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Open AccessArticle

On the Search for Supersingular Elliptic Curves and Their Applications

by Ismel Martinez-Diaz, Rashad Ali and Muhammad Kamran Jamil

Mathematics 2025, 13(2), 188; https://doi.org/10.3390/math13020188 - 8 Jan 2025

Cited by 1 | Viewed by 1145

Elliptic curves with the special quality known as supersingularity have gained much popularity in the rapidly developing field of cryptography. The conventional method of employing random search is quite ineffective in finding these curves. This paper analyzes the search of supersingular elliptic curves [...] Read more.

F_{p^{2}}

. We show that naive random search is unsuitable to easily find any supersingular elliptic curves when the space size is greater than

10^{13}

. We improve the random search using a necessary condition for supersingularity. As our main result, we define for the first time an objective function to measure the supersingularity in ordinary curves, and we apply local search and a genetic algorithm using that function. The study not only finds these supersingular elliptic curves but also investigates possible uses for them. These curves were used to create cycles inside the isogeny graph in one particular application. The research shows how the design of S-boxes may strategically use these supersingular elliptic curves. The key components of replacement, which is a fundamental step in the encryption process that shuffles and encrypts the data inside images, are S-boxes. This work represents a major advancement in effectively identifying these useful elliptic curves, eventually leading to their wider application and influence in the rapidly expanding field of cryptography. Full article

(This article belongs to the Special Issue New Advances in Coding Theory and Cryptography, 2nd Edition)

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13 pages, 379 KB

Open AccessArticle

Optimized CSIDH Implementation Using a 2-Torsion Point

by Donghoe Heo, Suhri Kim, Kisoon Yoon, Young-Ho Park and Seokhie Hong

Cryptography 2020, 4(3), 20; https://doi.org/10.3390/cryptography4030020 - 29 Jul 2020

Cited by 4 | Viewed by 3806

Abstract

The implementation of isogeny-based cryptography mainly use Montgomery curves, as they offer fast elliptic curve arithmetic and isogeny computation. However, although Montgomery curves have efficient 3- and 4-isogeny formula, it becomes inefficient when recovering the coefficient of the image curve for large degree isogenies. Because the Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) requires odd-degree isogenies up to at least 587, this inefficiency is the main bottleneck of using a Montgomery curve for CSIDH. In this paper, we present a new optimization method for faster CSIDH protocols entirely on Montgomery curves. To this end, we present a new parameter for CSIDH, in which the three rational two-torsion points exist. By using the proposed parameters, the CSIDH moves around the surface. The curve coefficient of the image curve can be recovered by a two-torsion point. We also proved that the CSIDH while using the proposed parameter guarantees a free and transitive group action. Additionally, we present the implementation result using our method. We demonstrated that our method is 6.4% faster than the original CSIDH. Our works show that quite higher performance of CSIDH is achieved while only using Montgomery curves. Full article

(This article belongs to the Special Issue Post-Quantum Cryptography: From Theoretical Foundations to Practical Deployments)

16 pages, 360 KB

Open AccessArticle

Memory Efficient Implementation of Modular Multiplication for 32-bit ARM Cortex-M4

by Hwajeong Seo

Appl. Sci. 2020, 10(4), 1539; https://doi.org/10.3390/app10041539 - 24 Feb 2020

Cited by 8 | Viewed by 5189

Abstract

In this paper, we present scalable multi-precision multiplication implementation and scalable multi-precision squaring implementation for 32-bit ARM Cortex-M4 microcontrollers. For efficient computation and scalable functionality, we present optimized Multiplication and ACcumulation (MAC) techniques for the target microcontrollers. In particular, we present the 64-bit wise MAC operation with the Unsigned Long Multiply with Accumulate Accumulate (UMAAL) instruction. The MAC is used to perform column-wise multiplication/squaring (i.e., product-scanning) with general-purpose registers in an optimal way. Second, the squaring algorithm is further optimized through an efficient doubling routine together with an optimized product-scanning method. Finally, the proposed implementations achieved a very small memory footprint and high scalability to cover algorityms ranging from well-known public key cryptography (i.e., Rivest–Shamir–Adleman (RSA) and Elliptic Curve Cryptography (ECC)) to post-quantum cryptography (i.e., Supersingular Isogeny Key Encapsulation (SIKE)). All SIKE round 2 protocols were evaluated with the proposed modular reduction implementations. The results demonstrate that the scalable implementation can achieve the smallest code size together with a reasonable performance. Full article

(This article belongs to the Special Issue Side Channel Attacks and Countermeasures)

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