How to compare the time of encryption, cracking and verification elliptic curve problem in the same framework？ everyone! As a beginner, I would like to ask you a question. The best algorithm known for cracking (done by anonymous snooper) this problem （Discrete logarithm problem of elliptic curve or ECDSA）is Pollard’s rho method, and takes about the square root of πp/2, where p is the modulus. My questions are:
(1) Elliptical encryption (done by designer and sender) also takes time. Can this time also be expressed by modulus p?
(2) After the other party receives the information, it needs to be verified. This verification (done by receiver of information) also takes time. Can the time for verification also be expressed by modulus?
That is to say, can the above three kinds of time be put into a frame for comparison, and they are all compared with the parameters of elliptic curve (it is best to use modulus or order of Base point). My intention is to know which of these three times is the slowest and which is the fastest. But there is no doubt that the time to crack is definitely the slowest, and the time for encryption and verification is definitely much shorter. But how to quantify them? Is there any literature recommendation that compares them? It would be best to give a simple example. Thank you very much!
Let $n$ be the number of points in the curve group and $p$ the size of the field. The quoted estimate for Pollard $\rho$ is $\sqrt {\pi n/2}$ which is the number of elliptic curve group operations required. To sign a ECDSA message (assuming that you already have your key pair) is an elliptic curve scalar point multiplication which takes (using windowed double-and-add) $(1+\epsilon)\log n$ elliptic curve group operations. To verify is 2 elliptic curve scalar multiplications which is $(2+\epsilon)\log n$ elliptic curve group operations. If you have to generate a key pair, that adds another elliptic curve scalar multiplication.
The elliptic curve group operation will get harder with $p$ as well and the exact complexity is subject to a range of variant algorithms. An estimate of $O(\log^2p)$ for each elliptic curve group would not be hugely inaccurate though.