Request for Comments: 7748
Category: Informational
ISSN: 20701721
M. Hamburg
Rambus Cryptography Research
S. Turner
sn3rd
January 2016
Elliptic Curves for Security
Abstract

This memo specifies two elliptic curves over prime fields that offer a high level of practical security in cryptographic applications, including Transport Layer Security (TLS). These curves are intended to operate at the ~128bit and ~224bit security level, respectively, and are generated deterministically based on a list of required properties.
Status of This Memo

This document is not an Internet Standards Track specification; it is published for informational purposes.
This document is a product of the Internet Research Task Force (IRTF). The IRTF publishes the results of Internetrelated research and development activities. These results might not be suitable for deployment. This RFC represents the consensus of the Crypto Forum Research Group of the Internet Research Task Force (IRTF). Documents approved for publication by the IRSG are not a candidate for any level of Internet Standard; see Section 2 of RFC 5741.
Information about the current status of this document, any errata, and how to provide feedback on it may be obtained at http://www.rfceditor.org/info/rfc7748.
Copyright Notice

Copyright © 2016 IETF Trust and the persons identified as the document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (http://trustee.ietf.org/licenseinfo) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document.
Table of Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Requirements Language . . . . . . . . . . . . . . . . . . . . 3 3. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4. Recommended Curves . . . . . . . . . . . . . . . . . . . . . 4 4.1. Curve25519 . . . . . . . . . . . . . . . . . . . . . . . 4 4.2. Curve448 . . . . . . . . . . . . . . . . . . . . . . . . 5 5. The X25519 and X448 Functions . . . . . . . . . . . . . . . . 7 5.1. SideChannel Considerations . . . . . . . . . . . . . . . 10 5.2. Test Vectors . . . . . . . . . . . . . . . . . . . . . . 11 6. DiffieHellman . . . . . . . . . . . . . . . . . . . . . . . 14 6.1. Curve25519 . . . . . . . . . . . . . . . . . . . . . . . 14 6.2. Curve448 . . . . . . . . . . . . . . . . . . . . . . . . 15 7. Security Considerations . . . . . . . . . . . . . . . . . . . 15 8. References . . . . . . . . . . . . . . . . . . . . . . . . . 16 8.1. Normative References . . . . . . . . . . . . . . . . . . 16 8.2. Informative References . . . . . . . . . . . . . . . . . 17 Appendix A. Deterministic Generation . . . . . . . . . . . . . . 19 A.1. p = 1 mod 4 . . . . . . . . . . . . . . . . . . . . . . . 20 A.2. p = 3 mod 4 . . . . . . . . . . . . . . . . . . . . . . . 21 A.3. Base Points . . . . . . . . . . . . . . . . . . . . . . . 21 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 22 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 22
1. Introduction

Since the initial standardization of Elliptic Curve Cryptography (ECC [RFC6090]) in [SEC1], there has been significant progress related to both efficiency and security of curves and implementations. Notable examples are algorithms protected against certain sidechannel attacks, various "special" prime shapes that allow faster modular arithmetic, and a larger set of curve models from which to choose. There is also concern in the community regarding the generation and potential weaknesses of the curves defined by NIST [NIST].
This memo specifies two elliptic curves ("curve25519" and "curve448") that lend themselves to constanttime implementation and an exceptionfree scalar multiplication that is resistant to a wide range of sidechannel attacks, including timing and cache attacks. They are Montgomery curves (where v^2 = u^3 + A*u^2 + u) and thus have birationally equivalent Edwards versions. Edwards curves support the fastest (currently known) complete formulas for the ellipticcurve group operations, specifically the Edwards curve x^2 + y^2 = 1 + d*x^2*y^2 for primes p when p = 3 mod 4, and the twisted Edwards curve x^2 + y^2 = 1 + d*x^2*y^2 when p = 1 mod 4. The maps to/from the Montgomery curves to their (twisted) Edwards equivalents are also given.
This memo also specifies how these curves can be used with the DiffieHellman protocol for key agreement.
2. Requirements Language

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in RFC 2119 [RFC2119].
3. Notation

