4. Recommended Curves

4.1. Curve25519

For the ~128-bit security level, the prime 2^255 - 19 is recommended for performance on a wide range of architectures. Few primes of the form 2^c-s with s small exist between 2^250 and 2^521, and other choices of coefficient are not as competitive in performance. This prime is congruent to 1 mod 4, and the derivation procedure in Appendix A results in the following Montgomery curve v^2 = u^3 + A*u^2 + u, called "curve25519":

p  2^255 - 19

A  486662

order  2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed

cofactor  8

U(P)  9

V(P)  14781619447589544791020593568409986887264606134616475288964881837755586237401

The base point is u = 9, v = 14781619447589544791020593568409986887264606134616475288964881837755586237401.

This curve is birationally equivalent to a twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2, called "edwards25519", where:

p  2^255 - 19

d  37095705934669439343138083508754565189542113879843219016388785533085940283555

order  2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed

cofactor  8

X(P)  15112221349535400772501151409588531511454012693041857206046113283949847762202

Y(P)  46316835694926478169428394003475163141307993866256225615783033603165251855960

The birational maps are:

(u, v) = ((1+y)/(1-y), sqrt(-486664)*u/x)
(x, y) = (sqrt(-486664)*u/v, (u-1)/(u+1))

The Montgomery curve defined here is equal to the one defined in [curve25519], and the equivalent twisted Edwards curve is equal to the one defined in [ed25519].

4.2. Curve448

For the ~224-bit security level, the prime 2^448 - 2^224 - 1 is recommended for performance on a wide range of architectures. This prime is congruent to 3 mod 4, and the derivation procedure in Appendix A results in the following Montgomery curve, called "curve448":

p  2^448 - 2^224 - 1

A  156326

order  2^446 - 0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d

cofactor  4

U(P)  5

V(P)  35529392678556817526412750206378333480897639938771427183188089843516908878696741000293267376586455091014274714726810583898559029060636262

This curve is birationally equivalent to the Edwards curve x^2 + y^2 = 1 + dx^2y^2 where:

p  2^448 - 2^224 - 1

d  61197585074452917616042322096555331754321969687101662632896893641508786004263647489178559928366602041476867897998937814706546281554501017

order  2^446 - 0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d

cofactor  4

X(P)  34539749303972951637400860415053741026665526007518329021640697028164569507367234443048178738443158358375934063322170839158342404178892412456700732

Y(P)  36341936214780344527466190394400226717682068034365903014074509959030616408336538634319819184933827296504444223092181868052674900918271809009182718

The birational maps are:

(u, v) = ((y-1)/(y+1), sqrt(156324)*u/x)
(x, y) = (sqrt(156324)*u/v, (1+u)/(1-u))

Both of those curves are also 4-isogenous to the following Edwards curve x^2 + y^2 = 1 + dx^2y^2, called "edwards448", where:

p  2^448 - 2^224 - 1

d  -39081

order  2^446 - 0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d

cofactor  4

X(P)  22458004029592430018760433409989603624678964163256413424612546168695041546740690902919286935795328257803207514644617367460263524771022458004029592430018760433409989603624678964163256413424612546168695041546740690902919286935795328257803207514644617367460263524771

Y(P)  29881921007848149267601793044393067343754404015408024209592824137233150618983587600353687865541878478473398230323350346250053154506250053154506250053154506250053154506283266

The 4-isogeny maps between the Montgomery curve and this Edwards curve are:

(u, v) = (y^2/x^2, (2 - x^2 - y^2)*y/x^3)
(x, y) = (4*v*(u^2 - 1)/(u^4 - 2*u^2 + 4*v^2 + 1),
          -(u^5 - 2*u^3 - 4*u*v^2 + u)/(u^5 - 2*u^2*v^2 - 2*u^3 - 2*v^2 + u))

The curve edwards448 defined here is also called "Goldilocks" and is equal to the one defined in [goldilocks].