Constructing elliptic curves of prime order

We present a very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N. Heuristically, this algorithm only takes polynomial time Otilde((\log N)^3), and it is so fast that it may profitably be used to tackle the related problem of finding elliptic curves with point groups of prime order of prescribed size. We also discuss the impact of the use of high level modular functions to reduce the run time by large constant factors and show that recent gonality bounds for modular curves imply limits on the time reduction that can be obtained.Comment: 13 page

Perancangan Busana Magnificent Of Modular Mode

Fast fashion can be interpreted as a quick response effort in providing the latest fashionable clothes according to consumer demand. This can lead to a accumulation of clothes which eventually becomes clothing waste. Clothing waste can be overcome with two opportunities, namely reuse and reduction, this reduction method uses the principle of sustainable design. Sustainable that is raised is a modular design. "Modular design" is a kind of design fashion that can not only make clothes more attractive, allow the wearer to participate in choices, increase the possibility of clothing styles, but also can extend the service cycle of clothes. In this "fast fashion" market, modular design ideas can be a breaking point, helping us find ways to balance low-carbon and eco-friendly needs and fashion. Therefore, there is a need for a ready-to-wear fashion modular design that inspires the Woloan Minahasa Stage House that can be disassembled. This will be the common thread in the creation of the work

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

Reduze 2 - Distributed Feynman Integral Reduction

Reduze is a computer program for reducing Feynman integrals to master integrals employing a variant of Laporta's reduction algorithm. This article describes version 2 of the program. New features include the distributed reduction of single topologies on multiple processor cores. The parallel reduction of different topologies is supported via a modular, load balancing job system. Fast graph and matroid based algorithms allow for the identification of equivalent topologies and integrals.Comment: 16 pages, 5 figure

3-D GPR survey with a modular system: reducing positioning inaccuracies and linear noise

Recently, the use of ground-penetrating radar (GPR) arrays with a large number of antenna elements in a fixed configuration has become more common. The investment needed for these systems is significant. Although gradually expandable modular systems, consisting of antennas which can be used independently, do not match the fast acquisition of detailed datasets by means of multi-channel arrays, they can help finding a compromise between increased acquisition speed and (limited) resources. In modular systems, the separation between transmitter-receiver pairs is often larger than the sampling distance prescribed by the Nyquist theorem. As a consequence, additional profiles have to be recorded in between, which requires a high positioning precision. As a completely identical response for the different antennas in an array is difficult to achieve, stripes can occur in the horizontal slices, especially when ringing occurs. This complicates the interpretation of features in the direction of the survey lines. In this paper, a three-dimensional frequency-wavenumber filter is proposed, consisting in a combination of a circular filter and a fan filter. The application of this filter to GPR data collected at the Roman town Mariana (Corsica, France) showed a reduction of the stripe patterns, allowing a more reliable characterization of subtle archaeological structures

Fast Modular Reduction for Large-Integer Multiplication

The work contained in this thesis is a representation of the successful attempt to speed-up the modular reduction as an independent step of modular multiplication, which is the central operation in public-key cryptosystems. Based on the properties of Mersenne and Quasi-Mersenne primes, four distinct sets of moduli have been described, which are responsible for converting the single-precision multiplication prevalent in many of today\u27s techniques into an addition operation and a few simple shift operations. A novel algorithm has been proposed for modular folding. With the backing of the special moduli sets, the proposed algorithm is shown to outperform (speed-wise) the Modified Barrett algorithm by 80% for operands of length 700 bits, the least speed-up being around 70% for smaller operands, in the range of around 100 bits

Faster 64-bit universal hashing using carry-less multiplications

Intel and AMD support the Carry-less Multiplication (CLMUL) instruction set in their x64 processors. We use CLMUL to implement an almost universal 64-bit hash family (CLHASH). We compare this new family with what might be the fastest almost universal family on x64 processors (VHASH). We find that CLHASH is at least 60% faster. We also compare CLHASH with a popular hash function designed for speed (Google's CityHash). We find that CLHASH is 40% faster than CityHash on inputs larger than 64 bytes and just as fast otherwise