Throughout this document, the following notation is used:
p Denotes the prime number defining the underlying field. GF(p) The finite field with p elements. A An element in the finite field GF(p), not equal to 2 or 2. d A nonzero element in the finite field GF(p), not equal to 1, in the case of an Edwards curve, or not equal to 1, in the case of a twisted Edwards curve. order The order of the primeorder subgroup. P A generator point defined over GF(p) of prime order. U(P) The ucoordinate of the elliptic curve point P on a Montgomery curve. V(P) The vcoordinate of the elliptic curve point P on a Montgomery curve. X(P) The xcoordinate of the elliptic curve point P on a (twisted) Edwards curve. Y(P) The ycoordinate of the elliptic curve point P on a (twisted) Edwards curve. u, v Coordinates on a Montgomery curve. x, y Coordinates on a (twisted) Edwards curve.
4. Recommended Curves
4.1. Curve25519

For the ~128bit security level, the prime 2^255  19 is recommended for performance on a wide range of architectures. Few primes of the form 2^cs 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) 147816194475895447910205935684099868872646061346164752889648818


The base point is u = 9, v = 1478161944758954479102059356840998688726 4606134616475288964881837755586237401.
This curve is birationally equivalent to a twisted Edwards curve x^2 + y^2 = 1 + d*x^2*y^2, called "edwards25519", where:
p 2^255  19 d 370957059346694393431380835087545651895421138798432190163887855330 85940283555 order 2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed cofactor 8
X(P) 151122213495354007725011514095885315114540126930418572060461132


Y(P) 463168356949264781694283940034751631413079938662562256157830336


The birational maps are:
(u, v) = ((1+y)/(1y), sqrt(486664)*u/x) (x, y) = (sqrt(486664)*u/v, (u1)/(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 ~224bit 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) 355293926785568175264127502063783334808976399387714271831880898

435169088786967410002932673765864550910142774147268105838985595290 606362
This curve is birationally equivalent to the Edwards curve x^2 + y^2 = 1 + d*x^2*y^2 where:
p 2^448  2^224  1 d 611975850744529176160423220965553317543219696871016626328968936415 087860042636474891785599283666020414768678979989378147065462815545 017 order 2^446  0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d cofactor 4
X(P) 345397493039729516374008604150537410266655260075183290216406970

281645695073672344430481787759340633221708391583424041788924124567 700732
Y(P) 363419362147803445274661903944002267176820680343659030140745099

590306164083365386343198191849338272965044442230921818680526749009 182718
The birational maps are:
(u, v) = ((y1)/(y+1), sqrt(156324)*u/x) (x, y) = (sqrt(156324)*u/v, (1+u)/(1u))
Both of those curves are also 4isogenous to the following Edwards curve x^2 + y^2 = 1 + d*x^2*y^2, called "edwards448", where:
p 2^448  2^224  1 d 39081 order 2^446  0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d cofactor 4
X(P) 224580040295924300187604334099896036246789641632564134246125461

686950415467406032909029192869357953282578032075146446173674602635 247710
Y(P) 298819210078481492676017930443930673437544040154080242095928241

372331506189835876003536878655418784733982303233503462500531545062 832660
The 4isogeny 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].
5. The X25519 and X448 Functions

