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This paper discusses an asymmetric cryptosystem C* which consists of public transformations of compIerity O(m2n3) and secret transformations of complexity O((mn)2(m + logn)), where each complexity is measured in the total number of bit-operations for processing an mn-bit message block. Each public key of C* is an n-tuple of quadratic n-variate polynomials over GF(2m) and can be used for both verifying signatures and encrypting plaintexts. This paper also shows that for C* it is practically infeasible to extract the n-tuple of n-variate polynomials representing the inverse of the corresponding public key.