Comparative Analysis of Calculations in Cryptographic Protocols Using a Combination of Different Bases of Finite Fields

The chapter introduces a comparative analysis of the complexity of the Tate pairing operation on a supersingular elliptic curve and the complexity of the final exponentiation in the tripartite key agreement cryptographic protocol. The analysis takes into account a possibility of using different bases of finite fields in combination. Operations of multiplication and multiple squaring in the field \( GF(2^{n} ) \) and its 4-degree extension, of Tate pairing on supersingular elliptic curve and of final exponentiation are considered separately and in combination. We conclude that the best complexity bound for the pairing and the final exponentiation in the cryptographically significant field \( GF(2^{191} ) \) is provided by the combination of the polynomial basis of this field and 1-type optimal basis of the field expansion.