14,346 research outputs found
On the key exchange and multivariate encryption with nonlinear polynomial maps of stable degree
We say that the sequence g^n, n 3, n ! 1 of polynomial transformation bijective mapsof free module K^n over commutative ring K is a sequence of stable degree if the order of g^n is growingwith n and the degree of each nonidentical polynomial map of kind g^n^k^ is an independent constant c.Transformation b = tgn
Quantum Theory of Reactive Scattering in Phase Space
We review recent results on quantum reactive scattering from a phase space
perspective. The approach uses classical and quantum versions of normal form
theory and the perspective of dynamical systems theory. Over the past ten years
the classical normal form theory has provided a method for realizing the phase
space structures that are responsible for determining reactions in high
dimensional Hamiltonian systems. This has led to the understanding that a new
(to reaction dynamics) type of phase space structure, a {\em normally
hyperbolic invariant manifold} (or, NHIM) is the "anchor" on which the phase
space structures governing reaction dynamics are built. The quantum normal form
theory provides a method for quantizing these phase space structures through
the use of the Weyl quantization procedure. We show that this approach provides
a solution of the time-independent Schr\"odinger equation leading to a (local)
S-matrix in a neighborhood of the saddle point governing the reaction. It
follows easily that the quantization of the directional flux through the
dividing surface with the properties noted above is a flux operator that can be
expressed in a "closed form". Moreover, from the local S-matrix we easily
obtain an expression for the cumulative reactio probability (CRP).
Significantly, the expression for the CRP can be evaluated without the need to
compute classical trajectories. The quantization of the NHIM is shown to lead
to the activated complex, and the lifetimes of quantum states initialized on
the NHIM correspond to the Gamov-Siegert resonances. We apply these results to
the collinear nitrogen exchange reaction and a three degree-of-freedom system
corresponding to an Eckart barrier coupled to two Morse oscillators.Comment: 59 pages, 13 figure
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
On the key expansion of D(n, K)-based cryptographical algorithm
The family of algebraic graphs D(n, K) defined over finite commutative ring K have been used in different cryptographical algorithms (private and public keys, key exchange protocols). The encryption maps correspond to special walks on this graph. We expand the class of encryption maps via the use of edge transitive automorphism group G(n, K) of D(n, K). The graph D(n, K) and related directed graphs are disconnected. So private keys corresponding to walks preserve each connected component. The group G(n, K) of transformations generated by an expanded set of encryption maps acts transitively on the plainspace. Thus we have a great difference with block ciphers, any plaintexts can be transformed to an arbitrarily chosen ciphertex by an encryption map. The plainspace for the D(n, K) graph based encryption is a free module P over the ring K. The group G(n, K) is a subgroup of Cremona group of all polynomial automorphisms. The maximal degree for a polynomial from G(n, K) is 3. We discuss the Diffie-Hellman algorithm based on the discrete logarithm problem for the group τ-1Gτ, where τ is invertible affine transformation of free module P i.e. polynomial automorphism of degree 1. We consider some relations for the discrete logarithm problem for G(n, K) and public key algorithm based on the D(n, K) graphs
Notes on nonabelian (0,2) theories and dualities
In this paper we explore basic aspects of nonabelian (0,2) GLSM's in two
dimensions for unitary gauge groups, an arena that until recently has largely
been unexplored. We begin by discussing general aspects of (0,2) theories,
including checks of dynamical supersymmetry breaking, spectators and weak
coupling limits, and also build some toy models of (0,2) theories for bundles
on Grassmannians, which gives us an opportunity to relate physical anomalies
and trace conditions to mathematical properties. We apply these ideas to study
(0,2) theories on Pfaffians, applying recent perturbative constructions of
Pfaffians of Jockers et al. We discuss how existing dualities in (2,2)
nonabelian gauge theories have a simple mathematical understanding, and make
predictions for additional dualities in (2,2) and (0,2) gauge theories.
Finally, we outline how duality works in open strings in unitary gauge
theories, and also describe why, in general terms, we expect analogous
dualities in (0,2) theories to be comparatively rare.Comment: 93 pages, LaTeX; v2: typos fixe
Equilibria, Fixed Points, and Complexity Classes
Many models from a variety of areas involve the computation of an equilibrium
or fixed point of some kind. Examples include Nash equilibria in games; market
equilibria; computing optimal strategies and the values of competitive games
(stochastic and other games); stable configurations of neural networks;
analysing basic stochastic models for evolution like branching processes and
for language like stochastic context-free grammars; and models that incorporate
the basic primitives of probability and recursion like recursive Markov chains.
It is not known whether these problems can be solved in polynomial time. There
are certain common computational principles underlying different types of
equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP.
Representative complete problems for these classes are respectively, pure Nash
equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria
in 2-player normal form games, and (mixed) Nash equilibria in normal form games
with 3 (or more) players. This paper reviews the underlying computational
principles and the corresponding classes
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
Dynamical systems as the main instrument for the constructions of new quadratic families and their usage in cryptography
Let K be a finite commutative ring and f = f(n) a bijective polynomial map f(n) of the Cartesian power K^n onto itself of a small degree c and of a large order. Let f^y be a multiple composition of f with itself in the group of all polynomial automorphisms, of free module K^n. The discrete logarithm problem with the pseudorandom base f(n) (solvef^y = b for y) is a hard task if n is sufficiently large. We will use families of algebraic graphs defined over K and corresponding dynamical systems for the explicit constructions of such maps f(n) of a large order with c = 2 such that all nonidentical powers f^y are quadratic polynomial maps. The above mentioned result is used in the cryptographical algorithms based on the maps f(n) – in the symbolic key exchange protocols and public keys algorithms
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