Network Working Group W. Ladd
Internet-Draft UC Berkeley
Intended status: Standards Track P. Longa
Expires: March 26, 2017 Microsoft Research
R. Barnes
Mozilla
September 22, 2016
Curve4Q
draft-ladd-cfrg-4q-00
Abstract
This document specifies a twisted Edwards curve that takes advantage
of arithmetic over the field GF(2^127-1) and two endomorphisms to
achieve the speediest Diffie-Hellman key agreements over a group of
order approximately 2^246, which provides around 128 bits of
security. Curve4Q implementations are more than two times faster
than those of Curve25519 and, when not using endomorphisms, are
between 1.2 and 1.6 times faster.
Status of This Memo
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Mathematical Prerequisites . . . . . . . . . . . . . . . . . 3
3. Representation of Curve Points . . . . . . . . . . . . . . . 4
4. Scalar multiplication . . . . . . . . . . . . . . . . . . . . 6
4.1. Alternative Point Representations and Addition Laws . . . 6
4.2. Multiplication without Endomorphisms . . . . . . . . . . 8
4.3. Multiplication with Endomorphisms . . . . . . . . . . . . 9
4.3.1. Endomorphisms . . . . . . . . . . . . . . . . . . . . 9
4.3.2. Scalar Decomposition and Recoding . . . . . . . . . . 11
4.3.3. Final Computation . . . . . . . . . . . . . . . . . . 12
5. Diffie-Hellman Key Agreement . . . . . . . . . . . . . . . . 13
6. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 14
7. Security Considerations . . . . . . . . . . . . . . . . . . . 14
8. Informative References . . . . . . . . . . . . . . . . . . . 15
Appendix A. Constants . . . . . . . . . . . . . . . . . . . . . 16
Appendix B. Point Decompression . . . . . . . . . . . . . . . . 18
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 20
1. Introduction
Public key cryptography continues to be computationally expensive,
particularly on less powerful devices. While recent advances in
efficient formulas for addition and doubling have substantially
reduced the cost of elliptic curve operations in terms of field
operations, the number of group operations involved in scalar
multiplication has not been reduced in the curves considered for IETF
use. Using curves with efficiently computable endomorphisms can
reduce the number of group operations by turning one long scalar
multiplication into the sum of several multiplications by smaller
scalars, which can be evaluated more efficiently.
For curves over quadratic extension fieldss, there are more
endomorphism families to choose from, and the field operations are
often more efficient compared to prime fields of the same size. The
ideal case is given by curves equipped with two distinct
endomorphisms, so that it becomes possible to divide scalars into
four parts. We focus on curves defined over the field GF(p^2) for
the Mersenne prime p = 2^127 - 1, which offers extremely efficient
arithmetic. Together, these improvements substantially reduce
computation time compared to other proposed Diffie-Hellman key
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exchange and digital signature schemes. However, the combined
availability of these features severely restricts the curves that can
be used for cryptographic applications.
As described in [Curve4Q], the elliptic curve "Curve4Q" defined in
this document is the only known elliptic curve that (1) permits a
four dimensional decomposition (using two endomorphisms) over GF(p^2)
and (2) has a large prime order subgroup. The order of this subgroup
is approximately 2^246, which provides around 128 bits of security.
No other known elliptic curve with such a decomposition has a larger
prime order subgroup over this field. This "uniqueness" allays
concerns about selecting curves vulnerable to undisclosed attacks.
Curve4Q can be used to implement Diffie-Hellman key exchange, as
described below. It is also possible to use Curve4Q as the basis for
digital signature scheme (e.g., [SchnorrQ]).
2. Mathematical Prerequisites
Curve4Q is defined over the finite field GF(p^2), where p is the
Mersenne prime 2^127 - 1. Elements of this finite field have the
form (a + b * i), where a and b are elements of the finite field
GF(p) (i.e., integers mod p) and i^2 = -1.
Let A = a0 + a1*i and B = b0 + b1*i be two elements of GF(p^2).
Below we present formulas for computing addition, subtraction,
multiplication, squaring, conjugation and inversion.
