Abstract
This chapter covers the essentials of elliptic curves as used in cryptography. The first section of the chapter gives the basics concepts of elliptic curves: the main defining equations for the curves of interest and an explanation of the arithmetic operations of “addition” and “multiplication” in the context of elliptic curves. We shall follow standard practice and first define elliptic curves over the field of real numbers, with geometric and algebraic interpretations of the arithmetic operations in relation to points on a curve.
Elliptic curves over the field of real numbers are not useful in cryptography, but the initial interpretations given are useful as a means of visualizing and understanding the arithmetic operations and the derivation of the relevant equations that are ultimately used in practice. In cryptography, the elliptic curves used are defined over finite fields, and the second section of the chapter covers that, with a focus on the two most commonly used fields: GF(p), with p prime, and GF(2m), with m a positive integer (The main aspect of the first two sections is the definition of point addition and point multiplication, the latter being the primary operation in elliptic-curve cryptosystems.) The third section is on the implementation of point multiplication. And the last section is on projective coordinates, which simply inversion relative to the “normal” affine coordinates.
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- 1.
These are obtained from Eqs. 8.1 by a change of variables.
- 2.
This ensures that the polynomial has distinct roots and no singularities.
- 3.
One may imagine that \(\varnothing \) is the point where parallel lines meet.
- 4.
“Extrapolate” from the case where P ≠ Q: if Q approaches P, then the line through the two points approaches the tangent and in the limit is that tangent.
- 5.
It would be awkward to have, say, either encryption or decryption yield different results according to different errors from different sequences of basic arithmetic operations for the same point operation.
- 6.
The general Discrete-Logarithm Problem is stated in Sect. 7.1.
- 7.
Except for GF(2) and GF(3), which need not concern us.
- 8.
And this has an effect on the choice of representation.
- 9.
Note that a polynomial-basis element could just as easily be represented by the string (a 0a 1⋯a m−1) and a normal-basis element by (a m−1a m−2⋯a 0). The choices here are simply a matter of convention.
- 10.
This seemingly unusual choice is explained in Sect. 11.1.
- 11.
We leave it to the reader to ascertain that here scanning the multiplier, k, from right to left gives an algorithm that is “awkward” for high-radix computation.
References
D. Hankerson, A. Menezes, and S. Vanstone. 2004. Guide to Elliptic Curve Cryptography. Springer-Verlag, New York.
H. Cohen, G. Frey, et al. 2005. Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman-Hall/CRC. Boca Raton, USA.
I. Blake, G. Seroussi, and N. Smart. 1999. Elliptic Curves in Cryptography. London Mathematical Society 265, Cambridge University Press.
L. Washington. 2003. Elliptic Curves: Number Theory and Cryptography. Chapman-Hall/CRC, Boca Raton, USA.
National Institute of Standards and Technology. 1999. Recommended Elliptic Curves for Federal Government Use. Gaithersburg, Maryland, USA.
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R. Omondi, A. (2020). Elliptic-Curve Basics. In: Cryptography Arithmetic. Advances in Information Security, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-030-34142-8_8
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