Elliptic Curve Cryptography

Summary

This documentary gives a brief introduction into elliptic curve cryptography

Elliptic Curve Cryptography

Elliptic curve cryptography is a part of asymmetric cryptography, it is based on the mathematical hard problem to find a solution for the elliptic curve discrete logarithm. The calculations are performed on the algebraic structure of elliptic curves over finite fields, which means we compute points on a elliptic curve over finite field by applying the group operations double and add. The scalar multiplication of a point on an elliptic curve over a finite field is equivalent to the exponentation of a number in a prime field, therefore the inversion is also called discrete logarithm.

First proposed application of elliptic curves in cryptography was random number generations, now ECC is widely used for key establishment and digital signature schemes.

Simple Weierstrass Elliptic Curve Presentation

• Simple Weierstrass form curve equation:

$y^{2}=x^{3}+ax+b$ The elliptic curve are all points in the $x,y$ coordinates $P(x,y)$ which fulfill the cubic curve equation, whereas a and b are called the characteristic of the curve. The curve needs to be smooth, which means that it will not contain any singularities such as a cusp or a self intersection,

This can be also described by the term: $4a^{3}+27b^{2}\neq 0$

Another characteristic we need to introduce is the point at infinity denoted by ${\mathcal {O}}$ (also known as ideal point), which can be thought as identity element infinitly raised on the y axis. Therefore our points on the elliptic curve over R² all fulfill this equation $\{(x,y)\in R^{2}|y^{2}=x^{3}+ax+b,4a^{3}+27b^{2}\neq 0\}\cup \{{\mathcal {O}}\}$. A valid curve over rational numbers is shown in the next image.

In cryptography elliptic curves over finite fields are used. The number of rational points $\#E$ of an elliptic curve over a finite field $K$ e.g. the prime field $GF(p)$ can be computed with the Schoof-Elkies-Alkin algorithm, which is implemented in the PARI/GP library. The Hasse theorem gives an estimation of the number of points in the intervall: $p+1-2{\sqrt {p}}\leq \#E\leq p+1+2{\sqrt {p}}$

Group Operations on Elliptic Curves

According to the group law all points support following operations:

• Point Addition: $P+Q=R$
• Case: $Q=P$ -> Point Doubling : $2P=R$
• Case: $Q=-P$ -> Inversion of a Point: $P+(-P)={\mathcal {O}}$

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Point Addition

Given the elliptic curve $E:y^{2}=x^{3}+ax+b$ and the points $P=(x_{1},y_{1})$ and $Q=(x_{2},y_{2})$, we can calculate the coordinates of the point $R(x_{3},y_{3})$ by adding this two points $R=P+Q$ as follows.

Geometric derivation of the point addition by the Tangent Chord Law

1. Calculation of $y_{3}$

• We know the equation from the line

$y=kx+d\qquad {\text{concising of}}\qquad k={\frac {\Delta y}{\Delta x}}$

• Therfore

$k={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$ and we can find the intersection with the y-axis and achieve d $d=y_{1}-kx_{1}={\frac {y_{1}\cdot x_{2}-y_{1}\cdot x_{1}-y_{2}\cdot x_{1}+y_{1}\cdot x_{1}}{x_{2}-x_{1}}}={\frac {y_{1}\cdot x_{2}-y_{2}\cdot x_{1}}{x_{2}-x_{1}}}$.

• $y_{3}$ we can find by insertion in the line equation:

$y_{3}=(-1){\bigg (}{\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\cdot x_{3}+y_{1}-{\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\cdot x_{1}{\bigg )}={\bigg (}{\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}{\bigg )}(x_{1}-x_{3})-y_{1}$

2. Calculation of $x_{3}$

• Now we insert the values of the line equation into the elliptic curve equation:

${\begin{aligned}(k\cdot x+d)^{2}&=x^{3}+a\cdot x+b\\k^{2}x^{2}+2k\cdot x\cdot d+d^{2}&=x^{3}+a\cdot x+b\\x^{3}-x^{2}k^{2}+x(a-2k\cdot d)+b-d^{2}&=0\end{aligned}}$

• The cross points can be searched by

${\begin{aligned}(x-x_{1})\cdot (x-x_{2})\cdot (x-x_{3})&=0\\(x^{2}-x\cdot x_{2}-x\cdot x_{1}+x_{1}\cdot x_{2})\cdot (x-x_{3})&=0\\x^{3}-x^{2}\cdot x_{2}-x^{2}\cdot x_{1}+x\cdot x_{1}x_{2}-x^{2}\cdot x_{3}+x\cdot x_{2}x_{3}+x\cdot x_{1}x_{3}-x_{1}x_{2}x_{3}&=0\\x^{3}-x^{2}(x_{1}+x_{2}+x_{3})+x(x_{1}x_{2}+x_{2}x_{3}+x_{1}x_{3})+x_{1}x_{2}x_{3}&=0\end{aligned}}$

• Now we can conclude from the second term

${\begin{aligned}x^{3}-x^{2}{\underline {k^{2}}}+x(a-2k\cdot d)+b-d^{2}&=0\\x^{3}-x^{2}{\underline {(x_{1}+x_{2}+x_{3})}}+x(x_{1}x_{2}+x_{2}x_{3}+x_{1}x_{3})+x_{1}x_{2}x_{3}&=0\\k^{2}&=x_{1}+x_{2}+x_{3}\end{aligned}}$

• and achieve a solution for $x_{3}$ by:

$x_{3}=k^{2}-x_{1}-x_{2}={\bigg (}{\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}{\bigg )}^{2}-x_{1}-x_{2}$

$x_{3}={\bigg (}{\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}{\bigg )}^{2}-x_{1}-x_{2}\ {\text{ and }}\ y_{3}={\bigg (}{\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}{\bigg )}(x_{1}-x_{3})-y_{1}.$

This operation is also called the point addition in affine coordinates and the computational costs are 1 x inversion, 2 x multiplications and 1 x squaring. Interactive tool to visualize point addition online at https://cdn.rawgit.com/andreacorbellini/ecc/920b29a/interactive/reals-add.html.

