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

2017 ◽  
Vol 70 (1) ◽  
pp. 107-117 ◽  
Author(s):  
Vasyl Ustimenko
Keyword(s):  
Computational Complexity ◽  
Commutative Ring ◽  
Finite Fields ◽  
Transformation Groups ◽  
Affine Spaces ◽  
Exponential Order ◽  
Cyclic Groups ◽  
Key Exchange Protocols ◽  
Polynomial Transformations ◽  
High Degree

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.

scholarly journals Construction of elliptic curves with cyclic groups over prime fields

2006 ◽  
Vol 73 (2) ◽  
pp. 245-254 ◽  
Author(s):  
Naoya Nakazawa
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Function Field ◽  
Modular Group ◽  
Modular Function ◽  
Invariant Function ◽  
Rational Points ◽  
Modular Invariant ◽  
Cyclic Groups ◽  
Prime Fields

The purpose of this article is to construct families of elliptic curves E over finite fields F so that the groups of F-rational points of E are cyclic, by using a representation of the modular invariant function by a generator of a modular function field associated with the modular group Γ0(N), where N = 5, 7 or 13.

scholarly journals Atomic Latin Squares based on Cyclotomic Orthomorphisms

10.37236/1919 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ian M. Wanless
Keyword(s):  
Finite Fields ◽  
Complete Graph ◽  
Prime Order ◽  
Automorphism Groups ◽  
Latin Square ◽  
Latin Squares ◽  
Established Method ◽  
Cyclic Groups ◽  
Complete Bipartite Graphs ◽  
First Time

Atomic latin squares have indivisible structure which mimics that of the cyclic groups of prime order. They are related to perfect $1$-factorisations of complete bipartite graphs. Only one example of an atomic latin square of a composite order (namely 27) was previously known. We show that this one example can be generated by an established method of constructing latin squares using cyclotomic orthomorphisms in finite fields. The same method is used in this paper to construct atomic latin squares of composite orders 25, 49, 121, 125, 289, 361, 625, 841, 1369, 1849, 2809, 4489, 24649 and 39601. It is also used to construct many new atomic latin squares of prime order and perfect $1$-factorisations of the complete graph $K_{q+1}$ for many prime powers $q$. As a result, existence of such a factorisation is shown for the first time for $q$ in $\big\{$529, 2809, 4489, 6889, 11449, 11881, 15625, 22201, 24389, 24649, 26569, 29929, 32041, 38809, 44521, 50653, 51529, 52441, 63001, 72361, 76729, 78125, 79507, 103823, 148877, 161051, 205379, 226981, 300763, 357911, 371293, 493039, 571787$\big\}$. We show that latin squares built by the 'orthomorphism method' have large automorphism groups and we discuss conditions under which different orthomorphisms produce isomorphic latin squares. We also introduce an invariant called the train of a latin square, which proves to be useful for distinguishing non-isomorphic examples.

scholarly journals Mixed-Degree Spherical Simplex-Radial Cubature Kalman Filter

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Shiyuan Wang ◽  
Yali Feng ◽  
Shukai Duan ◽  
Lidan Wang
Keyword(s):  
Kalman Filter ◽  
Computational Complexity ◽  
Kalman Filters ◽  
Order Moment ◽  
Low Degree ◽  
Cubature Kalman Filter ◽  
Highly Nonlinear ◽  
Spherical Simplex ◽  
The Cost ◽  
High Degree

Conventional low degree spherical simplex-radial cubature Kalman filters often generate low filtering accuracy or even diverge for handling highly nonlinear systems. The high-degree Kalman filters can improve filtering accuracy at the cost of increasing computational complexity; nevertheless their stability will be influenced by the negative weights existing in the high-dimensional systems. To efficiently improve filtering accuracy and stability, a novel mixed-degree spherical simplex-radial cubature Kalman filter (MSSRCKF) is proposed in this paper. The accuracy analysis shows that the true posterior mean and covariance calculated by the proposed MSSRCKF can agree accurately with the third-order moment and the second-order moment, respectively. Simulation results show that, in comparison with the conventional spherical simplex-radial cubature Kalman filters that are based on the same degrees, the proposed MSSRCKF can perform superior results from the aspects of filtering accuracy and computational complexity.

Computational Complexity over Finite Fields

1976 ◽  
Vol 5 (2) ◽  
pp. 324-331 ◽  
Author(s):  
Volker Strassen
Keyword(s):  
Computational Complexity ◽  
Finite Fields

Explicit constructions of extremal graphs and new multivariate cryptosystems

2015 ◽  
Vol 52 (2) ◽  
pp. 185-204 ◽  
Author(s):  
Vasyl Ustimenko
Keyword(s):  
Discrete Logarithm ◽  
Public Key Infrastructure ◽  
Public Key ◽  
Polynomial Degree ◽  
Affine Spaces ◽  
Private Key ◽  
Explicit Constructions ◽  
Diffie Hellman ◽  
Polynomial Transformations ◽  
Finite Commutative Ring

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).

Local multiplication maps on F[x]

2014 ◽  
Vol 14 (03) ◽  
pp. 1550029
Author(s):  
Kelly Aceves ◽  
Manfred Dugas
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Linear Transformation ◽  
Ring Of Polynomials

Let F be a field and A a F-algebra. The F-linear transformation φ : A → A is a local multiplication map if for all a ∈ A there exists some ua ∈ A such that φ(a) = aua. Let [Formula: see text] denote the F-algebra of all local multiplication maps of A. If F is infinite and F[x] is the ring of polynomials over F, then it is known Lemma 1 in [J. Buckner and M. Dugas, Quasi-Localizations of ℤ, Israel J. Math.160 (2007) 349–370] that [Formula: see text]. The purpose of this paper is to study [Formula: see text] for finite fields F. It turns out that in this case [Formula: see text] is a "very" non-commutative ring of cardinality 2ℵ0 with many interesting properties.

scholarly journals Periodic rings with commuting nilpotents

1984 ◽  
Vol 7 (2) ◽  
pp. 403-406
Author(s):  
Hazar Abu-Khuzam ◽  
Adil Yaqub
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Finite Fields ◽  
Boolean Ring ◽  
Fixed Integer ◽  
Integer Coefficients ◽  
Nilpotent Elements ◽  
Periodic Rings

LetRbe a ring (not necessarily with identity) and letNdenote the set of nilpotent elements ofR. Suppose that (i)Nis commutative, (ii) for everyxinR, there exists a positive integerk=k(x)and a polynomialf(λ)=fx(λ)with integer coefficients such thatxk=xk+1f(x), (iii) the setIn={x|xn=x}wherenis a fixed integer,n>1, is an ideal inR. ThenRis a subdirect sum of finite fields of at mostnelements and a nil commutative ring. This theorem, generalizes the “xn=x” theorem of Jacobson, and (takingn=2) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume thatInis a subring ofR.

Sign in / Sign up

Export Citation Format

Share Document