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The premise "we don't have a way of generating and verifying a 2048-bit prime number with 100% accuracy" is wrong (if we trust the computers performing the operations): it has long been known practicable ways to generate randomly-seeded provable primes, and it is a (somewhat marginal) practice in RSA key generation (see FIPS 186-4 appendix B.3.2). We can ...

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Here's a 2048-bit modulus $n$: ...

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Signature verification is failed because you are using a different public key in the verification method. Use the public key to verify the signature which is consistent with the private key that is used into rsaSign() method. Hope this will help you. Note that, this public key is consistent with the private key which is used in Signature Generation method : ...

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I don't think this problem is solvable as specified. With a small message space, and deterministic hashing (or encryption), a generic attack involves exhaustively searching all likely messages to find one that corresponds to the known hash / ciphertext. If all of the digits of the ID numbers were random, an exhaustive search would require about \$10^{10} ...

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The client generates a random symmetric key and encrypts it with the public key. This public key needs to be trusted. Make sure you use a good padding mode, OAEP should do it. Send to server, server decrypts it with the private key. Eh, that's it. No forward security though, the session can be decrypted if the RSA scheme is broken or if the private key is ...

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