Finite Field Fp Research Articles

Geometry of discrete quantum computing

Conventional quantum computing entails a geometry based on the description of an n-qubit state using 2n infinite precision complex numbers denoting a vector in a Hilbert space. Such numbers are in general uncomputable using any real-world resources, and, if we have the idea of physical law as some kind of computational algorithm of the universe, we would be compelled to alter our descriptions of physics to be consistent with computable numbers. Our purpose here is to examine the geometric implications of using finite fields Fp and finite complexified fields (based on primes p congruent to 3 (mod4)) as the basis for computations in a theory of discrete quantum computing, which would therefore become a computable theory. Because the states of a discrete n-qubit system are in principle enumerable, we are able to determine the proportions of entangled and unentangled states. In particular, we extend the Hopf fibration that defines the irreducible state space of conventional continuous n-qubit theories (which is the complex projective space ) to an analogous discrete geometry in which the Hopf circle for any n is found to be a discrete set of p + 1 points. The tally of unit-length n-qubit states is given, and reduced via the generalized Hopf fibration to , the discrete analogue of the complex projective space, which has irreducible states. Using a measure of entanglement, the purity, we explore the entanglement features of discrete quantum states and find that the n-qubit states based on the complexified field have pn(p − 1)n unentangled states (the product of the tally for a single qubit) with purity 1, and they have pn + 1(p − 1)(p + 1)n − 1 maximally entangled states with purity zero.

On the invertible cellular automata 150 over [formula omitted

In this work the reversibility problem for cellular automata with rule number 150 over the finite field Fp is tackled. It is shown that when null boundary conditions are stated, the reversibility is independent of the state set Fp and it appears when the number of cells of the cellular space satisfies some conditions. Furthermore, the explicit expressions of the inverse cellular automata are computed.

Identity-based key agreement protocol employing a symmetric balanced incomplete block design

Key agreement protocol is a fundamental protocol in cryptography whereby two or more participants can agree on a common conference key in order to communicate securely among themselves. In this situation, the participants can securely send and receive messages with each other. An adversary not having access to the conference key will not be able to decrypt the messages. In this paper, we propose a novel identity-based authenticated multi user key agreement protocol employing a symmetric balanced incomplete block design. Our protocol is built on elliptic curve cryptography and takes advantage of a kind of bilinear map called Weil pairing. The protocol presented can provide an identification (ID)-based authentication service and resist different key attacks. Furthermore, our protocol is efficient and needs only two rounds for generating a common conference key. It is worth noting that the com munication cost for generating a conference key in our protocol is only O(n√n) and the computation cost is only O(nm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), where n implies the number of participants and m denotes the extension degree of the finite field Fp <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> . In addition, in order to resist the different key attack from malicious participants, our protocol can be further extended to provide the fault tolerant property.

Lower central series of a free associative algebra over the integers and finite fields

Consider the free algebra An generated over Q by n generators x1,…,xn. Interesting objects attached to A=An are members of its lower central series, Li=Li(A), defined inductively by L1=A, Li+1=[A,Li], and their associated graded components Bi=Bi(A) defined as Bi=Li/Li+1. These quotients Bi for i⩾2, as well as the reduced quotient B¯1=A/(L2+AL3), exhibit a rich geometric structure, as shown by Feigin and Shoikhet (2007) [FS] and later authors (Dobrovolska et al., 1997 [DKM], Dobrovolska and Etingof, 2008 [DE], Arbesfeld and Jordan, 2010 [AJ], Bapat and Jordan, 2010 [BJ]).We study the same problem over the integers Z and finite fields Fp. New phenomena arise, namely, torsion in Bi over Z, and jumps in dimension over Fp. We describe the torsion in the reduced quotient B¯1 and B2 geometrically in terms of the De Rham cohomology of Zn. As a corollary we obtain a complete description of B¯1(An(Z)) and B¯1(An(Fp)), as well as of B2(An(Z[1/2])) and B2(An(Fp)), p>2. We also give theoretical and experimental results for Bi with i>2, formulating a number of conjectures and questions on their basis. Finally, we discuss the supercase, when some of the generators are odd and some are even, and provide some theoretical results and experimental data in this case.

