Arithmetic Progressions on Conics.

In this paper, we look at long arithmetic progressions on conics. By an arithmetic progression on a curve, we mean the existence of rational points on the curve whose x-coordinates are in arithmetic progression. We revisit arithmetic progressions on the unit circle, constructing 3-term progressions of points in the first quadrant containing an arbitrary rational point on the unit circle. We also provide infinite families of three term progressions on the unit hyperbola, as well as conics ax2 + cy2 = 1 containing arithmetic progressions as long as 8 terms.

Report on Pairing-based Cryptography.

This report summarizes study results on pairing-based cryptography. The main purpose of the study is to form NIST's position on standardizing and recommending pairing-based cryptography schemes currently published in research literature and standardized in other standard bodies. The report reviews the mathematical background of pairings. This includes topics such as pairing-friendly elliptic curves and how to compute various pairings. It includes a brief introduction to existing identity-based encryption (IBE) schemes and other cryptographic schemes using pairing technology. The report provides a complete study of the current status of standard activities on pairing-based cryptographic schemes. It explores different application scenarios for pairing-based cryptography schemes. As an important aspect of adopting pairing-based schemes, the report also considers the challenges inherent in validation testing of cryptographic algorithms and modules. Based on the study, the report suggests an approach for including pairing-based cryptography schemes in the NIST cryptographic toolkit. The report also outlines several questions that will require further study if this approach is followed.

Post-quantum cryptography and the quantum future of cybersecurity.

We review the current status of efforts to develop and deploy post-quantum cryptography on the Internet. Then we suggest specific ways in which quantum technologies might be used to enhance cybersecurity in the near future and beyond. We focus on two goals: protecting the secret keys that are used in classical cryptography, and ensuring the trustworthiness of quantum computations. These goals may soon be within reach, thanks to recent progress in both theory and experiment. This progress includes interactive protocols for testing quantumness as well as for performing uncloneable cryptographic computations; and experimental demonstrations of device-independent random number generators, device-independent quantum key distribution, quantum memories, and analog quantum simulators.

ON THE FAMILY OF ELLIPTIC CURVES X + 1/X + Y + 1/Y + t = 0.

We study various properties of the family of elliptic curves x+1/x+y+1/y+t = 0, which is isomorphic to the Weierstrass curve E t : Y 2 = X ( X 2 + ( t 2 4 - 2 ) X + 1 ) . . This equation arises from the study of the Mahler measure of polynomials. We show that the rank of E t ( Q ¯ ( t ) ) is 0 and the torsion subgroup of E t ( Q ( t ) ) is isomorphic to Z ∕ 4 Z . Over the rational field Q we obtain infinite subfamilies of ranks (at least) one and two, and find specific instances of Et with rank 5 and 6. We also determine all possible torsion subgroups of E t ( Q ) and conclude with some results regarding integral points in arithmetic progression on Et .

Isogenies on twisted Hessian curves.

Elliptic curves are typically defined by Weierstrass equations. Given a kernel, the well-known Vélu's formula shows how to explicitly write down an isogeny between Weierstrass curves. However, it is not clear how to do the same on other forms of elliptic curves without isomorphisms mapping to and from the Weierstrass form. Previous papers have shown some isogeny formulas for (twisted) Edwards, Huff, and Montgomery forms of elliptic curves. Continuing this line of work, this paper derives explicit formulas for isogenies between elliptic curves in (twisted) Hessian form. In addition, we examine the numbers of operations in the base field to compute the formulas. In comparison with other isogeny formulas, we note that our formulas for twisted Hessian curves have the lowest costs for processing the kernel and our X-affine formula has the lowest cost for processing an input point in affine coordinates.

NEW MISSION AND OPPORTUNITY FOR MATHEMATICS RESEARCHERS: CRYPTOGRAPHY IN THE QUANTUM ERA.

This article introduces the NIST post-quantum cryptography standardization process. We highlight the challenges, discuss the mathematical problems in the proposed post-quantum cryptographic algorithms and the opportunities for mathematics researchers to contribute.

On addition-subtraction chains of numbers with low Hamming weight.

An addition chain is a sequence of integers such that every element in the sequence is the sum of two previous elements. They have been much studied, and generalized to addition-subtraction chains, Lucas chains, and Lucas addition-subtraction chains. These various chains have been useful in finding efficient exponentiation algorithms in groups. As a consequence, finding chains of minimal length is critical. The main objective of this paper is to extend results known for addition chains to addition-subtraction chains with Lucas addition-subtraction as a tool to construct such minimal chains. Specifically, if we let ℓ -(n) stand for the minimal length of all the Lucas addition-subtraction chains for n, we prove |ℓ -(2n) - ℓ -(n)| ≤ 1 for all integers n of Hamming weight ≤ 4. Thus, to find a minimal addition-subtraction chain for low Hamming weight integers, it suffices to only consider odd integers.

ELLIPTIC CURVES ARISING FROM THE TRIANGULAR NUMBERS.

We study the Legendre family of elliptic curves Et : y 2 = x(x - 1)(x - Δ t ), parametrized by triangular numbers Δ t = t(t + 1)/2. We prove that the rank of Et over the function field Q ‒ ( t ) is 1, while the rank is 0 over Q ( t ) . We also produce some infinite subfamilies whose Mordell-Weil rank is positive, and find high rank curves from within these families.

Integral Isosceles Triangle-Parallelogram and Heron triangle-Rhombus Pairs with a Common Area and Common Perimeter.

In this paper we show that there are infinitely many pairs of integer isosceles triangles and integer parallelograms with a common (integral) area and common perimeter. We also show that there are infinitely many Heron triangles and integer rhombuses with common area and common perimeter. As a corollary, we show there does not exist any Heron triangle and integer square which have a common area and common perimeter.

GEOMETRIC PROGRESSIONS ON ELLIPTIC CURVES.

In this paper, we look at long geometric progressions on different model of elliptic curves, namely Weierstrass curves, Edwards and twisted Edwards curves, Huff curves and general quartics curves. By a geometric progression on an elliptic curve, we mean the existence of rational points on the curve whose x-coordinate (or y-coordinate) are in geometric progression. We find infinite families of twisted Edwards curves and Huff curves with geometric progressions of length 5, an infinite family of Weierstrass curves with 8 term progressions, as well as infinite families of quartic curves containing 10-term geometric progressions.

HERON QUADRILATERALS VIA ELLIPTIC CURVES.

A Heron quadrilateral is a cyclic quadrilateral whose area and side lengths are rational. In this work, we establish a correspondence between Heron quadrilaterals and a family of elliptic curves of the form y 2 = x 3 + αx 2 - n 2 x. This correspondence generalizes the notions of Goins and Maddox who established a similar connection between Heron triangles and elliptic curves. We further study this family of elliptic curves, looking at their torsion groups and ranks. We also explore their connection with congruent numbers, which are the α = 0 case. Congruent numbers are positive integers which are the area of a right triangle with rational side lengths.

High rank elliptic curves with torsion ℤ/4ℤ.

Working over the field ℚ(t), Kihara constructed an elliptic curve with torsion group ℤ/4ℤ and five independent rational points, showing the rank is at least five. Following his approach, we give a new infinite family of elliptic curves with torsion group ℤ/4ℤ and rank at least five. This matches the current record for such curves. In addition, we give specific examples of these curves with high ranks 10 and 11.