Cryptography and information security pdf size of the elliptic curve determines the difficulty of the problem. 256-bit elliptic curve public key should provide comparable security to a 3072-bit RSA public key.
Elliptic curve cryptography algorithms entered wide use in 2004 to 2005. 2 or 3, or the curve equation will be somewhat more complicated. ECC for digital signature generation and key exchange. The suite is intended to protect both classified and unclassified national security systems and information. As a result, several standard bodies published domain parameters of elliptic curves for several common field sizes. SECG test vectors are also available. NIST has approved many SECG curves, so there is a significant overlap between the specifications published by NIST and SECG.
EC domain parameters may be either specified by value or by name. However the public key may be smaller to accommodate efficient encryption, especially when processing power is limited. 112-bit key for the prime field case and a 109-bit key for the binary field case. 5 months using this cluster when running continuously. The binary field case was broken in April 2004 using 2600 computers over 17 months.
A current project is aiming at breaking the ECC2K-130 challenge by Certicom, by using a wide range of different hardware: CPUs, GPUs, FPGA. Fortunately, points on a curve can be represented in different coordinate systems which do not require an inversion operation to add two points. 2000 standard uses “projective coordinates” to refer to what is commonly called Jacobian coordinates. An additional speed-up is possible if mixed coordinates are used. 3, which improves addition in Jacobian coordinates. According to Bernstein and Lange, many of the efficiency-related decisions in NIST FIPS 186-2 are sub-optimal.
Other curves are more secure and run just as fast. In 1999, NIST recommended 15 elliptic curves. For each of the prime fields, one elliptic curve is recommended. 163, 233, 283, 409, and 571. The NIST recommendation thus contains a total of 5 prime curves and 10 binary curves.