Probability distributions over CSS codes: two-universality, QKD hashing, collision bounds, security

We characterize novel probability distributions for CSS codes. Such classes of error correcting codes, originally introduced by Calderbank, Shor, and Steane, are of great significance in advancing the fidelity of Quantum computation, with implications for future near term applications. Within the context of Quantum key distribution, such codes, as examined by Ostrev in arXiv: 2109.06709 along with two-universal hashing protocols, have greatly simplified Quantum phases of computation for unconditional security. To further examine novel applications of two-universal hashing protocols, particularly through the structure of parity check matrices, we demonstrate how being able to efficiently compute functions of the parity check matrices relates to marginals of a suitably defined probability measure supported over random matrices. The security of the two-universal QKD hashing protocol will be shown to depend upon the computation of purified states of random matrices, which relates to probabilistic collision bounds between two hashing functions. Central to our approach are the introduction of novel real, simulator, and ideal, isometries, hence allowing for efficient computations of functions of the two parity check matrices. As a result of being able to perform such computations involving parity check matrices, the security of the two-universal hashing protocol is a factor of $2^{ \frac{5}{2} ( 5 - \frac{3}{2} ) + \mathrm{log}_2 \sqrt{C}}$ less secure, for some strictly positive constant $C$.