Arithmetic of split Kummer surfaces: Montgomery endomorphism of Edwards products

International audienceLet $E$ be an elliptic curve, $\mathcal{K}_1$ its Kummer curve $E/\{\pm1\}$, $E^2$ its square product, and $\mathcal{K}_2$ the split Kummer surface $E^2/\{\pm1\}$. The addition law on $E^2$ gives a large endomorphism ring, which induce endomorphisms of $\mathcal{K}_2$. With a view to the practical applications to scalar multiplication on $\mathcal{K}_1$, we study the explicit arithmetic of $\mathcal{K}_2$

A New Encoding Algorithm for a Multidimensional Version of the Montgomery Ladder

We propose a new encoding algorithm for the simultaneous differential multidimensional scalar point multiplication algorithm $d$-MUL. Previous encoding algorithms are known to have major drawbacks in their efficient and secure implementation. Some of these drawbacks have been avoided in a recent paper in 2018 at a cost of losing the general functionality of the point multiplication algorithm. In this paper, we address these issues. Our new encoding algorithm takes the binary representations of scalars as input, and constructs a compact binary sequence and a permutation, which explicitly determines a regular sequence of group operations to be performed in $d$-MUL. Our algorithm simply slides windows of size two over the scalars and it is very efficient. As a result, while preserving the full generality of $d$-MUL, we successfully eliminate the recursive integer matrix computations in the originally proposed encoding algorithms. We also expect that our new encoding algorithm will make it easier to implement $d$-MUL in constant time. Our results can be seen as the efficient and full generalization of the one dimensional Montgomery ladder to arbitrary dimension

Efficient arithmetic on elliptic curves in characteristic 2

International audienceWe present normal forms for elliptic curves over a field of characteristic 2 analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient algorithms for point addition and scalar multiplication on these forms. The resulting algorithms apply to any elliptic curve over a field of characteristic 2 with a 4-torsion point, via an isomorphism with one of the normal forms. We deduce algorithms for duplication in time $2M + 5S + 2m_c$ and for addition of points in time $7M + 2S$, where $M$ is the cost of multiplication, $S$ the cost of squaring , and $m_c$ the cost of multiplication by a constant. By a study of the Kummer curves $\mathcal{K} = E/\{\pm1]\}$, we develop an algorithm for scalar multiplication with point recovery which computes the multiple of a point P with $4M + 4S + 2m_c + m_t$ per bit where $m_t$ is multiplication by a constant that depends on $P$

Elliptic Curve Scalar Multiplication Combining Yao’s Algorithm and Double Bases

Abstract. In this paper we propose to take one step back in the use of double base number systems for elliptic curve point scalar multiplication. Using a mod-ified version of Yao’s algorithm, we go back from the popular double base chain representation to a more general double base system. Instead of representing an integer k as Pn i=1 2 bi3ti where (bi) and (ti) are two decreasing sequences, we only set a maximum value for both of them. Then, we analyze the efficiency of our new method using different bases and optimal parameters. In particular, we pro-pose for the first time a binary/Zeckendorf representation for integers, providing interesting results. Finally, we provide a comprehensive comparison to state-of-the-art methods, including a large variety of curve shapes and latest point addition formulae speed-ups

Making Password Authenticated Key Exchange Suitable For Resource-Constrained Industrial Control Devices

Connectivity becomes increasingly important also for small embedded systems such as typically found in industrial control installations. More and more use-cases require secure remote user access increasingly incorporating handheld based human machine interfaces, using wireless links such as Bluetooth. Correspondingly secure operator authentication becomes of utmost importance. Unfortunately, often passwords with all their well-known pitfalls remain the only practical mechanism. We present an assessment of the security requirements for the industrial setting, illustrating that offline attacks on passwords-based authentication protocols should be considered a significant threat. Correspondingly use of a Password Authenticated Key Exchange protocol becomes desirable. We review the signif-icant challenges faced for implementations on resource-constrained devices. We explore the design space and shown how we succeeded in tailoring a partic-ular variant of the Password Authenticated Connection Establishment (PACE) protocol, such that acceptable user interface responsiveness was reached even for the constrained setting of an ARM Cortex-M0+ based Bluetooth low-energy transceiver running from a power budget of 1.5 mW without notable energy buffers for covering power peak transients

MoTE-ECC: Energy-Scalable Elliptic Curve Cryptography for Wireless Sensor Networks

