Elliptic Curve Cryptosystem: Neal Koblitz and Victor S. Miller

We are trying to design a designer of a secure and complex cryptographic system, but our opponents are constantly introducing new methods of tempering and attacking. Designers can only use the various methods successfully to adjust the data. Designers need to design security algorithms. In some cases, another countermeasure is useful for new attacks. Thus, a game that designs complex security algorithms and attacks new designs is an endless job. This article outlines a recent attack against elliptic curve cryptography and explains its countermeasure.

In 1985, elliptic curve cryptography (ECC) was independently presented by cryptographers Victor Miller (IBM) and Neal Koblitz (University of Washington). ECC is based on the difficulty of solving elliptic curve discrete logarithm problem (ECDLP). As with prime factorization problems, ECDLP is another seemingly simple "difficult" problem. Find the integer n from two points P and Q on the elliptic curve and if it exists, P = nQ. Elliptic curves combine number theory and algebraic geometry. These curves can be defined with any number of bodies (ie real numbers, integers, complex numbers), but they are usually used in finite field cryptography. An elliptic curve consists of real numbers (x, y) that satisfy the following equation.

Every public key cryptosystem is based on difficult calculation problems. Elliptic curve cryptosystems are no exception. Consider the equation Q = kP. Here, P and Q are two points on the elliptic curve E (Fp). Where p is a sufficiently large prime number (eg ~ 2 ^ 160). It is relatively easy to compute Q for k and P, but it is very difficult to determine k for a given Q and P. This is called elliptic curve discrete logarithm problem (ECDLP). From the simple exercise E (F 11) above, the discrete logarithm problem can be easily solved by table lookup. If P = (3, 3), you can calculate kP, k = 0, ..., 14 using the point addition formula and place the results in the following table.