Group law computations on Jacobians of hyperelliptic curves

We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form

Fast Scalar Multiplication on Elliptic Curve Cryptography in Selected Intervals Suitable For Wireless Sensor Networks

International audienceIn Wireless Sensor Networks (WSNs), providing a robust security mechanism with limited energy resources is very challenging because of sensor node's limited resources (computation, bandwidth, memory). Asymmetric-key can fulfill the requirement, but if the number of nodes is large, symmetric-key cryptography is the best natural method because of its scalability. Asymmetric-key cryptography is power-hungry; nevertheless, Elliptic Curve Cryptosystems (ECC) are feasible and more flexible for sensor nodes. Scalar multiplication is the most widely used operation on ECC. Various methods for fast scalar multiplication exist, but they are based on the binary/ternary representation of the scalar. In this paper, we present a novel technique to make fast scalar multiplication on Elliptic Curve Cryptosystems over prime field for light-weight embedded devices like sensor nodes. Our method significantly reduces the computation of scalar multiplication by an equivalent representation of points based on point order in a given interval. Since our technique can act as a support for most existing methods, after an analytical and efficiency analysis, we implement and evaluate its performance in different scenari

Speeding up Ate Pairing Computation in Affine Coordinates

At Pairing 2010, Lauter et al\u27s analysis showed that Ate pairing computation in affine coordinates may be much faster than projective coordinates at high security levels. In this paper, we further investigate techniques to speed up Ate pairing computation in affine coordinates. On the one hand, we improve Ate pairing computation over elliptic curves admitting an even twist by describing an $4$-ary Miller algorithm in affine coordinates. This technique allows us to trade one multiplication in the full extension field and one field inversion for several multiplications in a smaller field. On the other hand, we investigate pairing computations over elliptic curves admitting a twist of degree $3$. We propose new fast explicit formulas for Miller function that are comparable to formulas over even twisted curves. We further analyze pairing computation on cubic twisted curves by proposing efficient subfamilies of pairing-friendly elliptic curves with embedding degrees $k = 9$, and $15$. These subfamilies allow us not only to obtain a very simple form of curve, but also lead to an efficient arithmetic and final exponentiation

Software Implementation of Optimal Pairings on Elliptic Curves with Odd Prime Embedding Degrees

Pairing computations on elliptic curves with odd prime degrees are rarely studied as low efficiency. Recently, Clarisse, Duquesne and Sanders proposed two new curves with odd prime embedding degrees: \textit{BW13-P310} and \textit{BW19-P286}, which are suitable for some special cryptographic schemes. In this paper, we propose efficient methods to compute the optimal ate pairing on this types of curves, instantiated by the \textit{BW13-P310} curve. We first extend the technique of lazy reduction into the finite field arithmetic. Then, we present a new method to execute Miller\u27s algorithm. Compared with the standard Miller iteration formulas, the new ones provide a more efficient software implementation of pairing computations. At last, we also give a fast formula to perform the final exponentiation. Our implementation results indicate that it can be computed efficiently, while it is slower than that over the BLS-446 curve at the same security level

A Survey of Fast Scalar Multiplication on Elliptic Curve Cryptography for Lightweight Embedded Devices

Elliptic curve cryptography (ECC) is one of the most famous asymmetric cryptographic schemes which offers the same level of security with much shorter keys than the other widely used asymmetric cryptographic algorithm, Rivest, Shamir, and Adleman (RSA). In ECC, the main and most heavily used operation is the scalar multiplication kP, where the scalar value k is a private integer and must be secured. Various methods for fast scalar multiplication are based on the binary/ternary representation of the scalar. In this chapter, we present various methods to make fast scalar multiplication on ECC over prime field for lightweight embedded devices like wireless sensor network (WSN) and Internet of Things (IoT)

Don’t Forget Pairing-Friendly Curves with Odd Prime Embedding Degrees

Pairing-friendly curves with odd prime embedding degrees at the 128-bit security level, such as BW13-310 and BW19-286, sparked interest in the field of public-key cryptography as small sizes of the prime fields. However, compared to mainstream pairing-friendly curves at the same security level, i.e., BN446 and BLS12-446, the performance of pairing computations on BW13-310 and BW19-286 is usually considered ineffcient. In this paper we investigate high performance software implementations of pairing computation on BW13-310 and corresponding building blocks used in pairing-based protocols, including hashing, group exponentiations and membership testings. Firstly, we propose effcient explicit formulas for pairing computation on this curve. Moreover, we also exploit the state-of-art techniques to implement hashing in G1 and G2, group exponentiations and membership testings. In particular, for exponentiations in G2 and GT , we present new optimizations to speed up computational effciency. Our implementation results on a 64-bit processor show that the gap in the performance of pairing computation between BW13-310 and BN446 (resp. BLS12-446) is only up to 4.9% (resp. 26%). More importantly, compared to BN446 and BLS12-446, BW13- 310 is about 109.1% − 227.3%, 100% − 192.6%, 24.5% − 108.5% and 68.2% − 145.5% faster in terms of hashing to G1, exponentiations in G1 and GT , and membership testing for GT , respectively. These results reveal that BW13-310 would be an interesting candidate in pairing-based cryptographic protocols

New Representation Method For Integers And Its Application On Elliptic Curve Cryptography

Public-key cryptosystems are broadly used in security protocols such as key agreement, authentication, encryption and others. The two main operations in many public-key algorithms are multiplication and exponentiation of large numbers. The performance and efficiency of these cryptographic primitives are highly reliant on the efficiency of these operations. Improving the efficiency of multiplication and exponentiation by applying a recoding method or using a specific number system which can reduction the Hamming Weight of numbers is very common. This study proposes a new Radix-r representation for integers which is known as Modified Generalized Non-Adjacent Form (MGNAF)