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We study scheme (hardware) and program (software) methods of multiplication of polynomials over fields of characteristic 7 in order to apply them to cryptographic protocols on elliptic curves based on parings. We consider hardware and software implementations of arithmetic in GF(7); GF(7^2); GF(7^n); GF(7^{7n}); and GF(7^{14n}) and estimate the complexity of corresponding schemes and programs.