isomorphism of polynomials: The regular case

Let f=(f1,…,fm) and g=(g1,…,gm) be two sets of m≥1 nonlinear polynomials in K[x1,…,xn] (K being a field). We consider the computational problem of finding–if any–an invertible transformation on the variables mapping f to g. The corresponding equivalence problem is known as Isomorphism of Polynomials with one Secret (IP1S) and is a fundamental problem in multivariate cryptography. Amongst its applications, we can cite Graph Isomorphism (GI) which reduces to equivalence of cubic polynomials with respect to an invertible linear change of variables, according to Agrawal and Saxena. The main result is a randomized polynomial-time algorithm for solving IP1S for quadratic instances–a particular case of importance in cryptography.To this end, we show that IP1S for quadratic polynomials can be reduced to a variant of the classical module isomorphism problem in representation theory. We show that we can essentially linearize the problem by reducing quadratic-IP1S to test the orthogonal simultaneous similarity of symmetric matrices; this latter problem was shown by Chistov, Ivanyos and Karpinski (ISSAC 1997) to be equivalent to finding an invertible matrix in the linear space Kn×n of n×n matrices over K and to compute the square root in a certain representation in a matrix algebra. While computing square roots of matrices can be done efficiently using numerical methods, it seems difficult to control the bit complexity of such methods. However, we present exact and polynomial-time algorithms for computing a representation of the square root of a matrix in Kn×n, for various fields (including finite fields), as a product of two matrices. Each coefficient of these matrices lies in an extension field of K of polynomial degree. We then consider #IP1S, the counting version of IP1S for quadratic instances. In particular, we provide a (complete) characterization of the automorphism group of homogeneous quadratic polynomials. Finally, we also consider the more general Isomorphism of Polynomials (IP) problem where we allow an invertible linear transformation on the variables and on the set of polynomials. A randomized polynomial-time algorithm for solving IP when f=(x1d,…,xnd) is presented. From an algorithmic point of view, the problem boils down to factoring the determinant of a linear matrix (i.e. a matrix whose components are linear polynomials). This extends to IP a result of Kayal obtained for PolyProj.

Internal geometry of hypersurfaces in projectively metric space

In this paper, we study the internal geometry of a hypersurface Vn−1 embedded in a projectively metric space Kn, n ≥ 3, and equipped with fields of geometric-objects \( \left\{ {G_n^i,{G_i}} \right\} \) and \( \left\{ {H_n^i,{G_i}} \right\} \) in the sense of Norden and with a field of a geometric object \( \left\{ {H_n^i,{H_n}} \right\} \) in the sense of Cartan. For example, we have proved that the projective-connection space Pn−1,n−1 induced by the equipment of the hypersurface \( {V_{n - 1}}\; \subset \;{K_n},\;n \geq 3 \), in the sense of Cartan with the field of a geometrical object \( \left\{ {H_n^i,{H_n}} \right\} \) is flat if and only if its normalization by the field of the object \( \left\{ {H_n^i,{G_i}} \right\} \) in the tangent bundle induces a Riemannian space Rn−1 of constant curvature K = −1/c.

Dual Riemannian spaces of constant curvature on a normalized hypersurface

In this paper, the following results are obtained: 1) It is proved that, in the fourth order differential neighborhood, a regular hypersurface Vn−1 embedded into a projective-metric space Kn, n ≥ 3, intrinsically induces a dual projective-metric space \( \bar K_n \). 2) An invariant analytical condition is established under which a normalization of a hypersurface Vn−1 ⊂ Kn (a tangential hypersurface \( \bar V_{n - 1} \) ⊂ \( \bar K_n \)) by quasitensor fields Hni, Hi (\( \bar H_n^i \), \( \bar H_i \)) induces a Riemannian space of constant curvature. If the two conditions are fulfilled simultaneously, the spaces Rn−1 and \( \bar R_{n - 1} \) are spaces of the same constant curvature \( K = - \tfrac{1} {c} \). 3) Geometric interpretations of the obtained analytical conditions are given.

Dual affine connections on a quadratic hyperband distribution in a projective-metric space and their applications

In this paper, we develop fundamentals of the dual theory of quadratic hyperband distributions H of m-dimensional line elements in a projective-metric space Kn (m < n − 1). In particular, we show that, on a dual normalized distribution H, there are induced two dual affine connections and indicate some applications of these connections to the geometry of m-webs on H.

