Four$$\\mathbb {Q}$$: Four-Dimensional Decompositions on a $$\\mathbb {Q}$$-curve over the Mersenne Prime

We introduce Four$$\\mathbb {Q}$$, a high-security, high-performance elliptic curve that targets the 128-bit security level. At the highest arithmetic level, cryptographic scalar multiplications on Four$$\\mathbb {Q}$$ can use a four-dimensional Gallant-Lambert-Vanstone decomposition to minimize the total number of elliptic curve group operations. At the group arithmetic level, Four$$\\mathbb {Q}$$ admits the use of extended twisted Edwards coordinates and can therefore exploit the fastest known elliptic curve addition formulas over large prime characteristic fields. Finally, at the finite field level, arithmetic is performed modulo the extremely fast Mersenne prime $$p=2^{127}-1$$. We show that this powerful combination facilitates scalar multiplications that are significantly faster than all prior works. On Intel's Haswell, Ivy Bridge and Sandy Bridge architectures, our software computes a variable-base scalar multiplication in 59,000, 71,000 cycles and 74,000 cycles, respectively; and, on the same platforms, our software computes a Diffie-Hellman shared secret in 92,000, 110,000 cycles and 116,000 cycles, respectively.