```
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)])
``````
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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 little-endian. 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).
``````
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)
``````
To implement the X25519(k, u) and X448(k, u) functions (where k is
the scalar and u is the u-coordinate), 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.
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x_1 = u
x_2 = 1
z_2 = 0
x_3 = u
z_3 = 1
swap = 0
For t = bits-1 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.
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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 all-1 or all-0 word of the same length as x_2
and x_3, computed, e.g., as mask(swap) = 0 - swap.
5.1. Side-Channel Considerations
X25519 and X448 are designed so that fast, constant-time
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 single-word division, can have variable
timing based on their inputs.
Side-channel attacks are an active research area that still sees
significant, new results. Implementors are advised to follow this
research closely.
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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 u-coordinate:
e6db6867583030db3594c1a424b15f7c726624ec26b3353b10a903a6d0ab1c4c
Input u-coordinate as a number (base 10):
34426434033919594451155107781188821651
316167215306631574996226621102155684838
Output u-coordinate:
c3da55379de9c6908e94ea4df28d084f32eccf03491c71f754b4075577a28552
Input scalar:
4b66e9d4d1b4673c5ad22691957d6af5c11b6421e0ea01d42ca4169e7918ba0d
Input scalar as a number (base 10):
35156891815674817266734212754503633747
128614016119564763269015315466259359304
Input u-coordinate:
e5210f12786811d3f4b7959d0538ae2c31dbe7106fc03c3efc4cd549c715a493
Input u-coordinate as a number (base 10):
88838573511839298940907593866106493194
17338800022198945255395922347792736741
Output u-coordinate:
95cbde9476e8907d7aade45cb4b873f88b595a68799fa152e6f8f7647aac7957
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X448:
Input scalar:
3d262fddf9ec8e88495266fea19a34d28882acef045104d0d1aae121
700a779c984c24f8cdd78fbff44943eba368f54b29259a4f1c600ad3
Input scalar as a number (base 10):
599189175373896402783756016145213256157230856
085026129926891459468622403380588640249457727
683869421921443004045221642549886377526240828
Input u-coordinate:
06fce640fa3487bfda5f6cf2d5263f8aad88334cbd07437f020f08f9
814dc031ddbdc38c19c6da2583fa5429db94ada18aa7a7fb4ef8a086
Input u-coordinate as a number (base 10):
382239910814107330116229961234899377031416365
240571325148346555922438025162094455820962429
142971339584360034337310079791515452463053830
Output u-coordinate:
ce3e4ff95a60dc6697da1db1d85e6afbdf79b50a2412d7546d5f239f
e14fbaadeb445fc66a01b0779d98223961111e21766282f73dd96b6f
Input scalar:
203d494428b8399352665ddca42f9de8fef600908e0d461cb021f8c5
38345dd77c3e4806e25f46d3315c44e0a5b4371282dd2c8d5be3095f
Input scalar as a number (base 10):
633254335906970592779259481534862372382525155
252028961056404001332122152890562527156973881
968934311400345568203929409663925541994577184
Input u-coordinate:
0fbcc2f993cd56d3305b0b7d9e55d4c1a8fb5dbb52f8e9a1e9b6201b
165d015894e56c4d3570bee52fe205e28a78b91cdfbde71ce8d157db
Input u-coordinate as a number (base 10):
622761797758325444462922068431234180649590390
024811299761625153767228042600197997696167956
134770744996690267634159427999832340166786063
Output u-coordinate:
884a02576239ff7a2f2f63b2db6a9ff37047ac13568e1e30fe63c4a7
ad1b3ee3a5700df34321d62077e63633c575c1c954514e99da7c179d
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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
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6. Diffie-Hellman
6.1. Curve25519
The X25519 function can be used in an Elliptic Curve Diffie-Hellman
(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 u-coordinate 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 all-zero value and
abort if so (see below). Alice and Bob can then use a key-derivation
function that includes K, K_A, and K_B to derive a symmetric key.
The check for the all-zero 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 side-channels 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
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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 u-coordinate 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 all-zero 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
brute-force attack on a primitive) of curve25519 is slightly under
the standard 128-bit 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 power-of-two
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].
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The ~224-bit security level of curve448 is a trade-off 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 Diffie-Hellman 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, elliptic-curve libraries instead. These
implementations could reject points on the twist and could reject
non-minimal 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 non-canonical 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,
```.
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RFC 7748 Elliptic Curves for Security January 2016
8.2. Informative References
[brainpool]
ECC Brainpool, "ECC Brainpool Standard Curves and Curve
Generation", October 2005,
.
[bruteforce]
Bernstein, D., "Understanding brute force", April 2005,
.
[curve25519]
Bernstein, D., "Curve25519: new Diffie-Hellman speed
records", 2006,
.
[ed25519] Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B.
Yang, "High-Speed High-Security Signatures", 2011,
.
[goldilocks]
Hamburg, M., "Ed448-Goldilocks, a new elliptic curve",
2015, .
[montgomery]
Montgomery, P., "Speeding the Pollard and Elliptic Curve
Methods of Factorization", January 1987,
.
[NIST] National Institute of Standards, "Recommended Elliptic
Curves for Federal Government Use", July 1999,
.
[reducing] Menezes, A., Okamoto, T., and S. Vanstone, "Reducing
elliptic curve logarithms to logarithms in a finite
field", DOI 10.1109/18.259647, 1993,
.
[RFC6090] McGrew, D., Igoe, K., and M. Salter, "Fundamental Elliptic
Curve Cryptography Algorithms", RFC 6090,
DOI 10.17487/RFC6090, February 2011,
.
Langley, et al. Informational [Page 17]
RFC 7748 Elliptic Curves for Security January 2016
[safecurves]
Bernstein, D. and T. Lange, "SafeCurves: choosing safe
curves for elliptic-curve cryptography", Oct 2013,
.
[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, .
[semaev] Semaev, I., "Evaluation of discrete logarithms on some
elliptic curves", 1998, .
[smart] Smart, N., "The Discrete Logarithm Problem on Elliptic
Curves of Trace One", 1999,
.
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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 (A-2)/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
(A-2) is a small integer that is divisible by four.
For each curve at a specific security level:
1. 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].
2. MOV Degree [reducing]: the embedding degree MUST be greater than
(order - 1) / 100, as in [brainpool] and [safecurves].
3. CM Discriminant: discriminant D MUST be greater than 2^100, as in
[safecurves].
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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 (A-2) is divisible by four and where the
cofactors are as desired. The find1Mod4 function in the following
Sage script returns this value given p:
```
def findCurve(prime, curveCofactor, twistCofactor):
F = GF(prime)
for A in xrange(3, int(1e9)):
if (A-2) % 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)
``````
Generating a curve where p = 1 mod 4
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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 (A-2) is
divisible by four. The find3Mod4 function in the following Sage
script returns this value given p:
``````
def find3Mod4(prime):
assert((prime % 4) == 3)
return findCurve(prime, 4, 4)
``````
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:
``````
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
``````
Generating the base point
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Acknowledgements
This document is the result of a combination of draft-black-rpgecc-01
and draft-turner-thecurve25519function-01. 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
sn3rd
Email: sean@sn3rd.com
Langley, et al. Informational [Page 22]
```