The "X25519" and "X448" functions perform scalar multiplication on the Montgomery form of the above curves. (This is used when implementing DiffieHellman.) The functions take a scalar and a ucoordinate as inputs and produce a ucoordinate as output. Although the functions work internally with integers, the inputs and outputs are 32byte strings (for X25519) or 56byte strings (for X448) and this specification defines their encoding.
The ucoordinates are elements of the underlying field GF(2^255  19) or GF(2^448  2^224  1) and are encoded as an array of bytes, u, in littleendian order such that u[0] + 256*u[1] + 256^2*u[2] + ... + 256^(n1)*u[n1] is congruent to the value modulo p and u[n1] is minimal. When receiving such an array, implementations of X25519 (but not X448) MUST mask the most significant bit in the final byte. This is done to preserve compatibility with point formats that reserve the sign bit for use in other protocols and to increase resistance to implementation fingerprinting.
Implementations MUST accept noncanonical values and process them as if they had been reduced modulo the field prime. The noncanonical values are 2^255  19 through 2^255  1 for X25519 and 2^448  2^224  1 through 2^448  1 for X448.
The following functions implement this in Python, although the Python code is not intended to be performant nor sidechannel free. Here, the "bits" parameter should be set to 255 for X25519 and 448 for X448:
<CODE BEGINS>
def decodeLittleEndian(b, bits):return sum([b[i] << 8*i for i in range((bits+7)/8)])
def decodeUCoordinate(u, bits):
u_list = [ord(b) for b in u] # Ignore any unused bits. if bits % 8: u_list[1] &= (1<<(bits%8))1 return decodeLittleEndian(u_list, bits) def encodeUCoordinate(u, bits): u = u % p return ''.join([chr((u >> 8*i) & 0xff) for i in range((bits+7)/8)]) <CODE ENDS>
Scalars are assumed to be randomly generated bytes. For X25519, in order to decode 32 random bytes as an integer scalar, set the three least significant bits of the first byte and the most significant bit of the last to zero, set the second most significant bit of the last byte to 1 and, finally, decode as littleendian. This means that the resulting integer is of the form 2^254 plus eight times a value between 0 and 2^251  1 (inclusive). Likewise, for X448, set the two least significant bits of the first byte to 0, and the most significant bit of the last byte to 1. This means that the resulting integer is of the form 2^447 plus four times a value between 0 and 2^445  1 (inclusive).
<CODE BEGINS>
def decodeScalar25519(k):k_list = [ord(b) for b in k] k_list[0] &= 248 k_list[31] &= 127 k_list[31] = 64 return decodeLittleEndian(k_list, 255) def decodeScalar448(k): k_list = [ord(b) for b in k] k_list[0] &= 252 k_list[55] = 128 return decodeLittleEndian(k_list, 448) <CODE ENDS>
To implement the X25519(k, u) and X448(k, u) functions (where k is the scalar and u is the ucoordinate), first decode k and u and then perform the following procedure, which is taken from [curve25519] and based on formulas from [montgomery]. All calculations are performed in GF(p), i.e., they are performed modulo p. The constant a24 is (486662  2) / 4 = 121665 for curve25519/X25519 and (156326  2) / 4 = 39081 for curve448/X448.
x_1 = u x_2 = 1 z_2 = 0 x_3 = u z_3 = 1 swap = 0
For t = bits1 down to 0:
k_t = (k >> t) & 1 swap ^= k_t // Conditional swap; see text below. (x_2, x_3) = cswap(swap, x_2, x_3) (z_2, z_3) = cswap(swap, z_2, z_3) swap = k_t A = x_2 + z_2 AA = A^2 B = x_2  z_2 BB = B^2 E = AA  BB C = x_3 + z_3 D = x_3  z_3 DA = D * A CB = C * B x_3 = (DA + CB)^2 z_3 = x_1 * (DA  CB)^2 x_2 = AA * BB z_2 = E * (AA + a24 * E) // Conditional swap; see text below. (x_2, x_3) = cswap(swap, x_2, x_3) (z_2, z_3) = cswap(swap, z_2, z_3) Return x_2 * (z_2^(p  2))
(Note that these formulas are slightly different from Montgomery's original paper. Implementations are free to use any correct formulas.)
Finally, encode the resulting value as 32 or 56 bytes in little endian order. For X25519, the unused, most significant bit MUST be zero.
The cswap function SHOULD be implemented in constant time (i.e., independent of the swap argument). For example, this can be done as follows:
cswap(swap, x_2, x_3):


dummy = mask(swap) AND (x_2 XOR x_3)
x_2 = x_2 XOR dummy
x_3 = x_3 XOR dummy
Return (x_2, x_3)

Where mask(swap) is the all1 or all0 word of the same length as x_2 and x_3, computed, e.g., as mask(swap) = 0  swap.
5.1. SideChannel Considerations

X25519 and X448 are designed so that fast, constanttime implementations are easier to produce. The procedure above ensures that the same sequence of field operations is performed for all values of the secret key, thus eliminating a common source of side channel leakage. However, this alone does not prevent all side channels by itself. It is important that the pattern of memory accesses and jumps not depend on the values of any of the bits of k. It is also important that the arithmetic used not leak information about the integers modulo p, for example by having b*c be distinguishable from c*c. On some architectures, even primitive machine instructions, such as singleword division, can have variable timing based on their inputs.
Sidechannel attacks are an active research area that still sees significant, new results. Implementors are advised to follow this research closely.
5.2. Test Vectors