A + B = (a0 + b0) + (a1 + b1)*i
A - B = (a0 - b0) + (a1 - b1)*i
A * B = (a0*b0 - a1*b1) + ((a0 + a1)*(b0 + b1)-(a0*b0 - a1*b1))*i
= (a0*b0 - a1*b1) + (a0*b1 + a1*b0)*i
A * A = (a0 + a1)*(a0 - a1) + 2*a0*a1*i
conj(A) = a0 - a1*i
1/A = conj(A) / (a0^2 + a1^2)
The GF(p) division in the formula for 1/A can be computed using an
exponentiation via Fermat's little theorem: 1/a = a^(p - 2) =
a^(2^127 - 3) for any element a of GF(p). One can use a fixed
addition chain to compute a^(2^127 - 3) (e.g., see [FourQlib]).
Curve4Q is the twisted Edwards curve E over GF(p^2) defined by the
following curve equation:
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E: -x^2 + y^2 = 1 + d * x^2 * y^2, with
d = 0x00000000000000e40000000000000142 +
0x5e472f846657e0fcb3821488f1fc0c8d * i
Let E(GF(p^2)) be the set of pairs (x, y) of elements of GF(p^2)
satisfying this equation. This set forms a group with the addition
operation (x1, y1) + (x2, y2) = (x3, y3), where:
x1 * y2 + y1 * x2 y1 * y2 + x1 * x2
x3 = ---------------------------, y3 = ---------------------------
1 + d * x1 * y1 * x2 * y2 1 - d * x1 * y1 * x2 * y2
As d is not a square in GF(p^2), and -1 is, this formula never
involves a division by zero when applied to points on the curve.
That is, the formula is complete and works without exceptions for any
input in E(GF(p^2)). The identity element is (0, 1), and the inverse
of (x, y) is (-x, y). The order of this group is #E = 2^3 . 7^2 . N,
where N is the following 246-bit prime:
N = 0x29cbc14e5e0a72f05397829cbc14e5dfbd004dfe0f79992fb2540ec7768ce7
Points P on E such that [N]*P = (0, 1) are N-torsion points. Given a
point P and Q which are both N-torsion points, it is difficult to
find m such that Q = [m]*P. This is the elliptic curve discrete
logarithm problem, which is closely related to the security of
Diffie-Hellman key exchanges as the best known attacks on the Diffie-
Hellman problem involve solving the discrete logarithm problem. The
best known algorithms take approximately 2^123 group operations.
This group has two different efficiently computable endomorphisms, as
described in [Curve4Q]. As discussed in [GLV] and [GLS], these
endomorphisms allow a multiplication by a large scalar to be computed
using multiple multiplications by smaller scalars, which can be
evaluated in much less time overall.
3. Representation of Curve Points
Elements a in GF(p) are represented as 16 byte little endian integers
which are the numbers in the range [0, p). The 16 bytes b[0], b[1],
... b[15] represent b[0] + 256*b[1] + 256^2*b[2] + ... +
256^15*b[15]. Since we are representing numbers in the range [0,
2^127-1), the top bit of b[15] is always zero.
An element x0 + x1*i of GF(p^2) is represented on the wire by the
concatenation of the encodings for x0 and x1. A point (x, y) on
Curve4Q is serialized in a compressed form as the representation of y
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with a modified top bit. This top bit is used to disambiguate
between x and -x during decoding.
To carry out this disambiguation we use the lexicographic order of
elements in GF(p^2): define two elements a = a0 + a1*i and b = b0 +
b1*i with all their coordinates in [0, p); a is greater than b if a0
is greater than b0. If a0 and b0 are equal, a is greater than b if
a1 is greater than b1.
Set the coordinate value x and its negative -x. The top bit of a
compressed point is 0 if x is smaller than -x. Otherwise, the top
bit is 1.
|--------------- y ---------------|
| y0 |0| y1 |s|
|..............|.|..............|.|
To decode an encoded point from a 32-byte sequence B:
o Parse out the encoded values y = y0 + y1 * i and s
o Check that y0 and y1 are both less than p
o Solve x^2 = (y^2 - 1) * (d * y^2 + 1) for x
o If s is 0, return the smaller of x and -x (in the lexicographic
ordering)
o If s is 1, return the larger of x and -x
o Check that (x,y) is a valid point on the curve
The appendix Appendix B details an algorithm for decoding a point
following the steps above.
