ElGamal cryptosystems on Boolean functions

Here is a description of ElGamal public-key encryption and digital signature schemes constructed on the base of bijective systems of Boolean functions. The description is illustrated with a simple example in which the used Boolean functions are written in logical notation. In our encryption and signature schemes on Boolean functions, every one ciphertext or message signature is a pair of values, as in the basic ElGamal cryptosystem on a group. In our case, these values are Boolean vectors. Each vector in the pair depends on the value of a function on a plaintext or on a message, and this function is typically obtained from a given bijective vector Boolean function $g$ by applying some random and secret negation and permutation operations on the sets of variables and coordinate functions of $g$. For the pair of vectors in the ciphertext or in the message signature, the decryption algorithm produces the plaintext, and the signature verification algorithm accepts the signature, performing some computation on this pair. The signature is accepted for a message if and only if the computation results in this message. All the computations in the processes of encryption, decryption, signing and verification are logical and performed for Boolean values, promising their implementation efficiency to be more high than in the basic ElGamal schemes on groups.