Two types of tests are provided. The first is a pair of test vectors for each function that consist of expected outputs for the given inputs. The inputs are generally given as 64 or 112 hexadecimal digits that need to be decoded as 32 or 56 binary bytes before processing.
X25519:
Input scalar: a546e36bf0527c9d3b16154b82465edd62144c0ac1fc5a18506a2244ba449ac4 Input scalar as a number (base 10): 31029842492115040904895560451863089656 472772604678260265531221036453811406496 Input ucoordinate: e6db6867583030db3594c1a424b15f7c726624ec26b3353b10a903a6d0ab1c4c Input ucoordinate as a number (base 10): 34426434033919594451155107781188821651 316167215306631574996226621102155684838 Output ucoordinate: c3da55379de9c6908e94ea4df28d084f32eccf03491c71f754b4075577a28552 Input scalar: 4b66e9d4d1b4673c5ad22691957d6af5c11b6421e0ea01d42ca4169e7918ba0d Input scalar as a number (base 10): 35156891815674817266734212754503633747 128614016119564763269015315466259359304 Input ucoordinate: e5210f12786811d3f4b7959d0538ae2c31dbe7106fc03c3efc4cd549c715a493 Input ucoordinate as a number (base 10): 88838573511839298940907593866106493194 17338800022198945255395922347792736741 Output ucoordinate: 95cbde9476e8907d7aade45cb4b873f88b595a68799fa152e6f8f7647aac7957
X448:
Input scalar: 3d262fddf9ec8e88495266fea19a34d28882acef045104d0d1aae121 700a779c984c24f8cdd78fbff44943eba368f54b29259a4f1c600ad3 Input scalar as a number (base 10): 599189175373896402783756016145213256157230856 085026129926891459468622403380588640249457727 683869421921443004045221642549886377526240828 Input ucoordinate: 06fce640fa3487bfda5f6cf2d5263f8aad88334cbd07437f020f08f9 814dc031ddbdc38c19c6da2583fa5429db94ada18aa7a7fb4ef8a086 Input ucoordinate as a number (base 10): 382239910814107330116229961234899377031416365 240571325148346555922438025162094455820962429 142971339584360034337310079791515452463053830 Output ucoordinate: ce3e4ff95a60dc6697da1db1d85e6afbdf79b50a2412d7546d5f239f e14fbaadeb445fc66a01b0779d98223961111e21766282f73dd96b6f Input scalar: 203d494428b8399352665ddca42f9de8fef600908e0d461cb021f8c5 38345dd77c3e4806e25f46d3315c44e0a5b4371282dd2c8d5be3095f Input scalar as a number (base 10): 633254335906970592779259481534862372382525155 252028961056404001332122152890562527156973881 968934311400345568203929409663925541994577184 Input ucoordinate: 0fbcc2f993cd56d3305b0b7d9e55d4c1a8fb5dbb52f8e9a1e9b6201b 165d015894e56c4d3570bee52fe205e28a78b91cdfbde71ce8d157db Input ucoordinate as a number (base 10): 622761797758325444462922068431234180649590390 024811299761625153767228042600197997696167956 134770744996690267634159427999832340166786063 Output ucoordinate: 884a02576239ff7a2f2f63b2db6a9ff37047ac13568e1e30fe63c4a7 ad1b3ee3a5700df34321d62077e63633c575c1c954514e99da7c179d
The second type of test vector consists of the result of calling the function in question a specified number of times. Initially, set k and u to be the following values:
For X25519: 0900000000000000000000000000000000000000000000000000000000000000 For X448: 05000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000
For each iteration, set k to be the result of calling the function and u to be the old value of k. The final result is the value left in k.
X25519:
After one iteration: 422c8e7a6227d7bca1350b3e2bb7279f7897b87bb6854b783c60e80311ae3079 After 1,000 iterations: 684cf59ba83309552800ef566f2f4d3c1c3887c49360e3875f2eb94d99532c51 After 1,000,000 iterations: 7c3911e0ab2586fd864497297e575e6f3bc601c0883c30df5f4dd2d24f665424
X448:
After one iteration: 3f482c8a9f19b01e6c46ee9711d9dc14fd4bf67af30765c2ae2b846a 4d23a8cd0db897086239492caf350b51f833868b9bc2b3bca9cf4113 After 1,000 iterations: aa3b4749d55b9daf1e5b00288826c467274ce3ebbdd5c17b975e09d4 af6c67cf10d087202db88286e2b79fceea3ec353ef54faa26e219f38 After 1,000,000 iterations: 077f453681caca3693198420bbe515cae0002472519b3e67661a7e89 cab94695c8f4bcd66e61b9b9c946da8d524de3d69bd9d9d66b997e37
6. DiffieHellman
6.1. Curve25519

