What is discrete logarithm problem for elliptic curves?
Elliptic curves over finite fields contain finite cyclic groups that we can use for cryptography. There is no factor- ization problem for elliptic curves, but what is used is the discrete logarithm problem, which is to solve kB = P for k.
What is discrete logarithm problem in cryptography?
Discrete logarithms are logarithms defined with regard to multiplicative cyclic groups. The discrete logarithm problem is defined as: given a group G, a generator g of the group and an element h of G, to find the discrete logarithm to the base g of h in the group G. Discrete logarithm problem is not always hard.
What is ECDLP problem?
The ECDLP is a special case of the discrete logarithm problem in which the cyclic group G is represented by the group \langle P\rangle of points on an elliptic curve. It is of cryptographic interest because its apparent intractability is the basis for the security of elliptic curve cryptography.
What is the discrete logarithm K of Q 4 5 to the base P 16 5 )?
Because 9P = (4, 5) = Q, the discrete logarithm Q = (4, 5) to the base P = (16, 5) is k = 9.
What encryption uses discrete logarithms?
ElGamal encryption
Popular choices for the group G in discrete logarithm cryptography (DLC) are the cyclic groups (Zp)× (e.g. ElGamal encryption, Diffie–Hellman key exchange, and the Digital Signature Algorithm) and cyclic subgroups of elliptic curves over finite fields (see Elliptic curve cryptography).
How do you solve Ecdlp?
Multiple-ECDLP: Given Q1,…,QL ∈ E(Fq) to compute a1,…,aL such that Qi = ai P for all 1 ≤ i ≤ L. Well-known that Pollard rho solves ECDLP in (1.25 + o(1)) √ r group operations. One can solve multiple-ECDLP in O( √ rL) group operations. Let P have order r and Q = aP.
Is elliptic curve cryptography asymmetric?
ECC is an approach — a set of algorithms for key generation, encryption and decryption — to doing asymmetric cryptography. One of the keys (the public key) is used for encryption, and its corresponding private key must be used for decryption.
Why are elliptic curves important?
1) Elliptic Curves provide security equivalent to classical systems (like RSA), but uses fewer bits. 2) Implementation of elliptic curves in cryptography requires smaller chip size, less power consumption, increase in speed, etc.
How hard is the discrete logarithm problem?
The discrete logarithm problem is considered to be computationally intractable. That is, no efficient classical algorithm is known for computing discrete logarithms in general. A general algorithm for computing logb a in finite groups G is to raise b to larger and larger powers k until the desired a is found.
What is the importance of discrete logarithms?
Aside from the intrinsic interest that the problem of computing discrete logarithms has, it is of considerable importance in cryptography. An efficient algorithm for discrete logarithms would make several authentication and key-exchange systems insecure.