Cyclic Mutually Unbiased Bases and Quantum Public-Key Encryption

Based on quantum physical phenomena, quantum information theory has a potential which goes beyond the classical conditions. Equipped with the resource of complementary information as an intrinsic property it offers many new perspectives. The field of quantum key distribution, which enables the ability to implement unconditional security, profits directly from this resource. To measure the state of quantum systems itself for different purposes in quantum information theory, which may be related to the construction of a quantum computer, as well as to realize quantum key distribution schemes, a certain set of bases is necessary. A type of set which is minimal is given by a complete set of mutually unbiased bases. The construction of these sets is discussed in the first part of this work. We present complete sets of mutually unbiased bases which are equipped with the additional property to be constructed cyclically, which means, each basis in the set is the power of a specific generating basis of the set. Whereas complete sets of mutually unbiased bases are related to many mathematical problems, it is shown that a new construction of cyclic sets is related to Fibonacci polynomials. Within this context, the existence of a symmetric companion matrix over the finite field F_2 is conjectured. For all Hilbert spaces which have a finite dimension that is a power of two (d=2^m), the cyclic sets can be generated explicitely with the discussed methods. Results for m={1,..,600} are given. A generalization of this construction is able to generate sets with different entanglement structures. It is shown that for dimensions d=2^(2^k) with k being a positive integer, a recursive construction of complete sets exists at least for k in {0,..,11}, where for higher dimensions a direct connection to an open conjecture in finite field theory by Wiedemann is identified. All discussed sets can be implemented directly into a quantum circuit by an invented algorithm. The (unitary) equivalence of the considered sets is discussed in detail. In the second part of this work the security of a quantum public-key encryption protocol is discussed, which was recently published by Nikolopoulos, where the information of all published keys is taken into account. Lower bounds on two different security parameters are given and an attack on single qubits is introduced which is asymptotically equivalent to the optimal attack. Finally, a generalization of this protocol is given that permits a noisy-preprocessing step and leads to a higher security against the presented attack for two leaked copies of the public key and to first results for a non-optimal implementation of the original protocol