Koblitz Curves over Quadratic Fields

Oliveira, Thomaz; López, Julio; Cervantes-Vázquez, Daniel; Rodríguez-Henríquez, Francisco

doi:10.1007/s00145-018-9294-z

Cited by 7 publications

(5 citation statements)

References 69 publications

(116 reference statements)

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“…Requires more cycle to perform operations [6,39] Requires less cycle to perform operations [6, 16] 2 Slower Multiplication [6] Faster multiplication [6, 58] 3 Higher power consumption [36] Lower power consumption [7, 36] 4 Cannot be implemented directly on the mobile phone [23,54] Can be implemented directly on the mobile phone. 5…”

Section: Resultsmentioning

confidence: 99%

An Identity-Based Elliptic Curve Cryptography for Mobile Payment Security

et al. 2020

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Security breaches have been observed in different dimensions in mobile payment system. The violation of user's privacy is a common phenomenon in mobile payment transactions. This study presents an improved security scheme for a mobile payment system using elliptic curve cryptography over a binary field with International Mobile Equipment Identity to ensure higher security. The scheme uses a payment gateway for registration and maps all input text to elliptic curve points using ASCII values. Payment details are stored on the gateway, which is encrypted but decrypted only with merchant's decryption key. The proposed scheme was evaluated in terms of key size, security strength, computational power, memory capacity, encryption and decryption time and mobile phone battery. The result shows that the scheme provides integrity, confidentiality and privacy. The result also shows that the proposed scheme is time-efficient and computationally inexpensive for resourceconstrained environment like mobile payment system.

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Section: Resultsmentioning

confidence: 99%

An Identity-Based Elliptic Curve Cryptography for Mobile Payment Security

et al. 2020

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“…We would like to remark that, although simple and perhaps even trivial in retrospect, the mentioned approaches for squaring have been proposed before in the literature. Precisely, [9] and [17] for the shuffle-based squaring and [35] for the CLMUL squaring.…”

Section: Binary Field Arithmeticmentioning

confidence: 99%

Constant Time Algorithms for ROLLO-I-128

et al. 2021

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In this work, we propose different techniques that can be used to implement the rank-based key encapsulation methods and public key encryption schemes of the ROLLO, and partially RQC, family of algorithms in a standalone, efficient and constant time library. For simplicity, we focus our attention on one specific instance of this family, ROLLO-I-128. For each of these techniques, we present explicit code (including intrinsics), or pseudo-code and performance measures to show their impact. More precisely, we use a combination of original and known results and describe procedures for Gaussian reduction of binary matrices, generation of vectors of given rank, multiplication with lazy reduction and inversion of polynomials in a composite Galois field. We also carry out a global performance analysis to show the impact of these improvements on ROLLO-I-128. Through the SUPERCOP framework, we compare it to other 128-bit secure KEMs in the NIST competition. To our knowledge, this is the first optimized full constant time implementation of ROLLO-I-128.

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“…For the linear algebra task, Magma uses Lanczos' algorithm, with approximate running time L(s) ≈ d • (F (s) + ǫ) 2 where d denotes the per-row density of the matrix M . In fact, it can be shown that d ≤ g.…”

Section: Costs Of the Index-calculus Based Algorithmmentioning

confidence: 99%

“…Indeed, Pollard's Rho algorithm is the best known algorithm to solve DLP instances on a generic elliptic curve defined over a characteristic-two field of the form F q ℓ . This algorithm has an exponential computational complexity of (1 + o(1))O 1 2 π • q ℓ [1,2]. On the other hand, using an index-calculus strategy one can solve the DLP on the Jacobian of a curve H over F q with a subexponential complexity of L q g 1 2 , √ 2 + o(1) (as q and g tend to infinity).…”

Section: Introductionmentioning

confidence: 99%

Extending the GLS endomorphism to speed up GHS Weil descent using Magma

Chi-Domínguez

Rodríguez-Henríquez

Smith

2021

Finite Fields and Their Applications

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Let q = 2 n , and let E/F q ℓ be a generalized Galbraith-Lin-Scott (GLS) binary curve, with ℓ ≥ 2 and (ℓ, n) = 1. We show that the GLS endomorphism on E/F q ℓ induces an efficient endomorphism on the Jacobian Jac H (F q ) of the genus-g hyperelliptic curve H corresponding to the image of the GHS Weil-descent attack applied to E/F q ℓ , and that this endomorphism yields a factor-n speedup when using standard index-calculus procedures for solving the Discrete Logarithm Problem (DLP) on Jac H (F q ). Our analysis is backed up by the explicit computation of a discrete logarithm defined on a prime-order subgroup of a GLS elliptic curve over the field F 2 5•31 . A Magma implementation of our algorithm finds the aforementioned discrete logarithm in about 1, 035 CPU-days.

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Koblitz Curves over Quadratic Fields

Cited by 7 publications

References 69 publications

An Identity-Based Elliptic Curve Cryptography for Mobile Payment Security

An Identity-Based Elliptic Curve Cryptography for Mobile Payment Security

Constant Time Algorithms for ROLLO-I-128

Extending the GLS endomorphism to speed up GHS Weil descent using Magma

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