As far as I know, a secret key corresponding to a given public key is not unique. So, for example, there are actually multiple secret keys that can produce the same digital signature under RSA signature scheme.
What I wonder is, how can I compute the expected time to find a matching secret key of a given public key? I assume some details are given like the length of keys, the key generation algorithm etc.
Also, is there a public-key encryption scheme s.t. there's a unique secret key for each public key?