770 research outputs found
Discrete phase-space structure of -qubit mutually unbiased bases
We work out the phase-space structure for a system of qubits. We replace
the field of real numbers that label the axes of the continuous phase space by
the finite field \Gal{2^n} and investigate the geometrical structures
compatible with the notion of unbiasedness. These consist of bundles of
discrete curves intersecting only at the origin and satisfying certain
additional properties. We provide a simple classification of such curves and
study in detail the four- and eight-dimensional cases, analyzing also the
effect of local transformations. In this way, we provide a comprehensive
phase-space approach to the construction of mutually unbiased bases for
qubits.Comment: Title changed. Improved version. Accepted for publication in Annals
of Physic
Mutually Unbiased Bases, Generalized Spin Matrices and Separability
A collection of orthonormal bases for a complex dXd Hilbert space is called
mutually unbiased (MUB) if for any two vectors v and w from different bases the
square of the inner product equals 1/d: || ^{2}=1/d. The MUB problem is to
prove or disprove the the existence of a maximal set of d+1 bases. It has been
shown in [W. K. Wootters, B. D. Fields, Annals of Physics, 191, no. 2, 363-381,
(1989)] that such a collection exists if d is a power of a prime number p. We
revisit this problem and use dX d generalizations of the Pauli spin matrices to
give a constructive proof of this result. Specifically we give explicit
representations of commuting families of unitary matrices whose eigenvectors
solve the MUB problem. Additionally we give formulas from which the orthogonal
bases can be readily computed. We show how the techniques developed here
provide a natural way to analyze the separability of the bases. The techniques
used require properties of algebraic field extensions, and the relevant part of
that theory is included in an Appendix
On fast multiplication of a matrix by its transpose
We present a non-commutative algorithm for the multiplication of a
2x2-block-matrix by its transpose using 5 block products (3 recursive calls and
2 general products) over C or any finite field.We use geometric considerations
on the space of bilinear forms describing 2x2 matrix products to obtain this
algorithm and we show how to reduce the number of involved additions.The
resulting algorithm for arbitrary dimensions is a reduction of multiplication
of a matrix by its transpose to general matrix product, improving by a constant
factor previously known reductions.Finally we propose schedules with low memory
footprint that support a fast and memory efficient practical implementation
over a finite field.To conclude, we show how to use our result in LDLT
factorization.Comment: ISSAC 2020, Jul 2020, Kalamata, Greec
Hard Mathematical Problems in Cryptography and Coding Theory
In this thesis, we are concerned with certain interesting computationally hard problems and the complexities of their associated algorithms. All of these problems share a common feature in that they all arise from, or have applications to, cryptography, or the theory of error correcting codes. Each chapter in the thesis is based on a stand-alone paper which attacks a particular hard problem. The problems and the techniques employed in attacking them are described in detail. The first problem concerns integer factorization: given a positive integer . the problem is to find the unique prime factors of . This problem, which was historically of only academic interest to number theorists, has in recent decades assumed a central importance in public-key cryptography. We propose a method for factorizing a given integer using a graph-theoretic algorithm employing Binary Decision Diagrams (BDD). The second problem that we consider is related to the classification of certain naturally arising classes of error correcting codes, called self-dual additive codes over the finite field of four elements, . We address the problem of classifying self-dual additive codes, determining their weight enumerators, and computing their minimum distance. There is a natural relation between self-dual additive codes over and graphs via isotropic systems. Utilizing the properties of the corresponding graphs, and again employing Binary Decision Diagrams (BDD) to compute the weight enumerators, we can obtain a theoretical speed up of the previously developed algorithm for the classification of these codes. The third problem that we investigate deals with one of the central issues in cryptography, which has historical origins in the theory of geometry of numbers, namely the shortest vector problem in lattices. One method which is used both in theory and practice to solve the shortest vector problem is by enumeration algorithms. Lattice enumeration is an exhaustive search whose goal is to find the shortest vector given a lattice basis as input. In our work, we focus on speeding up the lattice enumeration algorithm, and we propose two new ideas to this end. The shortest vector in a lattice can be written as . where are integer coefficients and are the lattice basis vectors. We propose an enumeration algorithm, called hybrid enumeration, which is a greedy approach for computing a short interval of possible integer values for the coefficients of a shortest lattice vector. Second, we provide an algorithm for estimating the signs or of the coefficients of a shortest vector . Both of these algorithms results in a reduction in the number of nodes in the search tree. Finally, the fourth problem that we deal with arises in the arithmetic of the class groups of imaginary quadratic fields. We follow the results of Soleng and Gillibert pertaining to the class numbers of some sequence of imaginary quadratic fields arising in the arithmetic of elliptic and hyperelliptic curves and compute a bound on the effective estimates for the orders of class groups of a family of imaginary quadratic number fields. That is, suppose is a sequence of positive numbers tending to infinity. Given any positive real number . an effective estimate is to find the smallest positive integer depending on such that for all . In other words, given a constant . we find a value such that the order of the ideal class in the ring (provided by the homomorphism in Soleng's paper) is greater than for any . In summary, in this thesis we attack some hard problems in computer science arising from arithmetic, geometry of numbers, and coding theory, which have applications in the mathematical foundations of cryptography and error correcting codes
Construction of self-dual normal bases and their complexity
Recent work of Pickett has given a construction of self-dual normal bases for
extensions of finite fields, whenever they exist. In this article we present
these results in an explicit and constructive manner and apply them, through
computer search, to identify the lowest complexity of self-dual normal bases
for extensions of low degree. Comparisons to similar searches amongst normal
bases show that the lowest complexity is often achieved from a self-dual normal
basis
Discrete coherent and squeezed states of many-qudit systems
We consider the phase space for a system of identical qudits (each one of
dimension , with a primer number) as a grid of
points and use the finite field to label the corresponding axes.
The associated displacement operators permit to define -parametrized
quasidistribution functions in this grid, with properties analogous to their
continuous counterparts. These displacements allow also for the construction of
finite coherent states, once a fiducial state is fixed. We take this reference
as one eigenstate of the discrete Fourier transform and study the factorization
properties of the resulting coherent states. We extend these ideas to include
discrete squeezed states, and show their intriguing relation with entangled
states between different qudits.Comment: 11 pages, 3 eps figures. Submitted for publicatio
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