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Elliptic Curves Cryptosystems

  • Cryptosystems based on DLP in elliptic curves are very strong, and efficient.
  • Unlike RSA or DHKE, encryption that require 1024 bit length to achieve security, elliptic curves can achieve same level of security with just 160 bits, making it more efficient in time and space.
  • Chapter 9 of understand cryptography discusses one algorithm based on this, which is simply re-implementing DHKE using Elliptic Curve groups.

DHKE with Elliptic Curves

  • First step is the setup phase.
    setup phase
  • Key exchange step
    key exchange phase

Attacks

Generic attacks against DHKE are valid so it needs 160 bits to be secure.
However the non-generic attacks like Index Calculus Method that forces us to use 1024 bits for DHKE in Z*p won't work against elliptic curves, so 160 bits are enough to reach a high level of security.

one final but crucial Quote from the book:

It should be stressed that this security is only achieved if cryptographically strong elliptic curves are used. There are several families of curves that possess cryptographic weaknesses, e.g., supersingular curves. They are relatively easy to spot, however. In practice, often standardized curves such as ones proposed by the National Institute of Standards and Technology (NIST) are being used.