Example of elliptic point operations over an elliptic curve over a finite field

Given a curve E $E:y^{2}=x^{3}+3x+5{\text{ over }}\mathbb {Z} _{17}$.

• First determine that the curve is smooth by calculating the discriminant.

$\Delta =-16(4a^{3}+27b^{2})=-12528\equiv 1(\mod 17)$ therefore $\Delta \neq 0$ holds and this curve can be used for EC operations.

• Choosing one random point of the curve P(6,1) as generator G all points on the curve can be calculated with the formular given above over a the prime field 17. This means first the point gets doubled and a modulo 17 operation is applied. Then continuously all other points are calculated by adding the point P to the result each time applying the modulo 17 .

(6,1)	(4,8)	(15,5)	(9,9)	(11,14)	(2,6)	(1,14)	(12,1)
(16,16)	(10,10)	(5,14)	(5,3)	(10,7)	(16,1)	(12,16)	(1,3)
(2,11)	(11,3)	(9,8)	(15,12)	(4,9)	(6,16)	${\mathcal {O}}$

The point P(6,1) generates all 23 points of the elliptic curve.

Algebraic group properties of the points

The points further comply associative and commutative algebraic group laws and the handling of the neutral element:

• Closure: $P+Q=R\quad \forall \quad P,Q,R\in E$
• Associative law: $P+{\mathcal {O}}={\mathcal {O}}+P\quad \forall \quad P\in E$
• Identity element and inverse that: $P+(-P)={\mathcal {O}}\quad \forall \quad P\in E$
• Cummutative law: $P+(Q+R)=(P+Q)+R\quad \forall \quad P,Q,R\in E$

The inverse point of a point P(x,y) is therefore P(x,-y).

Scalar Multiplication

Is main used operation in cryptography, it adds n times the point P of the elliptic curve over a prime field.

$nP=\underbrace {P+P+P+P+...+P} _{\text{n times}}$

Standardization of elliptic curves

The domain parameters for ECC schemes are described in the form $E(q,a,b,G,n,h)$.

Parameter description
$q$	defines the field size, either a prime $p$ or $2^{m}$ where m is prime
$a$	first parameter of the curve equation
$b$	second parameter of the curve equation
$G$	generating point consisting of both point coordinates $(x_{G},y_{G})$
$n$	order of the point $G$
$h$	cofactor which is equal to the order of the curve divided by $n$

The generation of safe elliptic curves is an effort, hence it is recommended to use standardized known curves. First curves have been standardized in the ANSI X9.62 standard by the American National Standards Institute(ANSI) in 1999 , these have been extended or replaced by ANSI X9.63 in 2001 and ANSI FRP256V1 in 2011. The National Institute of Standard and Technology (NIST) defined their own curves in the NIST FIPS 186-2 in 2000. In the same year the Certicom published the widely-used Certicom SEC2 curves which have been continously updated in version 2 . In 2005 NIST published the NSA Suite B and the Federal Office for Information Security in Germany proposed their own randomly generated curves in the same year.

Curve25519

Curve25519 is a highly optimized curve proposed by Daniel J. Bernstein (djb) in 2005. The curve equation is $y^{2}=x^{3}+486662x^{2}+x$ over a prime field $2^{255}-19$

Edward curves

Applications of Elliptic Curve Cryptography

Example: Elliptic Curve Diffie Hellman Key Exchange (ECDH)

Is a key establishment protocol that allows two parties which know the common domain parameter of a curve $E(p,a,b,G,n,h)$ to calculate a common shared secret over an insecure channel. Assuming Alice and Bob like to establish a common shard secret for communication, both need to generate a keypair. A keypair consists of a private/secret key $d_{i}$ and a public key $Q_{i}$ . For the private key $d_{i}$ a integer less than $n$ is randomly selected, then both need to calculate their public key with $Q_{i}=d_{i}\cdot G$, so that Alice has a pair of $(d_{A},Q_{A})$. She sends her public key $Q_{A}$ to Bob and he sends back his public key $Q_{B}$. Now both are able to calculate the shared secret, Alice computes $K_{AB}=Q_{B}\cdot d_{A}$ and Bob $K_{BA}=Q_{A}\cdot d_{B}$. This works because $K_{AB}=Q_{B}\cdot d_{A}=(d_{B}\cdot G)\cdot d_{A}=(d_{A}\cdot G)\cdot d_{B}=K_{BA}$.

Online resources

• https://safecurves.cr.yp.to/
• https://andrea.corbellini.name/2015/05/17/elliptic-curve-cryptography-a-gentle-introduction/