In this article we produce the simplified SWU encoding to some Barreto–Naehrig curves, including BN512, BN638 from the standards ISO/IEC 15946-5 and TCG Algorithm Registry respectively. Moreover, we show (for any $j$-invariant) how to implement the simplified SWU encoding in constant time of one exponentiation in the basic field, namely without quadratic residuosity tests and inversions. Thus in addition to the protection against timing attacks, the new encoding turns out to be much more efficient than the (universal) SWU encoding, which generally requires to perform two quadratic residuosity tests.

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