Koblitz Elliptic Curves

Synonyms

Anomalous binary curves

Related Concepts

Elliptic Curve Cryptography; Elliptic Curves

Definition

Koblitz curves are elliptic curves defined over $\mathbb{F}[2]$.

Background

Koblitz curves, also known as anomalous binary curves, were proposed by Koblitz [2] for cryptographic use. Compared to random binary curves, significantly faster point multiplication methods are available.

Applications

Elliptic curves ${E}_{a} : {y}^{2} + xy = {x}^{3} + a{x}^{2} + 1$ defined over $\mathbb{F}[2]$ (with a ∈ { 0, 1}), known as Koblitz curves, permit computationally attractive point multiplication algorithms that replace point doubling by inexpensive field-squaring operations. The basic strategy is outlined here; see [3] for details.

Let m > 0 be an integer. Given a point $(x,y) \in {E}_{a}(\mathbb{F}[{2}^{m}])$, the Frobenius map $\tau : (x,y)\mapsto ({x}^{2},{y}^{2})$ gives another point in ${E}_{a}(\mathbb{F}[{2}^{m}])$, and this mapping can be exploited in point multiplication....