Optimal alphabets for noise-resistant quantum cryptography

Possibilities of improving critical error rate of quantum key distribution (QKD) protocols for different strategies of eavesdropping are investigated. QKD-protocols with discrete alphabets, letters of which form regular polyhedrons on the Bloch sphere (tetrahedron, octahedron, cube, icosahedron, and dodecahedron, which have 4, 6, 8, 12, and 20 vertexes respectively) and QKD-protocol with continuous alphabet, which corresponds to the limiting case of a polyhedron with infinitive number of vortexes are considered. Stability of such QKD-protocols to the noise in a quantum channel, which is due to the Eve's interference that apply either intercept-receipt or optimal eavesdropping strategy at the individual attacks, is studied in detail. It is shown that in case of optimal eavesdropping strategy, after bases reconciliation, the QKD-protocol with continuous alphabet surpasses all other protocols in terms of noise-resistance. Without basis reconciliation the highest critical error rate have the protocol with tetrahedron-type alphabet.