Jacobian Coordinates on Genus 2 Curves

Hışıl, Hüseyin; Costello, Craig

doi:10.1007/978-3-662-45611-8_18

Cited by 11 publications

(5 citation statements)

References 28 publications

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“…We compute the group operation ⊕ in J C (F q ) using a function ADD, which implements the algorithm found in [14] (after a change of coordinates to meet their Assumption 1) 3 at a cost of 28M + 2S + 11a + 24s + 1I.…”

Section: Elements Of J C Compressed and Decompressedmentioning

confidence: 99%

$$\mu $$Kummer: Efficient Hyperelliptic Signatures and Key Exchange on Microcontrollers

Renes¹,

Schwabe²,

Smith³

et al. 2016

Lecture Notes in Computer Science

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Abstract. We describe the design and implementation of efficient signature and key-exchange schemes for the AVR ATmega and ARM Cortex M0 microcontrollers, targeting the 128-bit security level. Our algorithms are based on an efficient Montgomery ladder scalar multiplication on the Kummer surface of Gaudry and Schost's genus-2 hyperelliptic curve, combined with the Jacobian point recovery technique of Costello, Chung, and Smith. Our results are the first to show the feasibility of software-only hyperelliptic cryptography on constrained platforms, and represent a significant improvement on the elliptic-curve state-of-the-art for both key exchange and signatures on these architectures. Notably, our key-exchange scalar-multiplication software runs in under 9740k cycles on the ATmega, and under 2650k cycles on the Cortex M0.

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Section: Elements Of J C Compressed and Decompressedmentioning

confidence: 99%

$$\mu $$Kummer: Efficient Hyperelliptic Signatures and Key Exchange on Microcontrollers

Renes¹,

Schwabe²,

Smith³

et al. 2016

Lecture Notes in Computer Science

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“…In a recent publication, the authors proposed HEC-DH 25 and implemented it over different genus curves. [26][27][28][29][30] The performance shows that, for constrained devices, Genus 2- 31 and Genus 3-based HECCs are highly recommendable, whereas Genus 4 and Genus 6 are suggestible for large-scale networks. Hence, in this paper, we used Genus 2 and Genus 3 curves in HEC-based key agreement algorithm to derive a common session key.…”

Section: Introductionmentioning

confidence: 99%

Lightweight secure communication system based on Message Queuing Transport Telemetry protocol for e‐healthcare environments

Naresh¹,

Reddi

Allavarpu

2021

Int J Communication

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In recent years, with the advent of communication technologies, healthcare sensing and remote monitoring have undergone a significant evolution to addressing almost all current e-health challenges. In view of this, the Internet of medical things (IoMT)-based applications are evolved. However, security and privacy are the primary concern as vast numbers of devices are connected and communicated through the wireless environment. The direct involvement of humans in IoMT-based healthcare applications made robust and secure communication among the sensors, actuators, and patients significant. In this direction, we proposed a novel security framework for Message Queuing Transport Telemetry (MQTT) protocol based on publish/subscribe messages, which is suitable for constrained and small devices in IoMT. In this paper, we proposed a lightweight hyper elliptic curve-multiple shared key algorithm to derive session keys in order to encrypt/decrypt health readings from the sensors connected to the patient body. The comparative analysis of performance shows that the proposed method outperforms different existing techniques in terms of computational time by reducing the computational times of broker and producer/subscriber by 0.084 and 0.0168, respectively, than the best performed existing method (Malina et al.). Finally, the security analysis shows that the proposed framework is secure against physical attacks, key control, machine-in-the-middle (MITM), non-repudiation, replay, and naming based attacks.

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“…For instance, it is known that all absolutely simple principally polarized abelian surfaces are isomorphic to the jacobian Jac(H) of an hyperelliptic curve H of genus 2. The addition law can then be computed using Cantor's algorithm [Can87] and these formulas have been optimized in [Lan05;HC]. Unfortunately, even with these formulas, genus 2 curves do not provide the same efficiency as elliptic curves for a similar level of security.…”

Section: Introductionmentioning

confidence: 99%

Arithmetic on abelian and Kummer varieties

Lubicz

Damien

2016

Finite Fields and Their Applications

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Abstract. A Kummer variety is obtained as the quotient of an abelian variety by the automorphism (−1) acting on it. Kummer varieties can be seen as a higher dimensional generalisation of the xcoordinate representation of a point of an elliptic curve given by its Weierstrass model. Although there is no group law on the set of points of a Kummer variety, the multiplication of a point by a scalar still makes sense, since it is compatible with the action of (−1), and can efficiently be computed with a Montgomery ladder. In this paper, we explain that the arithmetic of a Kummer variety is not limited to this scalar multiplication and is much richer than usually thought. We describe a set of composition laws which exhaust this arithmetic and explain how to compute them efficiently in the model of Kummer varieties provided by level 2 theta functions. Moreover, we present concrete example where these laws turn out to be useful in order to improve certain algorithms. As an application interesting for instance in cryptography, we explain how to recover the full group law of the abelian variety with a representation almost as compact and in many cases as efficient as the level 2 theta functions model of Kummer varieties.

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Jacobian Coordinates on Genus 2 Curves

Cited by 11 publications

References 28 publications

$$\mu $$Kummer: Efficient Hyperelliptic Signatures and Key Exchange on Microcontrollers

$$\mu $$Kummer: Efficient Hyperelliptic Signatures and Key Exchange on Microcontrollers

Lightweight secure communication system based on Message Queuing Transport Telemetry protocol for e‐healthcare environments

Arithmetic on abelian and Kummer varieties

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