Fast Mapping Method based on Matrix Approach For Elliptic Curve Cryptography

A new mapping method of the alphanumeric characters on to the x-y coordinate of the elliptic curve defined over a finite field Fp is provided. This paper attemps to utilize the properties of invertible matrices and elliptic curve to generate an algorithm that will guarantee the confidentiality of messages. The results points are encrypted by using elliptic curve encryption technique.The proposed method is efficient in its principale and has great potentiel to be applied to others situations where the exchange of messages is done confidentially. Finally, the use of matrices and their algebraic properties will provide better performance in this regard.

Rigid local systems and a question of Wootters

Recently we learned from Ron Evans of some fascinating questions raised by Wootters [A-S-S-W]. These questions, which concern exponential sums, arose from his investigations of a particular quantum state with special properties, where the underlying vector space is the space of functions on the finite field Fp := Z/pZ, p a prime which is 3 mod 4. Due to our ignorance of the underlying physics, we concentrate on the exponential sums themselves. In our approach, it costs us nothing to work over an arbitrary finite field Fq of odd characteristic. [Thus Fq is “the” finite field of q elements, q a power of some odd prime p.] We also introduce a parameter a ∈ Fq . In the Wootters setup, where q = p is 3 mod 4, the parameter a is simply a = −1. Ultimately we end up proving identities among exponential sums, but not at all in a straightforward way; we need to invoke the theory of Kloosterman sheaves and their rigidity properties, as well as the fundamental results of [De-Weil II] and [BBD]. It would be interesting to find direct proofs of these identities.

Fast arithmetics in Artin–Schreier towers over finite fields

An Artin–Schreier tower over the finite field Fp is a tower of field extensions generated by polynomials of the form Xp−X−α. Following Cantor and Couveignes, we give algorithms with quasi-linear time complexity for arithmetic operations in such towers. As an application, we present an implementation of Couveignes’ algorithm for computing isogenies between elliptic curves using the p-torsion.

Elliptic Divisibility Sequences over Finite Fields

In this work, we study elliptic divisibility sequences over finite fields. Morgan Ward in [14], [15] gave arithmetic theory of elliptic divisibility sequences and formulas for elliptic divisibility sequences with rank two over finite field Fp. We study elliptic divisibility sequences with rank three, four and five over a finite field Fp, where p > 3 is a prime and give general terms of these sequences and then we determine elliptic and singular curves associated with these sequences. Keywords—Elliptic divisibility sequences, singular elliptic divisibility sequences, elliptic curves, singular curves.

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

Elliptic and Hyper Elliptic Curve Cryptography Over Finite Field Fp

Elliptic and Hyper Elliptic Curve Cryptography Over Finite Field Fp

Frobenius distributions of Drinfeld modules over finite fields

We express the weighted class number of Drinfeld A A -modules of rank two with given characteristic polynomial over the finite field F p = A / p {\mathbb {F}} _{\mathfrak {p}}=A/{\mathfrak {p}} ( p ∈ Spec ⁡ A ({\mathfrak {p}} \in \operatorname {Spec}A , where A = F q [ T ] ) A=\mathbb {F} _q[T]) as an infinite product of local terms. Some auxiliary results of independent interest about characteristic polynomials of Drinfeld modules are given.

Representation of elements in F2m enabling unified field arithmetic for elliptic curve cryptography

A change of representation for elements in F2m is proposed. The proposed representation is useful for architectures that implement unified Montgomery multiplication in finite fields F2m and Fp used for elliptic curve cryptography since it transforms a standard F2m multiplication into a Montgomery multiplication and comes at virtually no cost in terms of conversion operations.

Predicting nonlinear pseudorandom number generators

Let p be a prime and let a and b be elements of the finite field Fp of p elements. The inversive congruential generator (ICG) is a sequence (u n ) of pseudorandom numbers defined by the relation u n+1 ≡ au -1 n +b mod p. We show that if sufficiently many of the most significant bits of several consecutive values u n of the ICG are given, one can recover the initial value u 0 (even in the case where the coefficients a and b are not known). We also obtain similar results for the quadratic congruential generator (QCG), v n+1 ≡ f(v n ) mod p, where f ∈ F p [X]. This suggests that for cryptographic applications ICG and QCG should be used with great care. Our results are somewhat similar to those known for the linear congruential generator (LCG), x n+1 ≡ ax n + b mod p, but they apply only to much longer bit strings. We also estimate limits of some heuristic approaches, which still remain much weaker than those known for LCG.

