Multivariate Cryptosystem
The basic idea of the HFE family of using this as a multivariate cryptosystem is to build the secret key starting from a polynomial in one unknown over some finite field (normally value is used). This polynomial can be easily inverted over, i.e. it is feasible to find any solutions to the equation when such solution exist. The secret transformation either decryption and/or signature is based on this inversion. As explained above can be identified with a system of equations using a fixed basis. To build a cryptosystem the polynomial must be transformed so that the public information hides the original structure and prevents inversion. This is done by viewing the finite fields as a vector space over and by choosing two linear affine transformations and . The triplet constitute the private key. The private polynomial is defined over . The public key is . Below is the diagram for MQ-trapdoor in HFE
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