The Bound of Post Quantum Ideal Lattice Cryptograph

This code is to simulate and calculate the hacking bound for post quantum cryptography with ideal lattice as an example

Zero-key or password free authentication is an authentication method in which a user can log in to a computer system without a password or any other secret, rather the users are required to enter their public identifiers backed up with tokens. The public key is provided during registration with authentication services, while the private key is saved on the user's token and can only be accessed by providing biometric signatures or zero-knowledge-proof authentication factors, where the user privacy is not jeopardized. Private key is embedded inside the publick key, the hacking is to recover the private key.