We propose a new suite of algorithms that significantly improve the performance of supersingular isogeny Diffie-Hellman (SIDH) key exchange. Subsequently, we present a full-fledged implementation of SIDH that is geared towards the 128-bit quantum and 192-bit classical security levels. Our library is the first constant-time SIDH implementation and is more than 2.5 times faster than the previous best (non-constant-time) SIDH software. The high speeds in this paper are driven by compact,

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... free point and isogeny arithmetic and fast SIDH-tailored field arithmetic: on an Intel Haswell processor, generating ephemeral public keys takes 51 million cycles for Alice and 59 million cycles for Bob while computing the shared secret takes 47 million and 57 million cycles, respectively. The size of public keys is only 751 bytes, which is significantly smaller than most of the popular post-quantum key exchange alternatives. Ultimately, the size and speed of our software illustrates the strong potential of SIDH as a post-quantum key exchange candidate and we hope that these results encourage a wider cryptanalytic effort.