We call the operation of compressing a point P into 32 bytes
Compress(P), and decompression Expand(S). Expand(Compress(P))=P for
all the points P on the curve, and Compress(Expand(S))=S if and only
if S is a valid representation of a point.
Not all 32 byte strings represent valid points. Implementations MUST
reject invalid strings and check that decompression is successful.
Strings are invalid if they are not possible outputs of the
compression operator. In particular the values of y0 and y1 MUST be
less then p.
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4. Scalar multiplication
Below, we present two algorithms for scalar multiplication on
Curve4Q. Each algorithm takes as input a 256-bit unsigned integer m
and an N-torsion point P and computes the product [m]*P.
The first algorithm uses a simple fixed-window exponentiation without
exploiting endomorphisms. The second algorithm uses endomorphisms to
accelerate computation. The execution of operations in both
algorithms has a regular pattern in order to enable constant-time
implementations and protect against timing and simple side channel
attacks. Both algorithms use the same addition and doubling
formulas.
First, we discuss explicit formulas and efficient projective
coordinate representations.
4.1. Alternative Point Representations and Addition Laws
We use coordinates based on extended twisted Edwards coordinates
introduced in [TwistedRevisited]: the tuple (X, Y, Z, T) with Z
nonzero and Z * T = X * Y corresponds to a point (x, y) satisfying x
= X/Z and y = Y/Z. The neutral point in this representation is (0,
1, 1, 0). The following slight variants are used in the optimized
scalar multiplication algorithm in order to save computations: point
representation R1 is given by (X, Y, Z, Ta, Tb), where T=Ta * Tb;
representation R2 is (N, D, E, F) = (X + Y, Y- X, 2Z, 2dT);
representation R3 is (N, D, Z, T) = (X + Y, Y - X, Z, T); and
representation R4 is (X, Y, Z). Similar "caching" techniques were
discussed in [TwistedRevisited] to accelerate repeated additions of
the same point. Converting between these representations is
straightforward.
o R1: (X, Y, Z, Ta, Tb), Ta * Tb = T, Z * T = X * Y
o R2: (N, D, E, F) = (X + Y, Y - X, 2 * Z, 2 * d * T)
o R3: (N, D, Z, T) = (X + Y, Y - X, Z, T)
o R4: (X, Y, Z)
A point doubling (DBL) takes an R4 point and produces an R1 point.
For addition, we first define an operation ADD_core that takes an R2
and an R3 point and produces an R1 point. This can be used to
implement an operation ADD which takes an R1 and an R2 point as
inputs (and produces an R1 point) by first converting the R1 point to
R3, and then executing ADD_core. Exposing these operations and the
multiple representations helps save time by avoiding redundant
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computations: the conversion of the first argument to ADD can be done
once if the argument will be used in multiple additions.
Below, we list the explicit formulas for the required point
operations. These formulas, which are adapted from [Twisted] and
[TwistedRevisited], are complete: they have no exceptional cases, and
therefore can be used in any algorithm for computing scalar multiples
without worrying about exceptional procedure attacks [Exceptional].
Note that we do not explicitly note the point format every time an
addition or doubling is used, and assume that conversions are done
when required.
DBL and ADD_core are computed as follows:
DBL(X1, Y1, Z1):
A = X1^2
B = Y1^2
C = 2 * Z1^2
D = A + B
E = (X1 + Y1)^2 - D
F = B - A
G = C - F
X3 = E * G
Y3 = D * F
Z3 = F * G
Ta3 = E
Tb3 = D
return(X3, Y3, Z3, Ta3, Tb3)
ADD\_core(N1, D1, E1, F1, N2, D2, Z2, T2):
A = D1 * D2
B = N1 * N2
C = T2 * F1
D = Z2 * E1
E = B - A
F = D - C
G = D + C
H = B + A
X3 = E * F
Y3 = G * H
Z3 = F * G
Ta3 = E
Tb3 = H
return (X3, Y3, Z3, Ta3, Tb3)
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4.2. Multiplication without Endomorphisms
We begin by taking our input point P, and computing a table of points
containing T[0] = [1]P, T[1] = [3]P, ... , T[7] = [15]P as follows:
Q = DBL(P)
Convert Q to R2 form
T[0] = P
Convert T[0] to R2 form
for i=1 to 7:
T[i] = ADD_core(Q, T[i-1])
Convert T[i] to R2 form
Next, take m and reduce it modulo N. Then, add N if necessary to
ensure that m is odd. At this point, we recode m into a signed digit
representation consisting of 63 signed, odd digits d[i] in base 16.
