scholarly journals High Speed Cryptoprocessor for η T Pairing on 128-bit Secure Supersingular Elliptic Curves over Characteristic Two Fields

2011 ◽  
pp. 442-458 ◽  
Author(s):  
Santosh Ghosh ◽  
Dipanwita Roychowdhury ◽  
Abhijit Das
Keyword(s):  
Elliptic Curves ◽  
High Speed ◽  
Supersingular Elliptic Curves ◽  
Characteristic Two

scholarly journals Construction of metrics on the set of elliptic curves over a finite field

2021 ◽  
Vol 109 (123) ◽  
pp. 125-141
Author(s):  
Keisuke Hakuta
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Multiplicative Group ◽  
Supersingular Elliptic Curves ◽  
Characteristic Two

We consider metrics on the set of elliptic curves in short Weierstrass form over a finite field of characteristic greater than three. The metrics have been first found by Mishra and Gupta (2008). Vetro (2011) constructs other metrics which are independent on the choice of a generator of the multiplicative group of the underlying finite field, whereas the metrics found by Mishra and Gupta, are dependent on the choice of a generator of the multiplicative group of the underlying finite field. Hakuta (2015, 2018) constructs metrics on the set of non-supersingular elliptic curves in shortWeierstrass form over a finite field of characteristic two and three, respectively. The aim of this paper is to point out that the metric found by Mishra and Gupta is in fact not a metric. We also construct new metrics which are slightly modified versions of the metric found by Mishra and Gupta.

Distance functions on the sets of ordinary elliptic curves in short Weierstrass form over finite fields of characteristic three

2018 ◽  
Vol 68 (4) ◽  
pp. 749-766
Author(s):  
Keisuke Hakuta
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Analogous Result ◽  
Distance Functions ◽  
Prime Field ◽  
Supersingular Elliptic Curves ◽  
Characteristic Two

Abstract We study distance functions on the set of ordinary (or non-supersingular) elliptic curves in short Weierstrass form (or simplified Weierstrass form) over a finite field of characteristic three. Mishra and Gupta (2008) firstly construct distance functions on the set of elliptic curves in short Weierstrass form over any prime field of characteristic greater than three. Afterward, Vetro (2011) constructs some other distance functions on the set of elliptic curves in short Weierstrass form over any prime field of characteristic greater than three. Recently, Hakuta (2015) has proposed distance functions on the set of ordinary elliptic curves in short Weierstrass form over any finite field of characteristic two. However, to our knowledge, no analogous result is known in the characteristic three case. In this paper, we shall prove that one can construct distance functions on the set of ordinary elliptic curves in short Weierstrass form over any finite field of characteristic three. A cryptographic application of our distance functions is also discussed.

scholarly journals Constructing supersingular elliptic curves with a given endomorphism ring

2014 ◽  
Vol 17 (A) ◽  
pp. 71-91 ◽  
Author(s):  
Ilya Chevyrev ◽  
Steven D. Galbraith
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Endomorphism Ring ◽  
Quaternion Algebra ◽  
Maximal Order ◽  
Running Time ◽  
Successive Minima ◽  
Supersingular Elliptic Curves ◽  
Class Polynomials ◽  
Theoretical Results

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathcal{O}$ be a maximal order in the quaternion algebra $B_p$ over $\mathbb{Q}$ ramified at $p$ and $\infty $. The paper is about the computational problem: construct a supersingular elliptic curve $E$ over $\mathbb{F}_p$ such that ${\rm End}(E) \cong \mathcal{O}$. We present an algorithm that solves this problem by taking gcds of the reductions modulo $p$ of Hilbert class polynomials.New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice $\mathcal{O}^T$ of $\mathcal{O}$, the order $\mathcal{O}$ is effectively characterized by the three successive minima and two other short vectors of $\mathcal{O}^T\! .$ The desired conditions turn out to hold whenever the $j$-invariant $j(E)$, of the elliptic curve with ${\rm End}(E) \cong \mathcal{O}$, lies in $\mathbb{F}_p$. We can then prove that our algorithm terminates with running time $O(p^{1+\varepsilon })$ under the aforementioned conditions.As a further application we present an algorithm to simultaneously match all maximal order types with their associated $j$-invariants. Our algorithm has running time $O(p^{2.5 + \varepsilon })$ operations and is more efficient than Cerviño’s algorithm for the same problem.

Faster MapToPoint on Supersingular Elliptic Curves in Characteristic 3

2011 ◽  
Vol E94-A (1) ◽  
pp. 150-155 ◽  
Author(s):  
Yuto KAWAHARA ◽  
Tetsutaro KOBAYASHI ◽  
Gen TAKAHASHI ◽  
Tsuyoshi TAKAGI
Keyword(s):  
Elliptic Curves ◽  
Supersingular Elliptic Curves ◽  
Characteristic 3

scholarly journals Fast Architectures for the \eta_T Pairing over Small-Characteristic Supersingular Elliptic Curves

2011 ◽  
Vol 60 (2) ◽  
pp. 266-281 ◽  
Author(s):  
Jean-Luc Beuchat ◽  
Jeremie Detrey ◽  
Nicolas Estibals ◽  
Eiji Okamoto ◽  
Francisco Rodriguez Henriquez
Keyword(s):  
Elliptic Curves ◽  
Supersingular Elliptic Curves
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