770 research outputs found
On the key exchange with new cubical maps based on graphs
Families of edge transitive algebraic graphs Fn(K), over the commutative ring K were used for the graph based cryptographic algorithms. We introduce a key exchange protocol defined in terms of bipartite graph An(K), n ≥ 2 with point set Pn and line set Ln isomorphic to n-dimensional free module Kn. Graphs A(n, K) are not vertex and edge transitive. There is a well defined projective limit lim A(n, K) = A(K), n → ∞ which is an infinite bipatrtite graph with point set P = lim Pn and line set L = limLn. Let K be a commutative ring contain at least 3 regular elements (not zero divisors). For each pair of (n, d), n ≥ 2, n ≥ 1 and sequence of elements α1, α2, …, α2d, such that α1, αi+αi+1, i = 1, 2, …, 2d, i = 1, 2, … 2d-1 and α2d+α1 are regular elements of the ring K. We define polynomial automorphism hn = hn (d, α1, α2, …, α2d) of variety Ln (or Pn). The existence of projective limit lim An(K) guarantees the existence of projective limit h = h(d, α1, α2, …, α2d) = lim hn, n → ∞ which is cubical automorphism of infinite dimensional varieties L (or P). We state that the order of h is an infinity. There is a constant n0 such that hn, n ≥ n0 is a cubical map. Obviously the order of hn is growing with the growth of n and the degree of polynomial map (hn)k from the Cremona group of all polynomial automorphisms of free module Kn with operation of composition is bounded by 3. Let τ be affine automorphism of Kn i.e. the element of Cremona group of degree 1. We suggest symbolic Diffie Hellman key exchange with the use of cyclic subgroup of Cremona group generated by τ-1hnτ. In the case of K = Fp, p is prime, the order of hn is the power of p. So the order is growing with the growth of p. We use computer simulation to evaluate the orders in some cases of K = Zm, where m is a composite integer.Show Reference
On the family of cubical multivariate cryptosystems based on the algebraic graph over finite commutative rings of characteristic 2
The family of algebraic graphs A(n;K) defined over the finite commutative ring K were used for the design of different multivariate cryptographical algorithms (private and public keys, key exchange protocols). The encryption map corresponds to a special walk on this graph. We expand the class of encryption maps via the use of an automorphism group of A(n;K). In the case of characteristic 2 the encryption transformation is a Boolean map. We change finite field for the commutative ring of characteristic 2 and consider some modifications of algorithm which allow to hide a ground commutative ring
On New Examples of Families of Multivariate Stable Maps and their Cryptographical Applications
Let K be a general finite commutative ring. We refer to a familyg^n, n = 1; 2;... of bijective polynomial multivariate maps of K^n as a family with invertible decomposition gn = g^1^n g^2^n...g^k^n , such that the knowledge of the composition of g^2^nallows computation of g^2^n for O(n^s) (s > 0) elementary steps. Apolynomial map g is stable if all non-identical elements of kind g^t, t > 0 are of the same degree.We construct a new family of stable elements with invertible decomposition.This is the first construction of the family of maps based on walks on the bipartitealgebraic graphs defined over K, which are not edge transitive. We describe theapplication of the above mentioned construction for the development of streamciphers, public key algorithms and key exchange protocols. The absence of edgetransitive group essentially complicates cryptanalysis
Internal Parametricity for Cubical Type Theory
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, and we give an account of the identity extension lemma for internal parametricity
Homology groups for particles on one-connected graphs
We present a mathematical framework for describing the topology of
configuration spaces for particles on one-connected graphs. In particular, we
compute the homology groups over integers for different classes of
one-connected graphs. Our approach is based on some fundamental combinatorial
properties of the configuration spaces, Mayer-Vietoris sequences for different
parts of configuration spaces and some limited use of discrete Morse theory. As
one of the results, we derive a closed-form formulae for ranks of the homology
groups for indistinguishable particles on tree graphs. We also give a detailed
discussion of the second homology group of the configuration space of both
distinguishable and indistinguishable particles. Our motivation is the search
for new kinds of quantum statistics.Comment: 26 pages, 16 figure