Gaussian hypergeometric functions and traces of Hecke operators

We establish a simple inductive formula for the trace Trknew(Γ0(8),p) of the pth Hecke operator on the space Sknew(Γ0(8)) of newforms of level 8 and weight k in terms of the values of 3F2-hypergeometric functions over the finite field Fp. Using this formula when k = 6, we prove a conjecture of Koike relating Tr6new(Γ0(8),p) to the values 6F5(1)p and 4F3(1)p. Furthermore, we find new congruences between Trknew(Γ0(8),p) and generalized Apéry numbers.

Exploring invariant linear codes through generators and centralizers

We investigate a H-invariant linear code C over the finite field Fp where H is a group of linear transformations. We show that in the case H is a noncyclic abelian group and (|H|, p) = 1, then the code C is the sum of the centralizer codes Cc (h) where h is a nonidentity element of H. Moreover if A is subgroup of H such that A ≅ Zq × Zq , q ≠ p, then dim C is known when the dimension of Cc (K) is known for each subgroup K ≠ 1 of A. In the last few sections we restrict our scope of investigation to a special class of invariant codes, namely affine codes and their centralizers. New results concerning the dimensions of these codes and their centralizers are obtained.

Average normalisations of elliptic curves

Ciet, Quisquater, and Sica have recently shown that every elliptic curve E over a finite field 𝔽p is isomorphic to a curve y2 = x3 + ax + b with a and b of size O (p¾). In this paper, we show that almost all elliptic curves satisfy the stronger bound O (p⅔). The problem is motivated by cryptographic considerations.

Incomplete Character Sums and Polynomial Interpolation of the Discrete Logarithm

In the first part of the paper, certain incomplete character sums over a finite field Fpr are considered which in the case of finite prime fields Fp are of the form ∑A+N−1n=Aχ(g(n))ψ(f(n)), where A and N are integers with 1≤N<p, g and f are polynomials over Fp, and χ denotes a multiplicative and ψ an additive character of Fp. Excluding trivial cases, it is shown that the above sums are at most of the order of magnitude N1/2pr/4. Recently, Shparlinski showed that a polynomial f over the integers which coincides with the discrete logarithm of the finite prime field Fp for N consecutive elements of Fp must have a degree at least of the order of magnitude Np−1/2. In this paper this result is extended to arbitrary Fpr. The proof is based on the above new bound for incomplete hybrid character sums.

Security of the most significant bits of the Shamir message passing scheme

Boneh and Venkatesan have recently proposed a polynomial time algorithm for recovering a hidden element α of a finite field Fp of p elements from rather short strings of the most significant bits of the remainder modulo p of αt for several values of t selected uniformly at random from Fp*. Unfortunately the applications to the computational security of most significant bits of private keys of some finite field exponentiation based cryptosystems given by Boneh and Venkatesan are not quite correct. For the Diffie-Hellman cryptosystem the result of Boneh and Venkatesan has been corrected and generalized in our recent paper. Here a similar analysis is given for the Shamir message passing scheme. The results depend on some bounds of exponential sums.

Uniform Families of Polynomial Equations Over a Finite Field and Structures Admitting an Euler Characteristic of Definable Sets

Over a fixed finite field Fp, families of polynomial equations f i ( x 1 , … , x n N ) = 0 for i = 1,…, kN, that are uniformly determined by a parameter N, are considered. The notion of a uniform family is defined in terms of first-order logic. A notion of an abstract Euler characteristic is used to give sense to a statement that the system has a solution for infinite N, and a statement linking the solvability of a linear system for infinite N with its solvability for finite N is proved. This characterisation is used to formulate a criterion yielding degree lower bounds for various ideal membership proof systems (for example, Nullstellensatz and the polynomial calculus). Further, several results about Euler structures (structures with an abstract Euler characteristic) are proved, and the case of fields, in particular, is investigated more closely. 1991 Mathematics Subject Classification: primary 03F20, 12L12, 15A06; secondary 03C99, 12E12, 68Q15, 13L05.

Complexity for modules over finite Chevalley groups and classical Lie algebras

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,