We describe fast new algorithms to implement recent cryptosystems based on the Tate pairing. In particular, our techniques improve pairing evaluation speed by a factor of about 55 compared to previously known methods in characteristic 3, and attain performance comparable to that of RSA in larger characteristics. We also propose faster algorithms for scalar multiplication in characteristic 3 and square root extraction over Fpm , the latter technique being also useful in contexts other than that

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... f pairing-based cryptography. Mathematical Preliminaries Let p be a prime number, m a positive integer and F p m the finite field with p m elements; p is said to be the characteristic of F p m , and m is its extension