The following algorithm accomplishes this task.
for i=0 to 61:
d[i] = (m mod 32) - 16
m = (m - d[i]) / 16
d[62] = m
Finally, the computation of the multiplication is as follows.
Let ind = (abs(d[62]) - 1) / 2
Let sign = sgn(d[62])
Q = sign * T[ind]
Convert Q into R4 form
for i from 61 to 0:
Q = DBL(Q)
Q = DBL(Q)
Q = DBL(Q)
Q = DBL(Q)
ind = (abs(d[i]) - 1) / 2
sign = sgn(d[i])
S = sign * T[ind]
Q = ADD(Q, S)
return Q
As sign is either -1 or 1, the multiplication sign * T[ind] is simply
a conditional negation. To negate a point (N, D, E, F) in R2 form
one computes (D, N, E, -F). The table lookups and conditional
negations must be carefully implemented as described in ``Security
Considerations'' to avoid side-channel attacks. This algorithm MUST
NOT be applied to points which are not N-torsion points; it will
produce the wrong answer.
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4.3. Multiplication with Endomorphisms
This algorithm makes use of the identity [m]*P = [a_1]*P +
[a_2]*phi(P) + [a_3]*psi(P) + [a_4]*psi(phi(P)), where a_1, a_2, a_3,
and a_4 are 64-bit scalars that depend on m. The overall product can
then can be computed using a small table of 8 precomputed points and
64 doublings and additions. This is considerably fewer operations
than the number of operations required by the algorithm above, at the
cost of a more complicated implementation.
We describe each phase of the computation separately: the computation
of the endomorphisms, the scalar decomposition and recoding, the
creation of the table of precomputed points and, lastly, the
computation of the final results. Each section refers to constants
listed in an appendix in order of appearance.
4.3.1. Endomorphisms
The two endomorphisms phi and psi used to accelerate multiplication
are computed as phi(Q) = tau_dual(upsilon(tau(Q)) and psi(Q) =
tau_dual(chi(tau(Q))). Below, we present procedures for tau,
tau_dual, upsilon and chi, adapted from [FourQlib]. Tau_dual
produces an R1 point, while the other procedures produce R4 points.
Note: Tau produces points on a different curve, while upsilon and chi
are endomorphisms of that different curve. Tau and tau_dual are the
isogenies mentioned in the mathematical background above. As a
result the intermediate results do not satisfy the equations of the
curve E. Implementers who wish to check the correctness of these
intermediate results are referred to [Curve4Q].
tau(X1, Y1, Z1):
A = X1^2
B = Y1^2
C = A + B
D = A - B
X2 = ctau * X1 * Y1 * D
Y2 = -(2 * Z1^2 + D) * C
Z2 = C * D
return(X2, Y2, Z2)
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tau_dual(X1, Y1, Z1):
A = X1^2
B = Y1^2
C = A + B
Ta2 = B - A
D = 2 * Z1^2 - Ta2
Tb2 = ctaudual * X1 * Y1
X2 = C * Tb2
Y2 = D * Ta2
Z2 = C * D
return(X2, Y2, Z2, Ta2, Tb2)
upsilon(X1, Y1, Z1):
A = cphi0 * X1 * Y1
B = Y1 * Z1
C = Y1^2
D = Z1^2
F = D^2
G = B^2
H = C^2
I = cphi1 * B
J = C + cphi2 * D
K = cphi8 * G + H + cphi9 * F
X2 = conj(A * K * (I + J) * (I - J))
L = C + cphi4 * D
M = cphi3 * B
N = (L + M) * (L - M)
Y2 = conj(cphi5 * D * N * (H + cphi6 * G + cphi7 * F))
Z2 = conj(B * K * N)
return(X2, Y2, Z2)
chi(X1, Y1, Z1):
A = conj(X1)
B = conj(Y1)
C = conj(Z1)^2
D = A^2
F = B^2
G = B * (D + cpsi2 * C)
H = -(D + cpsi4 * C)
X2 = cpsi1 * A * C * H
Y2 = G * (D + cpsi3 * C)
Z2 = G * H
return(X2, Y2, Z2)
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4.3.2. Scalar Decomposition and Recoding
This stage has two parts. The first one consists in decomposing the
scalar into four 64-bit integers, and the second one consists in
recoding these integers into a form that can be used to efficiently
and securely compute the scalar multiplication.
