Fast computation of canonical lifts of elliptic curves and its application to point counting

Satoh, Takakazu; Skjernaa, Berit; Taguchi, Yuichiro

doi:10.1016/s1071-5797(02)00013-8

Cited by 29 publications

(50 citation statements)

References 4 publications

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“…A more elaborated implementation strategy (with indexes) yields a O(n) time complexity [7]. 2 ) time complexity with space equal to O(nm) with the clever algorithm described in [12].…”

Section: Lifting the Frobeniusmentioning

confidence: 99%

“…More specifically, Satoh et al underlined the importance in the so-called SST algorithm to solve this problem when φ is equal to Φ p , the p-th modular polynomial (to compute the j-invariant of the canonical lift) [12], or equal to x p − y (for computing the Teichmüller lift) [14]. Since the idea behind the SST algorithm is to use the Taylor expansion of φ to increment by one the precision of the root of equation (3) computed at the intermediate steps in the algorithm, it is not difficult to generalize it to a more general φ.…”

Section: Lifting Algorithmsmentioning

confidence: 99%

“…In recent papers, Satoh, Skjernaa and Taguchi [12] improved further the complexity of Satoh's algorithm to reach a O(n 2µ+ 1 2 ) time complexity and a O(n 2 ) space complexity after a O(n 3 ) bit operations precomputation. Finally, H. Kim et al [7] showed that if the base field of the elliptic curve has a Gaussian Normal Basis (which is the case for numerous fields in a cryptographic context), the complexity in time of this algorithm can be reduced to O(n 2µ+ 1 1+µ ).…”

Section: Introductionmentioning

confidence: 99%

“…Formally speaking, images by π m of these elements of Z p will be in these algorithms said to be the mathematical objects computed "at precision m" or "at accuracy m" or "modulo p m ". Following the conventions of [12], we will denote by T m,n the complexity of multiplying two elements of Z p n with accuracy m. Furthermore, in the following S m,n will be the complexity in time to compute at accuracy m the image of an element of Z q by a power of the Frobenius substitution.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Counting Points on Elliptic Curves over Finite Fields of Small Characteristic in Quasi Quadratic Time

Lercier¹,

Lubicz²

2003

Lecture Notes in Computer Science

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Abstract. Let p be a small prime and q = p n . Let E be an elliptic curve over Fq. We propose an algorithm which computes without any preprocessing the j-invariant of the canonical lift of E with the cost of O(log n) times the cost needed to compute a power of the lift of the Frobenius. Let µ be a constant so that the product of two n-bit length integers can be carried out in O(n µ ) bit operations, this yields an algorithm to compute the number of points on elliptic curves which reaches, at the expense of a O(n 5 2 ) space complexity, a theoretical time complexity bound equal to O(n max(1.19,µ)+µ+ 1 2 log n). When the field has got a Gaussian Normal Basis of small type, we obtain furthermore an algorithm with O(log(n)n 2µ ) time and O(n 2 ) space complexities. From a practical viewpoint, the corresponding algorithm is particularly well suited for implementations. We outline this by a 100002-bit computation.

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“…A more elaborated implementation strategy (with indexes) yields a O(n) time complexity [7]. 2 ) time complexity with space equal to O(nm) with the clever algorithm described in [12].…”

Section: Lifting the Frobeniusmentioning

confidence: 99%

Section: Lifting Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Counting Points on Elliptic Curves over Finite Fields of Small Characteristic in Quasi Quadratic Time

Lercier¹,

Lubicz²

2003

Lecture Notes in Computer Science

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“…The security of EC depends on the size of the EC order. The common methods for computing order include Schoof's algorithm [5], SEA (Schoof-Elkies-Atkin) algorithm [6]， Satoh algorithm [7] and so on.…”

Section: Koblitz Elliptic Curvesmentioning

confidence: 99%

Hybrid Differential Evolutionary Algorithms for Koblitz Elliptic Curves Generating

Ku¹,

Cai²,

Yang³

2014

Proceedings of the 2014 International Conference on Mechatronics, Control and Electronic Engineering

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Karatsuba-Block-Comb technique for elliptic curve cryptography over binary fields

Seo

Liu

Choi

et al. 2015

Security Comm. Networks

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Efficient implementation of elliptic curve cryptography on resource‐constrained microcontroller is considered to be one of the hot and challenging research topics because of the limited computing power and storages of target platforms and high computational costs of elliptic curve cryptography. In this paper, we focus on enhancing the performance of scalar multiplication over GF(2m) by suggesting a new technique for speeding up the performance of multiplication, called Karatsuba–Block‐Comb (KBC) multiplication. KBC method combines both the advantages of Karatsuba algorithm and Block‐Comb method. This technique replaces the part of expensive Block‐Comb binary field multiplications with several cheap additions by following Karatsuba rule. In case of squaring, we describe an optimized squaring algorithm with 8‐bit look‐up table that is significantly faster than previous works with 4‐bit look‐up table. Both of the proposed approaches improve the best known results by a factor of 24.6% and 16.8% (160‐bit operand) over 8‐bit AVR processor (Atmel Corporation, San Jose, CA, USA), respectively. Finally, we realize the scalar multiplication over GF(2163), which only requires 0.29 s for a full scalar multiplication when the processor runs at 7.37 MHz. This result outperforms the previous best implementation by a factor of 9.3%. The research results presented in this paper prove that it is also possible to achieve high performance over binary fields by combing the algorithm with sub‐quadratic complexity. Furthermore, we suggest constant time KBC method. Block‐Comb method does not provide constant time, and look‐up table method is also vulnerable to memory address side channel attack. However, our method is establishing the scalar multiplication in 0.35 s with high security against both attacks. Copyright © 2015 John Wiley & Sons, Ltd.

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Fast computation of canonical lifts of elliptic curves and its application to point counting

Cited by 29 publications

References 4 publications

Counting Points on Elliptic Curves over Finite Fields of Small Characteristic in Quasi Quadratic Time

Counting Points on Elliptic Curves over Finite Fields of Small Characteristic in Quasi Quadratic Time

Hybrid Differential Evolutionary Algorithms for Koblitz Elliptic Curves Generating

Karatsuba-Block-Comb technique for elliptic curve cryptography over binary fields

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