The X25519 function can be used in an Elliptic Curve DiffieHellman (ECDH) protocol as follows:
Alice generates 32 random bytes in a[0] to a[31] and transmits K_A = X25519(a, 9) to Bob, where 9 is the ucoordinate of the base point and is encoded as a byte with value 9, followed by 31 zero bytes.
Bob similarly generates 32 random bytes in b[0] to b[31], computes K_B = X25519(b, 9), and transmits it to Alice.
Using their generated values and the received input, Alice computes X25519(a, K_B) and Bob computes X25519(b, K_A).
Both now share K = X25519(a, X25519(b, 9)) = X25519(b, X25519(a, 9)) as a shared secret. Both MAY check, without leaking extra information about the value of K, whether K is the allzero value and abort if so (see below). Alice and Bob can then use a keyderivation function that includes K, K_A, and K_B to derive a symmetric key.
The check for the allzero value results from the fact that the X25519 function produces that value if it operates on an input corresponding to a point with small order, where the order divides the cofactor of the curve (see Section 7). The check may be performed by ORing all the bytes together and checking whether the result is zero, as this eliminates standard sidechannels in software implementations.
Test vector:
Alice's private key, a: 77076d0a7318a57d3c16c17251b26645df4c2f87ebc0992ab177fba51db92c2a Alice's public key, X25519(a, 9): 8520f0098930a754748b7ddcb43ef75a0dbf3a0d26381af4eba4a98eaa9b4e6a Bob's private key, b: 5dab087e624a8a4b79e17f8b83800ee66f3bb1292618b6fd1c2f8b27ff88e0eb Bob's public key, X25519(b, 9): de9edb7d7b7dc1b4d35b61c2ece435373f8343c85b78674dadfc7e146f882b4f Their shared secret, K: 4a5d9d5ba4ce2de1728e3bf480350f25e07e21c947d19e3376f09b3c1e161742
6.2. Curve448

The X448 function can be used in an ECDH protocol very much like the X25519 function.
If X448 is to be used, the only differences are that Alice and Bob generate 56 random bytes (not 32) and calculate K_A = X448(a, 5) or K_B = X448(b, 5), where 5 is the ucoordinate of the base point and is encoded as a byte with value 5, followed by 55 zero bytes.
As with X25519, both sides MAY check, without leaking extra information about the value of K, whether the resulting shared K is the allzero value and abort if so.
Test vector:
Alice's private key, a: 9a8f4925d1519f5775cf46b04b5800d4ee9ee8bae8bc5565d498c28d d9c9baf574a9419744897391006382a6f127ab1d9ac2d8c0a598726b Alice's public key, X448(a, 5): 9b08f7cc31b7e3e67d22d5aea121074a273bd2b83de09c63faa73d2c 22c5d9bbc836647241d953d40c5b12da88120d53177f80e532c41fa0 Bob's private key, b: 1c306a7ac2a0e2e0990b294470cba339e6453772b075811d8fad0d1d 6927c120bb5ee8972b0d3e21374c9c921b09d1b0366f10b65173992d Bob's public key, X448(b, 5): 3eb7a829b0cd20f5bcfc0b599b6feccf6da4627107bdb0d4f345b430 27d8b972fc3e34fb4232a13ca706dcb57aec3dae07bdc1c67bf33609 Their shared secret, K: 07fff4181ac6cc95ec1c16a94a0f74d12da232ce40a77552281d282b b60c0b56fd2464c335543936521c24403085d59a449a5037514a879d
7. Security Considerations

