Elliptic Curve Discrete Logarithm Problem Sample Clauses

Elliptic Curve Discrete Logarithm Problem. Let G be a cyclic additive elliptic curve group with the prime order q and P is a generator of G. Suppose that the multiplication and inversion operation in G can be computed efficiently, the two intractable problems in G are defined as follows: • Elliptic curve discrete logarithm (ECDL) problem: Given P, aP ∈ G for unknown a ∈ Zq∗, to find a. • Elliptic curve computational Xxxxxx-Xxxxxxx (ECCDH) problem: Given P, aP, bP ∈ G for unknown a, b ∈ Zq∗, to compute abP.
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Elliptic Curve Discrete Logarithm Problem. The security of ECC relies on the difficulty of Elliptic Curve Discrete Logarithm Problem. Let P and Q be two points on an elliptic curve such that kP = Q, where k is a scalar. Given P and Q, it is infeasible to find k computationally if k is enough large. The scaler k is the discrete logarithm of Q to base P. Therefore, point multiplication is a principle operation in ECC. In other words, multiplication of a scalar k with any point P on the curve will find another point Q on the curve.
Elliptic Curve Discrete Logarithm Problem. The elliptic curve discrete logarithm problem (ECDLP) can be stated as follows. underlying field, GF ( p) or GF (2k ) , the elliptic curve E with parameters a, b, and the base point P. The order of the base point P is equal to n. Also we ensure that the selected elliptic curve has a prime order, in order to comply with the appropriate security standards [10]. At the end of the protocol the communicating parties end up with the same value K which represent a unique point on the curve. A part of this value can be used as a secret key to a secret-key encryption algorithm. We give a brief description of the protocol. • Entity A selects an integer, dA : d A ∈[2, n − 2]. (10) • • Entity B selects an integer Fix a prime p and an elliptic curve E . Let xP represent dB : dB ∈[2, n − 2]. (11) the point P added to itself x times. Suppose Q is a • multiple of P , so that Q = xP for some

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