Generalised Mersenne Numbers Revisited

Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and feature in the NIST (FIPS 186-2) and SECG standards for use in elliptic curve cryptography. Their form is such that modular reduction is extremely efficient, thus making them an attractive choice for modular multiplication implementation. However, the issue of residue multiplication efficiency seems to have been overlooked. Asymptotically, using a cyclic rather than a linear convolution, residue multiplication modulo a Mersenne number is twice as fast as integer multiplication; this property does not hold for prime GMNs, unless they are of Mersenne's form. In this work we exploit an alternative generalisation of Mersenne numbers for which an analogue of the above property --- and hence the same efficiency ratio --- holds, even at bitlengths for which schoolbook multiplication is optimal, while also maintaining very efficient reduction. Moreover, our proposed primes are abundant at any bitlength, whereas GMNs are extremely rare. Our multiplication and reduction algorithms can also be easily parallelised, making our arithmetic particularly suitable for hardware implementation. Furthermore, the field representation we propose also naturally protects against side-channel attacks, including timing attacks, simple power analysis and differential power analysis, which is essential in many cryptographic scenarios, in constrast to GMNs.Comment: 32 pages. Accepted to Mathematics of Computatio

On isogeny classes of Edwards curves over finite fields

We count the number of isogeny classes of Edwards curves over finite fields, answering a question recently posed by Rezaeian and Shparlinski. We also show that each isogeny class contains a {\em complete} Edwards curve, and that an Edwards curve is isogenous to an {\em original} Edwards curve over \F_q if and only if its group order is divisible by 8 if $q \equiv -1 \pmod{4}$, and 16 if $q \equiv 1 \pmod{4}$. Furthermore, we give formulae for the proportion of d \in \F_q \setminus \{0,1\} for which the Edwards curve $E_d$ is complete or original, relative to the total number of $d$ in each isogeny class.Comment: 27 page

On the discrete logarithm problem in finite fields of fixed characteristic

For $q$ a prime power, the discrete logarithm problem (DLP) in $\mathbb{F}_{q}$ consists in finding, for any $g \in \mathbb{F}_{q}^{\times}$ and $h \in \langle g \rangle$, an integer $x$ such that $g^x = h$. We present an algorithm for computing discrete logarithms with which we prove that for each prime $p$ there exist infinitely many explicit extension fields $\mathbb{F}_{p^n}$ in which the DLP can be solved in expected quasi-polynomial time. Furthermore, subject to a conjecture on the existence of irreducible polynomials of a certain form, the algorithm solves the DLP in all extensions $\mathbb{F}_{p^n}$ in expected quasi-polynomial time.Comment: 15 pages, 2 figures. To appear in Transactions of the AM

Estimates of genetic variability in the F4 generation of three populations of common wheat (Triticum aestivum L. Em. Thell.)

Three populations of common bread wheat (Triticum aestivum L. em. Thell.) were studied to gain an estimate of the genetic variability associated with four agronomic characters in each population. The primary objective was to determine if homozygous lines could be isolated from these populations that would be equal to or better than the F1 of the respective population. Two populations (population 1 and 3) exhibited heterosis for yield in the F1 generation while the other population (population 2) showed no heterosis for yield in the F1. Each population consisted of random F4 lines, each of which could be traced to a single F2 plant. Population 1 originated from a cross of the cultivars \u27Seneca\u27 x \u27Knox 62\u27; population 2 from a cross of \u27Monon\u27 x \u27Triumph\u27; and population 3 from a cross of \u27Tenn. 9\u27 x \u27Knox 62\u27. The characters studied were: (1) mature plant height (centimeters), (2) heading date (days past March 31), (3) yield (grams per plot), and (4) kernel weight (grams per 1000 kernels). The experimental design was a randomized com-plete block with two replications and the experiment was grown at three locations. Estimates of broad sense heritability were obtained via variance component analyses. These heritability estimates were used to calculate expected advance through selection for each character in each population. Phenotypic and genotypic correlation coefficients for all characters in all combinations were calculated. Coefficients of variation and genetic coefficients of variation were also computed. Analyses of the data revealed that the non-heterotic population (population 2) had significant (P. = .05) variability for all four characters. Population 1 showed significant (P. = .05) variability for heading date and kernel weight only, and population 3, for only heading date and mature plant height. Heading date in population 3, mature plant height and kernel weight in population 2 were the only characters whose expected F5 means would be better than the best parent. Evidence was found to support a two major gene hypothesis for the inheritance of heading time. From the results of this study, it appears that homozygous lines that equal or better the F1 could not be found in later generations

Milton\u27s Christ, as Seen by the Critics of Paradise Lost and Paradise Regained Since 1900

The over-all purpose of this thesis is to present the primary investigations and commentaries of the twentieth century critics upon Milton\u27s Christ and to arrive at conclusions which pertain to these critical findings

Processing AIRS Scientific Data Through Level 3

The Atmospheric Infra-Red Sounder (AIRS) Science Processing System (SPS) is a collection of computer programs, known as product generation executives (PGEs). The AIRS SPS PGEs are used for processing measurements received from the AIRS suite of infrared and microwave instruments orbiting the Earth onboard NASA's Aqua spacecraft. Early stages of the AIRS SPS development were described in a prior NASA Tech Briefs article: Initial Processing of Infrared Spectral Data (NPO-35243), Vol. 28, No. 11 (November 2004), page 39. In summary: Starting from Level 0 (representing raw AIRS data), the AIRS SPS PGEs and the data products they produce are identified by alphanumeric labels (1A, 1B, 2, and 3) representing successive stages or levels of processing. The previous NASA Tech Briefs article described processing through Level 2, the output of which comprises geo-located atmospheric data products such as temperature and humidity profiles among others. The AIRS Level 3 PGE samples selected information from the Level 2 standard products to produce a single global gridded product. One Level 3 product is generated for each day s collection of Level 2 data. In addition, daily Level 3 products are aggregated into two multiday products: an eight-day (half the orbital repeat cycle) product and monthly (calendar month) product