Elliptic Curve Cryptography (Ecc) Mathematical Basis of Ecc
Elliptic Curve Cryptography (ECC)
Mathematical basis of ECC
Elliptic Curve is a set of solutions (x, y) to an equation of the form y2=x3+ax+b where 4a3+27b2≠0, together with a point at infinity denoted O. Elliptic Curve originally developed to measure circumference of an ellipse and now have been proposed for applications in cryptography due to their group law and because so far no sub exponential attack on their discrete logarithm problem. Cryptography based on elliptic curves depends on arithmetic involving the points of the curve.
Definition: An elliptic curve E over a field K is defined by following equation which is called Weiestress equation.
E:y2+a1xy+a3y=x3+a2x2+a4x+a6
y2=x3+ax+b is the simplified version of the Weiestress equation.
Figure 1. Group law on elliptic curve y2=fx over R
Group Law
The definition of Group Law is where the chord-and-tangent rule of adding two points in the curve to give third point which reflects across the x-axis. It is this group that is used in the construction of elliptic curve cryptographic systems.
Closure, Inverse, Commutative, Identity and Associativity are conditions that the set and operation must satisfy to be qualify as a group which also known as group axioms.
Addition Formulae
Let P1=(x1,y1) and P2=(x2,y2) be non-inverses. Then P1+P2=(x3,y3)
Scalar multiplication
Scalar multiplication is repeated group addition: cP=P+…+P (c times)where c is an integer
The Elliptic Curve Discrete Logarithm Problem (ECDLP)
The security of all ECC schemes are depends on the hardness of the elliptic curve discrete logarithm problem.
Problem: Given two points W, G find s such that W=sG
The elliptic curve parameters for cryptographic schemes should be carefully chosen with appropriate cryptographic restriction in order to resist all known attacks on the ECDLP which is believed to take exponential time.
O(sqrtr) time, where r is the order of W
By comparison, factoring and ordinary discrete logarithms can be solved in
Mathematical basis of ECC
Elliptic Curve is a set of solutions (x, y) to an equation of the form y2=x3+ax+b where 4a3+27b2≠0, together with a point at infinity denoted O. Elliptic Curve originally developed to measure circumference of an ellipse and now have been proposed for applications in cryptography due to their group law and because so far no sub exponential attack on their discrete logarithm problem. Cryptography based on elliptic curves depends on arithmetic involving the points of the curve.
Definition: An elliptic curve E over a field K is defined by following equation which is called Weiestress equation.
E:y2+a1xy+a3y=x3+a2x2+a4x+a6
y2=x3+ax+b is the simplified version of the Weiestress equation.
Figure 1. Group law on elliptic curve y2=fx over R
Group Law
The definition of Group Law is where the chord-and-tangent rule of adding two points in the curve to give third point which reflects across the x-axis. It is this group that is used in the construction of elliptic curve cryptographic systems.
Closure, Inverse, Commutative, Identity and Associativity are conditions that the set and operation must satisfy to be qualify as a group which also known as group axioms.
Addition Formulae
Let P1=(x1,y1) and P2=(x2,y2) be non-inverses. Then P1+P2=(x3,y3)
Scalar multiplication
Scalar multiplication is repeated group addition: cP=P+…+P (c times)where c is an integer
The Elliptic Curve Discrete Logarithm Problem (ECDLP)
The security of all ECC schemes are depends on the hardness of the elliptic curve discrete logarithm problem.
Problem: Given two points W, G find s such that W=sG
The elliptic curve parameters for cryptographic schemes should be carefully chosen with appropriate cryptographic restriction in order to resist all known attacks on the ECDLP which is believed to take exponential time.
O(sqrtr) time, where r is the order of W
By comparison, factoring and ordinary discrete logarithms can be solved in