More Discriminants with the Brezing-Weng Method

The Brezing-Weng method is a general framework to generate families of pairing-friendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number

Solving discrete logarithms on a 170-bit MNT curve by pairing reduction

Pairing based cryptography is in a dangerous position following the breakthroughs on discrete logarithms computations in finite fields of small characteristic. Remaining instances are built over finite fields of large characteristic and their security relies on the fact that the embedding field of the underlying curve is relatively large. How large is debatable. The aim of our work is to sustain the claim that the combination of degree 3 embedding and too small finite fields obviously does not provide enough security. As a computational example, we solve the DLP on a 170-bit MNT curve, by exploiting the pairing embedding to a 508-bit, degree-3 extension of the base field.Comment: to appear in the Lecture Notes in Computer Science (LNCS

A short-list of pairing-friendly curves resistant to Special TNFS at the 128-bit security level

https://www.iacr.org/docs/pub_2013-16.htmlThis paper is the IACR version. It can be made freely available on the homepages of authors, on their employer's institutional page, and in non-commercial archival repositories such as the Cryptology ePrint Archive, ArXiv/CoRR, HAL, etc.International audienceThere have been notable improvements in discrete logarithm computations in finite fields since 2015 and the introduction of the Tower Number Field Sieve algorithm (TNFS) for extension fields. The Special TNFS is very efficient in finite fields that are target groups of pairings on elliptic curves, where the characteristic is special (e.g.~sparse). The key sizes for pairings should be increased, and alternative pairing-friendly curves can be considered.We revisit the Special variant of TNFS for pairing-friendly curves. In this case the characteristic is given by a polynomial of moderate degree (between 4 and 38) and tiny coefficients, evaluated at an integer (a seed). We present a polynomial selection with a new practical trade-off between degree and coefficient size. As a consequence, the security of curves computed by Barbulescu, El~Mrabet and Ghammam in 2019 should be revised: we obtain a smaller estimated cost of STNFS for all curves except BLS12 and BN.To obtain TNFS-secure curves, we reconsider the Brezing--Weng generic construction of families of pairing-friendly curves and estimate the cost of our new Special TNFS algorithm for these curves. This improves on the work of Fotiadis and Konstantinou, Fotiadis and Martindale, and Barbulescu, El~Mrabet and Ghammam. We obtain a short-list of interesting families of curves that are resistant to the Special TNFS algorithm, of embedding degrees 10 to 16 for the 128-bit security level. We conclude that at the 128-bit security level, BLS-12 and Fotiadis--Konstantinou--Martindale curves with $k=12$ over a 440 to 448-bit prime field seem to be the best choice for pairing efficiency. We also give hints at the 192-bit security level

"I think one of the ways they will help is to create awareness": primary school teachers’ perceptions of cardiovascular diseases in Nigeria

The increasing incidence and prevalence of non-communicable diseases is a major global health concern. Cardiovascular diseases (CVDs) account for the highest percentage of deaths related to non-communicable diseases, and low and middle-income countries (LMIC) face the highest burden of CVDs. Understanding the knowledge and perception of CVDs and their risk factors in an LMIC such as Nigeria may play an important role in cardiovascular health promotion and improvement plans to reduce CVD-related deaths. A qualitative study was conducted using semi-structured interviews to gain an in-depth understanding of some personal and sociocultural views on CVDs and their risk factors. The participants were purposively sampled primary school teachers in South-Eastern Nigeria. Thematic analysis approach was used for data analysis. The study findings include knowledge of heart disease, perceived causes and risk factors of CVDs, spirituality, and the way forward. Overall, the knowledge of CVDs in the setting was found to be related to the psychosocial nature of the participants; the effectiveness of any intervention needs to take these factors into consideration. For example, health policies for CVD health education and awareness should be tailored to address some of the issues of belief, values, and religion, as mentioned in the study

Faster pairings on special Weierstrass curves

This paper presents efficient formulas for computing cryptographic pairings on the curve y 2 = c x 3 + 1 over fields of large characteristic. We provide examples of pairing-friendly elliptic curves of this form which are of interest for efficient pairing implementations

Faster pairing computations on curves with high-degree twists

Research on efficient pairing implementation has focussed on reducing the loop length and on using high-degree twists. Existence of twists of degree larger than 2 is a very restrictive criterion but luckily constructions for pairing-friendly elliptic curves with such twists exist. In fact, Freeman, Scott and Teske showed in their overview paper that often the best known methods of constructing pairing-friendly elliptic curves over fields of large prime characteristic produce curves that admit twists of degree 3, 4 or 6. A few papers have presented explicit formulas for the doubling and the addition step in Miller’s algorithm, but the optimizations were all done for the Tate pairing with degree-2 twists, so the main usage of the high- degree twists remained incompatible with more efficient formulas. In this paper we present efficient formulas for curves with twists of degree 2, 3, 4 or 6. These formulas are significantly faster than their predecessors. We show how these faster formulas can be applied to Tate and ate pairing variants, thereby speeding up all practical suggestions for efficient pairing implementations over fields of large characteristic

High Security Pairing-Based Cryptography Revisited

The security and performance of pairing based cryptography has provoked a large volume of research, in part because of the exciting new cryptographic schemes that it underpins. We re-examine how one should implement pairings over ordinary elliptic curves for various practical levels of security. We conclude, contrary to prior work, that the Tate pairing is more e#cient than the Weil pairing for all such security levels. This is achieved by using e#cient exponentiation techniques in the cyclotomic subgroup backed by e#cient squaring routines within the same subgroup