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The Claude C Compiler illustrates the other side: it optimizes for passing tests, not for correctness. It hard-codes values to satisfy the test suite. It will not generalize. Property-based testing would likely catch this particular case, but the general problem remains: for any fixed testing strategy, a sufficiently adversarial system can overfit to it. A proof cannot be gamed. It covers all inputs by construction.
In real systems we do need to be a bit careful to ensure that the key K is chosen honestly and at random, to avoid a user duplicating another user’s key or doing something tricky. Often real-world issuance protocols will have K chosen jointly by the Issuer and User, but this is a bit too technically deep for a blog post.