Geodesic mappings between Kahlerian spaces

Geodesic mappings from a Kahlerian space Kn onto a Kahlerian space Kn will be investigated in this paper. We present a construction of Kahlerian space Kn which admits non-trivial geodesic mapping onto Kahlerian space Kn.

Formule d'homotopie entre les complexes de Hochschild et de De Rham

Let k be the field ℂ or ℝ let M be the space kn and let A be the algebra of polynomials over M. We know from Hochschild and co-workers that the Hochschild homology H·(A,A) is isomorphic to the de Rham differential forms over M: this means that the complexes (C·(A,A),b) and (Ω·(M), 0) are quasi-isomorphic. In this work, I produce a general explicit homotopy formula between those two complexes. This formula can be generalized when M is an open set in a complex manifold and A is the space of holomorphic functions over M. Then, by taking the dual maps, I find a new homotopy formula for the Hochschild cohomology of the algebra of smooth fonctions over M (when M is either a complex or a real manifold) different from the one given by De Wilde and Lecompte. I will finally show how this formula can be used to construct an homotopy for the cyclic homology.

On singular critical points of positive operators in Krein spaces

We give an example of a positive operator B in a Krein space with the following properties: the nonzero spectrum of B consists of isolated simple eigenvalues, the norms of the orthogonal spectral projections in the Krein space onto the eigenspaces of B are uniformly bounded and the point ∞ is a singular critical point of B. An operator A in the Krein space (K, [ · , · ]) is said to be positive if [Ax, x] > 0 for all nonzero x in the domain of A. A bounded positive operator A in the Krein space (K, [ · , · ]) has a projection valued spectral function E with 0 being its only possible critical point (see [1, Theorem IV.1.5] or [5, Section II.3.]). Recall that, by [5, Proposition 5.6], the condition ‖E((−∞, α])‖ ≤ C− 0 (2) is equivalent to the existence of the limit limβ↓0E([β,+∞)) in the strong operator topology. Since 0 is not an eigenvalue of a positive operator A, [5, Proposition 3.2] implies that (1) and (2) are equivalent. Also, if 0 is a critical point, it is said to be regular if one of the conditions (1) or (2) is fulfilled. If the critical point 0 is not regular, it is called singular. In the sequel the operator A considered will have a discrete spectrum outside 0. Examples of bounded positive operators inK having 0 as a singular critical point can be constructed as follows (see also the examples in [2, Section 1], [3], [4]). Consider a sequence of two-dimensional Krein spaces Kn = C with fundamental symmetry Jn = ( 1 0 0 −1 ) and positive operators An in Kn; denote by λn (λn , respectively) its positive (negative, respectively) eigenvalues and by P n (P − n , respectively) the orthogonal (in Kn) projection onto the corresponding eigenspace. Received by the editors October 15, 1998. 2000 Mathematics Subject Classification. Primary 47B50, 46C50.

A public key system with signature and master key functions

Let K be a finite field of 2m elements. Let Ø4Ø3Ø2Ø1be tame automorphisms of the n + r-dimensional affine space Kn+r . Let the composition Ø4Ø3Ø2Ø1 be π. The automorphism π and some of the Ø i ’S will be hidden. Let the component expression of π be . Let the restriction of π to a subspace be as . The field K and the polynomial map will be announced as the public key. Given a plaintext , then the ciphertext will be . Given Ø i and , it is easy to find . Therefore the plaintext can be recovered by . The private key will be the set of maps Ø4Ø3Ø2Ø1. The security of the system rests in part on the difficulty of finding the map π from the partial informations provided by the map and the factorization of the map π into a product (i.e., composition) of tame automorphisms Ø i 's.

On weak solutions of random differential inclusions

In the paper we study the existence of solutions of the random differential inclusion urn:x-wiley:20903332:media:ista802365:ista802365-math-0001 where G is a given set‐valued mapping value in the space Kn of all nonempty, compact and convex subsets of the space ℝn, and μ is some probability measure on the Borel σ‐algebra in ℝn. Under certain restrictions imposed on F and μ, we obtain weak solutions of problem (I), where the initial condition requires that the solution of (I) has a given distribution at time t = 0.