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In this paper ways to efficiently implement public-key schemes based on ultivariate uadratic polynomials ($\mathcal{MQ}$-schemes for short) are investigated. In particular, they are claimed to resist quantum computer attacks. It is shown that such schemes can have a much better time-area product than elliptic curve cryptosystems. For instance, an optimised FPGA implementation of amended TTS is estimated to be over 50 times more efficient with respect to this parameter. Moreover, a general framework for implementing small-field $\mathcal{MQ}$-schemes in hardware is proposed which includes a systolic architecture performing Gaussian elimination over composite binary fields.