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AcceptableShort Weierstrass

numsp384d1

numsp384d1 is a 384-bit 'nothing up my sleeve' Weierstrass curve proposed by Microsoft Research in 2014.

numsp384d1 is a short Weierstrass curve y² = x³ + ax + b over a 384-bit prime field with a 384-bit prime-order subgroup and cofactor 1; its estimated security level is 192 bits. Status: secure, but regional or niche.

The NUMS curves (Bos, Costello, Longa and Naehrig, 2014; draft-black-numscurves) were generated deterministically so that no constant needs explaining: p = 2^384 − c with the smallest c that makes p prime and p ≡ 3 (mod 4), then the smallest b for which y² = x³ − 3x + b and its twist both have prime order. numsp384d1 is that prime-order Weierstrass curve, at about 192 bits of security.

The Internet-Draft expired in 2015 without becoming an RFC and no protocol registry lists the curves; Microsoft Research's MSR ECCLib implements them. The twisted Edwards NUMS curve over the same prime is numsp384t1.

Names in standards

StandardName
draft-black-numscurves-02numsp384d1

Domain parameters

384-bit prime field · about 192-bit security

Field prime p
0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffec3
Coefficient a
0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffec0
Coefficient b
0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff77bb
Base point x
0x2
Base point y
0x3c9f82cb4b87b4dc71e763e0663e5dbd8034ed422f04f82673330dc58d15ffa2b4a3d0bad5d30f865bcbbf503ea66f43
Order n of the base point
0xffffffffffffffffffffffffffffffffffffffffffffffffd61eaf1eeb5d6881beda9d3d4c37e27a604d81f67b0e61b9
Cofactor h
0x1

Used in

  • MSR ECCLib (Microsoft Research)

Standards and references

All curves