# Question about discrete logarithm problem

DSA and ECDSA both are based on discrete logarithm problem (and elliptic curve based DLP)

And ECDSA protocols resembles DSA.

My question is,

Can every protocol based on DLP be changed (substituted) to a protocol that is based on ECDLP?

• Does any one person know every DLP based protocol? – mikeazo Mar 8 '17 at 2:46
• Define "Can (..) be changed (substituted)". Would you consider that the much larger size of the shared secret obtained by Diffie-Hellman key exchange when operating on $\mathbb Z_p^*$ with $p$ and $(p-1)/2$ primes of about 3072-bit, compared to what it is when operating on an Elliptic curve of comparable security such as P-384, counts towards stating that for this usage the ECDLP can't be used as a substitute of the DLP? – fgrieu Mar 8 '17 at 14:00

To convert every protocol based on DPL to ECDPL based protocol there is a limitation as EC uses generators from a very large prime(or binary) fields. For an instance consider ECDSA which is an EC analogy of DSA, in ECDSA there is a Generator G (used in calculating public key) is from a large prime subgroup of EC where as in DSA the private key (x) is just a random number such that 0 < x < q where q is N-bit prime number.

• You are comparing ECDSA generator with DSA private key which doesn't make sense. In DSA you select a generator $g$ which usually generate a large prime order subgroup of $\mathbb{Z_p^*}$, how does that differentiate from ECDSA ? – Ruggero Mar 8 '17 at 12:48