The security level (i.e., the number of "operations" needed for a bruteforce attack on a primitive) of curve25519 is slightly under the standard 128bit level. This is acceptable because the standard security levels are primarily driven by much simpler, symmetric primitives where the security level naturally falls on a power of two. For asymmetric primitives, rigidly adhering to a poweroftwo security level would require compromises in other parts of the design, which we reject. Additionally, comparing security levels between types of primitives can be misleading under common threat models where multiple targets can be attacked concurrently [bruteforce].
The ~224bit security level of curve448 is a tradeoff between performance and paranoia. Large quantum computers, if ever created, will break both curve25519 and curve448, and reasonable projections of the abilities of classical computers conclude that curve25519 is perfectly safe. However, some designs have relaxed performance requirements and wish to hedge against some amount of analytical advance against elliptic curves and thus curve448 is also provided.
Protocol designers using DiffieHellman over the curves defined in this document must not assume "contributory behaviour". Specially, contributory behaviour means that both parties' private keys contribute to the resulting shared key. Since curve25519 and curve448 have cofactors of 8 and 4 (respectively), an input point of small order will eliminate any contribution from the other party's private key. This situation can be detected by checking for the all zero output, which implementations MAY do, as specified in Section 6. However, a large number of existing implementations do not do this.
Designers using these curves should be aware that for each public key, there are several publicly computable public keys that are equivalent to it, i.e., they produce the same shared secrets. Thus using a public key as an identifier and knowledge of a shared secret as proof of ownership (without including the public keys in the key derivation) might lead to subtle vulnerabilities.
Designers should also be aware that implementations of these curves might not use the Montgomery ladder as specified in this document, but could use generic, ellipticcurve libraries instead. These implementations could reject points on the twist and could reject nonminimal field elements. While not recommended, such implementations will interoperate with the Montgomery ladder specified here but may be trivially distinguishable from it. For example, sending a noncanonical value or a point on the twist may cause such implementations to produce an observable error while an implementation that follows the design in this text would successfully produce a shared key.
8. References
8.1. Normative References

[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, <http://www.rfceditor.org/info/rfc2119>.
8.2. Informative References

[brainpool]
ECC Brainpool, "ECC Brainpool Standard Curves and Curve Generation", October 2005, <http://www.eccbrainpool.org/download/ Domainparameters.pdf>.
[bruteforce]
Bernstein, D., "Understanding brute force", April 2005, <http://cr.yp.to/snuffle/bruteforce20050425.pdf>.
[curve25519]
Bernstein, D., "Curve25519: new DiffieHellman speed records", 2006, <http://www.iacr.org/cryptodb/archive/2006/ PKC/3351/3351.pdf>. [ed25519] Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B. Yang, "HighSpeed HighSecurity Signatures", 2011, <http://link.springer.com/ chapter/10.1007/9783642239519_9>.
[goldilocks]
Hamburg, M., "Ed448Goldilocks, a new elliptic curve", 2015, <http://eprint.iacr.org/2015/625.pdf>.
[montgomery]
Montgomery, P., "Speeding the Pollard and Elliptic Curve Methods of Factorization", January 1987, <http://www.ams.org/journals/mcom/198748177/ S00255718198708661137/S00255718198708661137.pdf>. [NIST] National Institute of Standards, "Recommended Elliptic Curves for Federal Government Use", July 1999, <http://csrc.nist.gov/groups/ST/toolkit/documents/dss/ NISTReCur.pdf>. [reducing] Menezes, A., Okamoto, T., and S. Vanstone, "Reducing elliptic curve logarithms to logarithms in a finite field", DOI 10.1109/18.259647, 1993, <http://ieeexplore.ieee.org/xpl/ articleDetails.jsp?arnumber=259647>. [RFC6090] McGrew, D., Igoe, K., and M. Salter, "Fundamental Elliptic Curve Cryptography Algorithms", RFC 6090, DOI 10.17487/RFC6090, February 2011, <http://www.rfceditor.org/info/rfc6090>.
[safecurves]
Bernstein, D. and T. Lange, "SafeCurves: choosing safe curves for ellipticcurve cryptography", Oct 2013, <http://safecurves.cr.yp.to/>. [satoh] Satoh, T. and K. Araki, "Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves", 1998. [SEC1] Certicom Research, "SEC 1: Elliptic Curve Cryptography", September 2000, <http://www.secg.org/sec1v2.pdf>. [semaev] Semaev, I., "Evaluation of discrete logarithms on some elliptic curves", 1998, <http://www.ams.org/journals/ mcom/199867221/S0025571898008874/ S0025571898008874.pdf>. [smart] Smart, N., "The Discrete Logarithm Problem on Elliptic Curves of Trace One", 1999, <http://www.hpl.hp.com/techreports/97/HPL97128.pdf>.
Appendix A. Deterministic Generation

