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ECDLPElliptic Curve Discrete Logarithm Problem
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The client's ID is encrypted by secure symmetric encryption with a random secrete key in our proposed scheme, i.e., [e.sub.i] = [E.sub.R.sub.2x]([HID.sub.i], [ID.sub.i]), and the value of [R.sub.2x] is obtained from [R.sub.2] = [d.sub.c]P = ([R.sub.2x], [R.sub.2y]) with a random integer [[d.sub.c] [member of] [Z*.sub.P], so [R.sub.2x] and [e.sub.i] are different in each session.[R.sub.2x] is protected under elliptic curve discrete logarithm problem (DLP), and only can be obtained by the one who has the server private key [k.sub.s].
Elliptic Curve Discrete Logarithm Problem (ECDLP): Given the equation P = kG where P, G [member of] [E.sub.p](a, b) and k < p, it is relatively easy to compute P when the values of k and G are known, but it is hard to evaluate k given the values of P and G.
Most of the ECC schemes rely on the hard problem of Elliptic Curve Discrete Logarithm Problem (ECDLP) which is impossible to compromise using any polynomial time algorithm.
Mehta, "A stamped blind signature scheme based on elliptic curve discrete logarithm problem," International Journal of Network Security, vol.
Discussion: our scheme's security mainly depends on the difficulty of solving the elliptic curve discrete logarithm problem (ECDLP).
Definition 1 (Elliptic Curve Discrete Logarithm Problem).
Elliptic Curve Discrete Logarithm Problem (ECDL Problem): Given an elliptic curve E defined over a finite field GF(p), and two points Q, P [member of] E, it is hard to find an integer k [member of] [Z.sub.q.sup.*] such that Q = k x P.
The largest Elliptic Curve Discrete Logarithm Problem (EC-DLP) to be solved so far had a key size of 109 bits, that is, over the finite field [GF.sub.109] it took 17 months to break [14].
Definition 1: Elliptic curve discrete logarithm problem (ECDLP) states that if the elliptic curve E is defined over the field [F.sub.q], and the point P [member of] E([F.sub.q]) with the order n and the point Q [member of] &lt;P&gt; are known, then an integer number l [member of] [0, n - 1] will be found so that Q = lP holds.
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