The existing construction methods of quantum boolean functions(QBFs) are extended and simplified. All QBFs with one qubit and all local QBFs with any qubits are constructed. And we propose the concept of Generalized q...
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The existing construction methods of quantum boolean functions(QBFs) are extended and simplified. All QBFs with one qubit and all local QBFs with any qubits are constructed. And we propose the concept of Generalized quantum boolean functions(GQBFs). We find all GQBFs with one qutrit and all kinds of local GQBFs with any qutrits. The number of each of the four kinds of functions above is uncountably infinitely many. By using diagonal matrices, we obtain uncountably infinitely many non-local QBFs with any qubits and GQBFs with any qutrits. Infinitely many families of GQBFs with any qudits are obtained from the properties of projection matrices of known saturated orthogonal arrays.
While performing cryptanalysis, it is of interest to approximate a boolean function in n variables f : F-2(n) -> F-2 by affine functions. Usually, it is assumed that all the input vectors to a boolean function are ...
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While performing cryptanalysis, it is of interest to approximate a boolean function in n variables f : F-2(n) -> F-2 by affine functions. Usually, it is assumed that all the input vectors to a boolean function are equiprobable while mounting affine approximation attack or fast correlation attacks. In this paper we consider a more general case when each component of the input vector to f is independent and identically distributed Bernoulli variates with the parameter p. Since our scope is within the area of cryptography, we initiate an analysis of cryptographic booleanfunctions under the previous considerations and derive expression of the analogue of Walsh-Hadamard transform and nonlinearity in the case under consideration. We observe that if we allow p to take up complex values then a framework involving quantum boolean functions can be introduced, which provides a connection between Walsh-Hadamard transform, nega-Hadamard transform and booleanfunctions with biased inputs.
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