On the Generator of Stable Cubical Multivariate Encryption Maps Over Boolean Rings for Protection of Large Information System
Encryption based on Walks in Algebraic GRAphs (EWAGRA) is used for protection of authors' rights, access to electronic books or documents located at a certain knowledge base (Information Quality Assurance Support Systems of a university, digital library supporting distance education, various digital archives and etc). The method allows generating nonlinear stream ciphers, which have some similarities with a one-time pad: different keys produce distinct ciphertexts from the same plaintext. In contrast to the case of a one-time pad, the length of the key is flexible and the encryption map is a nonlinear poly- nomial map, which order is growing with the growth of the dimension n of the plaintext space. The encryption has good resistance to attacks of the adversary when he has no access to plaintext space or has a rather small number of intercepted plaintext- ciphertext pairs. It is known that encryption and decryption maps are cubical maps. So, interception of n3 + O(n) plaintext-ciphertext pairs allows conducting a plain linearization attack for finding the inverse map. We consider the idea of the modification of this encryption algorithm after sending each message without using key exchange protocols. So the new algorithm is resistant to plain linearization attacks
Reconstructibility of matroid polytopes
We specify what is meant for a polytope to be reconstructible from its graph
or dual graph. And we introduce the problem of class reconstructibility, i.e.,
the face lattice of the polytope can be determined from the (dual) graph within
a given class. We provide examples of cubical polytopes that are not
reconstructible from their dual graphs. Furthermore, we show that matroid
(base) polytopes are not reconstructible from their graphs and not class
reconstructible from their dual graphs; our counterexamples include
hypersimplices. Additionally, we prove that matroid polytopes are class
reconstructible from their graphs, and we present a algorithm that
computes the vertices of a matroid polytope from its -vertex graph.
Moreover, our proof includes a characterisation of all matroids with isomorphic
basis exchange graphs.Comment: 22 pages, 5 figure
On the vanishing of discrete singular cubical homology for graphs
We prove that if G is a graph without 3-cycles and 4-cycles, then the
discrete cubical homology of G is trivial in dimension d, for all d\ge 2. We
also construct a sequence { G_d } of graphs such that this homology is
non-trivial in dimension d for d\ge 1. Finally, we show that the discrete
cubical homology induced by certain coverings of G equals the ordinary singular
homology of a 2-dimensional cell complex built from G, although in general it
differs from the discrete cubical homology of the graph as a whole.Comment: Minor changes, background information adde
Reconstructibility of matroid polytopes
We specify what is meant for a polytope to be reconstructible from its graph or dual graph, and we introduce the problem of class reconstructibility; i.e., the face lattice of the polytope can be determined from the (dual) graph within a given class. We provide examples of cubical polytopes that are not reconstructible from their dual graphs. Furthermore, we show that matroid (base) polytopes are not reconstructible from their graphs and not class reconstructible from their dual graphs; our counterexamples include hypersimplices. Additionally, we prove that matroid polytopes are class reconstructible from their graphs, and we present an O(n3) algorithm that computes the vertices of a matroid polytope from its n-vertex graph. Moreover, our proof includes a characterization of all matroids with isomorphic basis exchange graphs. © 2022 Society for Industrial and Applied Mathematic
Quasi-isometric groups with no common model geometry
A simple surface amalgam is the union of a finite collection of surfaces with
precisely one boundary component each and which have their boundary curves
identified. We prove if two fundamental groups of simple surface amalgams act
properly and cocompactly by isometries on the same proper geodesic metric
space, then the groups are commensurable. Consequently, there are infinitely
many fundamental groups of simple surface amalgams that are quasi-isometric,
but which do not act properly and cocompactly on the same proper geodesic
metric space.Comment: v2: 19 pages, 6 figures; minor changes. To appear in Journal of the
London Mathematical Societ
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