136 research outputs found
Random Integer Lattice Generation via HNF
Lattices used in cryptography are integer lattices. Defining and generating a random integer lattice are interesting topics. A generation algorithm for random integer lattice can be used to serve as a random input of all the lattice algorithms. In this paper, we recall the definition of random integer lattice given by G.Hu et al. and present an improved generation algorithm for it via Hermite Normal Form. It can be proved that with probability >= 0.99, this algorithm outputs an n-dim random integer lattice within O(n^2) operations
Practical improvements to class group and regulator computation of real quadratic fields
We present improvements to the index-calculus algorithm for the computation
of the ideal class group and regulator of a real quadratic field. Our
improvements consist of applying the double large prime strategy, an improved
structured Gaussian elimination strategy, and the use of Bernstein's batch
smoothness algorithm. We achieve a significant speed-up and are able to compute
the ideal class group structure and the regulator corresponding to a number
field with a 110-decimal digit discriminant
A-Tint: A polymake extension for algorithmic tropical intersection theory
In this paper we study algorithmic aspects of tropical intersection theory.
We analyse how divisors and intersection products on tropical cycles can
actually be computed using polyhedral geometry. The main focus of this paper is
the study of moduli spaces, where the underlying combinatorics of the varieties
involved allow a much more efficient way of computing certain tropical cycles.
The algorithms discussed here have been implemented in an extension for
polymake, a software for polyhedral computations.Comment: 32 pages, 5 figures, 4 tables. Second version: Revised version, to be
published in European Journal of Combinatoric
A Coefficient-Embedding Ideal Lattice can be Embedded into Infinitely Many Polynomial Rings
Many lattice-based crypstosystems employ ideal lattices for high efficiency.
However, the additional algebraic structure of ideal lattices usually makes us
worry about the security, and it is widely believed that the algebraic
structure will help us solve the hard problems in ideal lattices more
efficiently. In this paper, we study the additional algebraic structure of
ideal lattices further and find that a given ideal lattice in some fixed
polynomial ring can be embedded as an ideal in infinitely many different
polynomial rings. We explicitly present all these polynomial rings for any
given ideal lattice. The interesting phenomenon tells us that a single ideal
lattice may have more abundant algebraic structures than we imagine, which will
impact the security of corresponding crypstosystems. For example, it increases
the difficulties to evaluate the security of crypstosystems based on ideal
lattices, since it seems that we need consider all the polynomial rings that
the given ideal lattices can be embedded into if we believe that the algebraic
structure will contribute to solve the corresponding hard problem. It also
inspires us a new method to solve the ideal lattice problems by embedding the
given ideal lattice into another well-studied polynomial ring. As a by-product,
we also introduce an efficient algorithm to identify if a given lattice is an
ideal lattice or not
A suite of quantum algorithms for the shortestvector problem
Crytography has come to be an essential part of the cybersecurity infrastructure that provides a safe environment for communications in an increasingly connected world. The advent of quantum computing poses a threat to the foundations of the current widely-used cryptographic model, due to the breaking of most of the cryptographic algorithms used to provide confidentiality, authenticity, and more. Consequently a new set of cryptographic protocols have been designed to be secure against quantum computers, and are collectively known as post-quantum cryptography (PQC). A forerunner among PQC is lattice-based cryptography, whose security relies upon the hardness of a number of closely related mathematical problems, one of which is known as the shortest vector problem (SVP).
In this thesis I describe a suite of quantum algorithms that utilize the energy minimization principle to attack the shortest vector problem. The algorithms outlined span the gate-model and continuous time quantum computing, and explore methods of parameter optimization via variational methods, which are thought to be effective on near-term quantum computers. The performance of the algorithms are analyzed numerically, analytically, and on quantum hardware where possible. I explain how the results obtained in the pursuit of solving SVP apply more broadly to quantum algorithms seeking to solve general real-world problems; minimize the effect of noise on imperfect hardware; and improve efficiency of parameter optimization.Open Acces
On Crystal-Structure Matches in Solid-Solid Phase Transitions
The exploration of solid-solid phase transition (SSPT) suffers from the
uncertainty of how two crystal structures match. We devised a theoretical
framework to describe and classify crystal-structure matches (CSM). Such
description fully exploits the translational and rotational symmetries and is
independent of the choice of supercells. This is enabled by the use of the
Hermite normal form, an analog of reduced echelon form for integer matrices.
With its help, exhausting all CSMs is made possible, which goes beyond the
conventional optimization schemes. As a demonstration, our enumeration
algorithm unveils the long-sought concerted mechanisms in the martensitic
transformation of steel accounting for the most commonly observed
Kurdjumov-Sachs (KS) orientation relationship (OR) and the Nishiyama-Wassermann
OR. Especially, the predominance of KS OR is explained. Given the unprecedented
comprehensiveness and efficiency, our enumeration scheme provide a promising
strategy for SSPT mechanism research.Comment: main text: 6 pages, 4 figures; supplemental materials: 14 pages, 6
figure
A provably secure variant of NTRU cryptosystem
In 1996 Hoffstein, Pipher ad Silverman presented NTRUEncrypt, which is to
date the fastest known lattice-based encryption scheme. Its moderate
key-sizes, excellent asymptotic performance and conjectured resistance to
quantum attacks make it a perfect candidate to succeed where factorization and discrete log fail. Unfortunately, no security proof has been produced for NTRUEncrypt nor for its signature counterpart NTRUSign.
In 2013 Stehlé and Steinfield proposed to apply some mild modification to
the encryption and signature scheme to make them provably secure, under the assumed quantum hardness of standard worst-case lattice problems, restricted to a family of lattices related to some cyclotomic fields. In particular they showed that if the secret key polynomials of the encryption scheme are chosen from discrete Gaussians, then the public key, i.e their ratio, is statistically indistinguishable from uniform. The security will then follow from the hardness of the R-LWE problem.The aim of this thesis is to present Stehlé's and Steinfield's work in a slightly more accessi-ble form, providing some more background and details in some points
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