Efficient arithmetic in (pseudo-)mersenne prime order fields

<p style='text-indent:20px;'>Elliptic curve cryptography is based upon elliptic curves defined over finite fields. Operations such require arithmetic the underlying field. In particular, fast implementations of multiplication and squaring field are required for performing efficient cryptography. The present work considers problem obtaining algorithms squaring. From a theoretical point view, we number multiplication/squaring reduction which appropriate different settings. Our collect together generalize ideas scattered across various papers codes. At same time, also introduce new to improve existing works. A key feature our that provide formal statements detailed proofs correctness describe. On implementation aspect, total fourteen primes considered, covering all previously proposed cryptographically relevant (pseudo-)Mersenne prime order fields at security levels. For each these fields, 64-bit assembly targeted towards two modern Intel architectures. We were able find previous six considered in this work. Haswell Skylake processors Intel, where available, outperform implementations.</p>