Making NTRU as Secure as Worst-Case Problems over Ideal Lattices

In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gröbner bases. Univariate ideal lattices are ideals in the residue class ring, [Formula: see text] (here [Formula: see text] is a monic polynomial) and cryptographic primitives have been built based on these objects. Ideal lattices in the univariate case are generalizations of cyclic lattices. We introduce the notion of multivariate cyclic lattices and show that ideal lattices are a generalization of them in the multivariate case too. Based on multivariate ideal lattices, we construct hash functions using Gröbner basis techniques. We define a worst case problem, shortest substitution problem with respect to an ideal in [Formula: see text], and use its computational hardness to establish the collision resistance of the hash functions.

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Enhanced Provably Secure NTRU Encryption Based on Hard Learning over Rings

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.236-237.1139 ◽

2012 ◽

Vol 236-237 ◽

pp. 1139-1144

Author(s):

Wu Yan Fang ◽

Zheng Huang ◽

Wang Li Cheng ◽

Wen Qiao Yan

Keyword(s):

High Speed ◽

Random Oracle Model ◽

Random Oracle ◽

Public Key ◽

Quantum Computers ◽

Public Key Cryptosystem ◽

Excellent Performance ◽

Worst Case ◽

Ideal Lattices ◽

Provably Secure

Since the presentation of NTRU public-key cryptosystem by Hoffstein, Pipher and Silverman, its favorable properties, such as easily created keys, high speed, excellent performance and conjectured resistance to quantum computers, have made it to be of great use. This paper proposes an enhanced scheme based on the hard learning with error over ring (R-LWE) problem to improve the security of the modified NTRUEncrypt presented by Stehle and Steinfled. We used part of the padding ideas of Fujisaki and Okamoto to obtain this scheme. It is semantically secure in strong sense of indistinguishability against adaptive chosen-ciphertext attacks in the random oracle model assuming the quantum hardness of standard worst-case problem over ideal lattices. It is also possible to arbitrarily decrease the error probability, and even to eliminate it completely. We gave the detailed analysis using the known results from classic works. Furthermore, this scheme owns many advantages such as the uniformity of public key, usual assumptions and the freedom for coding messages.

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Mosaic Mapping: A Means of Tiling Images to Shorten Acquisition Speed for Lower - Magnification SEM Images and Wavelength Dispersive Spectrometers (WDS) X-Ray Maps

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100164659 ◽

1996 ◽

Vol 54 ◽

pp. 438-439

Author(s):

J.D. Geller ◽

C.R. Herrington

Keyword(s):

Limiting Factors ◽

X Rays ◽

Angular Range ◽

Acceptance Angle ◽

Sem Images ◽

Worst Case ◽

X Ray ◽

Focusing Distance ◽

Layered Crystals ◽

Working Distance

The minimum magnification for which an image can be acquired is determined by the design and implementation of the electron optical column and the scanning and display electronics. It is also a function of the working distance and, possibly, the accelerating voltage. For secondary and backscattered electron images there are usually no other limiting factors. However, for x-ray maps there are further considerations. The energy-dispersive x-ray spectrometers (EDS) have a much larger solid angle of detection that for WDS. They also do not suffer from Bragg’s Law focusing effects which limit the angular range and focusing distance from the diffracting crystal. In practical terms EDS maps can be acquired at the lowest magnification of the SEM, assuming the collimator does not cutoff the x-ray signal. For WDS the focusing properties of the crystal limits the angular range of acceptance of the incident x-radiation. The range is dependent upon the 2d spacing of the crystal, with the acceptance angle increasing with 2d spacing. The natural line width of the x-ray also plays a role. For the metal layered crystals used to diffract soft x-rays, such as Be - O, the minimum magnification is approximately 100X. In the worst case, for the LEF crystal which diffracts Ti - Zn, ˜1000X is the minimum.

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