New answers tagged elliptic-curves
2
You're essentially correct. Index calculus is impractical on elliptic curves because there is not a straightforward notion of smoothness in these groups.
In prime fields, there is the easy mapping from the multiplicative group to the integers, where smoothness is well-defined. Similarly, in extension fields there is the mapping to polynomials over the ...
3
I haven't worked through all the (boring) details but it should be fairly easy. Just start with the normal affine addition and doubling formulae but when it comes to the stage where you do any division, don't bother and just keep track of the numerator/denominator separately. The whole point of Jacobian and other projective representations is that you don't ...
1
Yes, TLS supports ECDSA for signing and ECDH for key exchange. You can read about it RFC 4492. Notice you don't get RSA, but you get signing and key exchange which are major functions of TLS.
Related material:
How do I create an ECDSA certificate with the OpenSSL command-line
SSH key-type, rsa, dsa, ecdsa, are there easy answers for which to choose when?
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3
If you are restricted to Diffie-Helman, RSA, and ElGamal, I believe you cannot do pairing based cryptography which has applications to attributed based encryption and identity based encryption. These applications are not used heavily in practice, but could be in the next decade or two.
For more information on speed benefits of ECC, see this question and ...
1
ECC allows the use of way smaller keys than necessary in traditional public key cyptography. According to this paper, these are just a fraction of the size of the keys with comparable strength of traditional RSA.
the computional effort for an equivalent level of security ranks from 1:3 (ECC vs DH, 80 bit) to 1:60 (ECC vs DH, 256 bit)
This also means, that ...
1
My understanding is that ECC is mostly lighter-weight than RSA. That is to say, for similar key strengths, the ECC key is significantly smaller. For example, a 2048-bit RSA key is believed to be about as strong as a 224-bit ECC key.
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