Algebra Seminar talk

Maciej Maczynski
Probability and cryptographic algebras

A probability algebra L is defined as a set of mappings from a set S into the interval [0,1], satisfying three axioms implying that L is a partially ordered orthocomplemented set (orthoposet). Probability algebras represent classical and non-classical systems of probabilities (called S-probabilities).

A cryptographic algebra is an algebra (A, +1, +2, p, s) of type (2,2,0,0) satisfying the axiom (x +1 p) +2 s = x. If +1 = +2 = +, and p = s = y (the axiom then takes the form (x + y) + y = x), then A is called a completely symmetric cryptographic algebra. If we assume that a probability algebra L is simultaneously a cryptographic algebra with respect to a suitably defined operation +, then L is a Boolean algebra (representing a classical logic), and conversely.

This provides a cryptographical characterization of classical probability systems: the system is classical if and only if it admits a completely symmetric cryptographic operation such that the secret key coincides with the public key. In other words, in classical system there are no hidden keys. On the other hand, non-classical system of probabilities (such as quantum mechanics in Hilbert space) do not admit completely symmetric cryptographic interpretation. If they are to be considered as cryptographic algebras, the public key must be different from secret key, which may be hidden. There is an open problem to find a secret key to a non-classical probability system (in particular to quantum mechanics. This problem is similar to the problem of hidden variables in quantum mechanics.