  We know the classical public cryptographic algorithms are based on certain
NP-hard problems such as the integer factoring in RSA and the discrete
logarithm in Diffie-Hellman. They are going to be vulnerable with
fault-tolerant quantum computers. We also know that the uncertainty principle
for quantum bits or qubits such as quantum key distribution or QKD based on the
quantum uncertainty principle offers the information theoretical security. The
interesting implication with the paradigm shifts from classical computing to
quantum computing is that the NP-hardness used for classical cryptography may
shift to the uncertainty principles for quantum cryptography including quantum
symmetric encryption, post-quantum cryptography, as well as quantum encryption
in phase space for coherent optical communications. This paper would like to
explore those so-called generalized uncertainty principles and explain what
their implications are for quantum security. We identified three generalized
uncertainty principles offering quantum security: non-commutability between
permutation gates, non-commutability between the displacement and phase shift
operators for coherent states, and the modular Diophantine Equation Problem in
general linear algebra for post-quantum cryptography.

Cryptography and Security