> not secret
Clearly, no smaller primes are secret either, nor are they so numerous that iterating over them is computationally difficult. And mathematically, anyone can discover primes. Why does it matter whether primes are secret?
Clearly, no smaller primes are secret either, nor are they so numerous that iterating over them is computationally difficult. And mathematically, anyone can discover primes. Why does it matter whether primes are secret?
That's not correct; for a given bound x, the expected number of primes is x/ln(x).
For example, if you need to choose a prime of 100 digits, there are something like 10^96 of them; you can find a random one using probabilistic methods, do your cryptography, and know that your attacker would have to try 10^96 primes to break it. Since they can't really do this, this is not the weak point of your cryptosystem.
If you chose a Mersenne prime, there are only 51 of them. Your attacker would have to try less than 10^2 of them.
Some cryptosystems like ECDSA are based on secret elements selected from a field, and the primes there are public.
Otherwise, reductio ad absurdum, Following the same line of logic, all cryptography is "security through obscurity" because there exists a secret key, which is not incorrect. The practice of hiding the design of a system is obscurity, the practice of hiding a specific key to a ciphertext is security, not obscurity, as long as the method of generating and using it is publicly available. If there's nothing to reverse engineer, there can be no obscurity. The method of finding a symmetric key or finding a RSA key is well-known, not obscured, it's simply the size of the problem is not computationally tractable in practice by today's computers.