This section specifies the procedure that was used to generate the above curves; specifically, it defines how to generate the parameter A of the Montgomery curve y^2 = x^3 + A*x^2 + x. This procedure is intended to be as objective as can reasonably be achieved so that it's clear that no untoward considerations influenced the choice of curve. The input to this process is p, the prime that defines the underlying field. The size of p determines the amount of work needed to compute a discrete logarithm in the elliptic curve group, and choosing a precise p depends on many implementation concerns. The performance of the curve will be dominated by operations in GF(p), so carefully choosing a value that allows for easy reductions on the intended architecture is critical. This document does not attempt to articulate all these considerations.
The value (A2)/4 is used in several of the elliptic curve point arithmetic formulas. For simplicity and performance reasons, it is beneficial to make this constant small, i.e., to choose A so that (A2) is a small integer that is divisible by four.
For each curve at a specific security level:
 The trace of Frobenius MUST NOT be in {0, 1} in order to rule out the attacks described in [smart], [satoh], and [semaev], as in [brainpool] and [safecurves].
 MOV Degree [reducing]: the embedding degree MUST be greater than (order  1) / 100, as in [brainpool] and [safecurves].
 CM Discriminant: discriminant D MUST be greater than 2^100, as in [safecurves].
A.1. p = 1 mod 4

For primes congruent to 1 mod 4, the minimal cofactors of the curve and its twist are either {4, 8} or {8, 4}. We choose a curve with the latter cofactors so that any algorithms that take the cofactor into account don't have to worry about checking for points on the twist, because the twist cofactor will be the smaller of the two.
To generate the Montgomery curve, we find the minimal, positive A value such that A > 2 and (A2) is divisible by four and where the cofactors are as desired. The find1Mod4 function in the following Sage script returns this value given p:
<CODE BEGINS>
def findCurve(prime, curveCofactor, twistCofactor):F = GF(prime)

for A in xrange(3, int(1e9)):

if (A2) % 4 != 0:
continue try: E = EllipticCurve(F, [0, A, 0, 1, 0]) except: continue groupOrder = E.order() twistOrder = 2*(prime+1)groupOrder


if (groupOrder % curveCofactor == 0 and

is_prime(groupOrder // curveCofactor) and twistOrder % twistCofactor == 0 and is_prime(twistOrder // twistCofactor)): return A def find1Mod4(prime): assert((prime % 4) == 1) return findCurve(prime, 8, 4) <CODE ENDS>

Generating a curve where p = 1 mod 4
A.2. p = 3 mod 4

For a prime congruent to 3 mod 4, both the curve and twist cofactors can be 4, and this is minimal. Thus, we choose the curve with these cofactors and minimal, positive A such that A > 2 and (A2) is divisible by four. The find3Mod4 function in the following Sage script returns this value given p:
<CODE BEGINS> def find3Mod4(prime): assert((prime % 4) == 3) return findCurve(prime, 4, 4) <CODE ENDS>
Generating a curve where p = 3 mod 4
A.3. Base Points

The base point for a curve is the point with minimal, positive u value that is in the correct subgroup. The findBasepoint function in the following Sage script returns this value given p and A:
<CODE BEGINS>
def findBasepoint(prime, A):F = GF(prime) E = EllipticCurve(F, [0, A, 0, 1, 0])

for uInt in range(1, 1e3):
u = F(uInt) v2 = u^3 + A*u^2 + u if not v2.is_square(): continue v = v2.sqrt() point = E(u, v) pointOrder = point.order() if pointOrder > 8 and pointOrder.is_prime(): return point <CODE ENDS>

Generating the base point
Acknowledgements

This document is the result of a combination of draftblackrpgecc01 and draftturnerthecurve25519function01. The following authors of those documents wrote much of the text and figures but are not listed as authors on this document: Benjamin Black, Joppe W. Bos, Craig Costello, Patrick Longa, Michael Naehrig, Watson Ladd, and Rich Salz.
The authors would also like to thank Tanja Lange, Rene Struik, Rich Salz, Ilari Liusvaara, Deirdre Connolly, Simon Josefsson, Stephen Farrell, Georg Nestmann, Trevor Perrin, and John Mattsson for their reviews and contributions.
The X25519 function was developed by Daniel J. Bernstein in [curve25519].
Authors' Addresses

Adam Langley Google 345 Spear Street San Francisco, CA 94105 United States
Email:
agl@google.com Mike Hamburg Rambus Cryptography Research 425 Market Street, 11th Floor San Francisco, CA 94105 United States
Email:
mike@shiftleft.org
Sean Turner
sn3rdEmail:
sean@sn3rd.com