Wireless Sensor Networks (WSNs) are susceptible to a wide range of malicious attacks, which has stimulated a body of research on "light-weight" security protocols and cryptographic primitives that are suitable for resource-restricted sensor nodes. In this paper we introduce MoTE-ECC, a highly optimized yet scalable ECC library for Memsic's MICAz motes and other sensor nodes equipped with an 8-bit AVR processor. MoTE-ECC supports scalar multiplication on Montgomery and twisted Edwards curves over Optimal Prime Fields (OPFs) of variable size, e.g. 160, 192, 224, and 256 bits, which allows for various trade-offs between security and execution time (resp. energy consumption). OPFs are a special family of "low-weight" prime fields that, in contrast to the NIST-specified fields, facilitate a parameterized implementation of the modular arithmetic so that one and the same software function can be used for operands of different length. To demonstrate the performance of MoTE-ECC, we take (ephemeral) ECDH key exchange between two nodes as example, which requires each node to execute two scalar multiplications. The first scalar multiplication is performed on a fixed base point (to generate a key pair), whereas the second scalar multiplication gets an arbitrary point as input. Our implementation uses a fixed-base comb method on a twisted Edwards curve for the former and a simple ladder approach on a birationally-equivalent Montgomery curve for the latter. Both scalar multiplications require about 9*10^6 clock cycles in total and occupy only 380 bytes in RAM when the underlying OPF has a length of 160 bits. We also describe our efforts to harden MoTE-ECC against side-channel attacks (e.g. simple power analysis) and introduce a highly regular implementation of the comb method

Computing supersingular isogenies on Kummer surfaces

We apply Scholten\u27s construction to give explicit isogenies between the Weil restriction of supersingular Montgomery curves with full rational 2-torsion over $GF(p^2)$ and corresponding abelian surfaces over $GF(p)$. Subsequently, we show that isogeny-based public key cryptography can exploit the fast Kummer surface arithmetic that arises from the theory of theta functions. In particular, we show that chains of 2-isogenies between elliptic curves can instead be computed as chains of Richelot (2,2)-isogenies between Kummer surfaces. This gives rise to new possibilities for efficient supersingular isogeny-based cryptography

FourQ on Embedded Devices with Strong Countermeasures Against Side-Channel Attacks

This work deals with the energy-efficient, high-speed and high-security implementation of elliptic curve scalar multiplication, elliptic curve Diffie-Hellman (ECDH) key exchange and elliptic curve digital signatures on embedded devices using FourQ and incorporating strong countermeasures to thwart a wide variety of side-channel attacks. First, we set new speed records for constant-time curve-based scalar multiplication, DH key exchange and digital signatures at the 128-bit security level with implementations targeting 8, 16 and 32-bit microcontrollers. For example, our software computes a static ECDH shared secret in 6.9 million cycles (or 0.86 seconds @8MHz) on a low-power 8-bit AVR microcontroller which, compared to the fastest Curve25519 and genus-2 Kummer implementations on the same platform, offers 2x and 1.4x speedups, respectively. Similarly, it computes the same operation in 496 thousand cycles on a 32-bit ARM Cortex-M4 microcontroller, achieving a factor-2.9 speedup when compared to the fastest Curve25519 implementation targeting the same platform. A similar speed performance is observed in the case of digital signatures. Second, we engineer a set of side-channel countermeasures taking advantage of FourQ\u27s rich arithmetic and propose a secure implementation that offers protection against a wide range of sophisticated side-channel attacks, including differential power analysis (DPA). Despite the use of strong countermeasures, the experimental results show that our FourQ software is still efficient enough to outperform implementations of Curve25519 that only protect against timing attacks. Finally, we perform a differential power analysis evaluation of our software running on an ARM Cortex-M4, and report that no leakage was detected with up to 10 million traces. These results demonstrate the potential of deploying FourQ on low-power applications such as protocols for the Internet of Things

Cofactorization on Graphics Processing Units

We show how the cofactorization step, a compute-intensive part of the relation collection phase of the number field sieve (NFS), can be farmed out to a graphics processing unit. Our implementation on a GTX 580 GPU, which is integrated with a state-of-the-art NFS implementation, can serve as a cryptanalytic co-processor for several Intel i7-3770K quad-core CPUs simultaneously. This allows those processors to focus on the memory-intensive sieving and results in more useful NFS-relations found in less time