The decomposition step uses four fixed vectors called b1, b2, b3, b4,
with four 64 bit entries each. In addition, we have integer
constants L1, L2, L3, L4, which are used to implement rounding. All
these values are listed in Appendix A. In addition, we use two
constant vectors derived from these inputs:
o c = 5 * b2 - 3 * b3 + 2 * b4
o c' = 5 * b2 - 3 * b3 + 3 * b4 = c + b4
Given m, first compute t[i] = floor(L[i] * m / 2^256) for i between 1
and 4. Then compute the vector sum a = (m, 0, 0, 0) - t1 b1 - t2 b2
- t3 b3 - t4 b4. Precisely one of a + c and a + c' has an odd first
coordinate: this is the vector v that is fed into the scalar recoding
step. Note that the entries of this vector are 64 bits, so
intermediate values in the calculation above can be truncated to this
width.
The recoding step takes the vector v=(v1, v2, v3, v4) from the
previous step and outputs two arrays m[0]..m[64] and d[0]..d[64].
Each entry of d is between 0 and 7, and each entry in m is -1 or 0.
The recoding algorithm is detailed below. bit(x, n) denotes the nth
bit of x, counting from least significant to most, starting with 0.
m[64] = -1
for i = 0 to 63 do:
b1 = bit(v1, i+1)
d[i] = 0
m[i] = b1
for j = 2 to 4 do:
bj = bit(vj, 0)
d[i] = d[i] + bj * 2^(j-2)
c = (b1 or bj) xor b1
vj = vj / 2 + c
d[64] = v2 + 2 * v3 + 4 * v4
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4.3.3. Final Computation
We now describe the last step in the endomorphism based algorithm for
computing scalar multiplication. On inputs m and P, the algorithm
first precomputes a table of images of P under the endomorphisms,
then recodes m, then uses these intermediate artifacts to compute the
scalar product.
First, compute a table T of 8 points in representation R2 as shown
below. Computations Q = psi(P), R = phi(P) and S = psi(phi(P)) are
carried out using formulas from Section 4.3.1.
Q is phi(P) in R3
R is psi(P) in R3
S is psi(Q) in R2
T[0] is P in R2
T[1] is ADD_core(Q, T[0]) # (P + Q)
Convert T[1] to R2
T[2] is ADD_Core(R, T[0]) # (P + R)
Convert T[2] to R2
T[3] is ADD_Core(R, T[1]) # (P + Q + R)
Convert T[3] to R2
T[4] is ADD_Core(S, T[0]) # (P + S)
Convert T[4] to R2
T[5] is ADD_Core(S, T[1]) # (P + Q + S)
Convert T[5] to R2
T[6] is ADD_Core(S, T[2]) # (P + R + S)
Convert T[6] to R2
T[7] is ADD_Core(S, T[3]) # (P + Q + R + S)
Convert T[7] to R2
Second, apply the scalar decomposition and recoding algorithm from
Section 4.3.2 to m, to produce the two arrays m[0]..m[64] and
d[0]..d[64].
Define s[i] to be 1 if m[i] is 1 and -1 if m[i] is 0. Then the
multiplication is completed as follows:
Q = s[64] * T[d[64]]
Convert Q to R4
for i=63 to 0 do:
Q = DBL(Q)
Q = ADD(Q, s[i] * T[di])
return Q = (X/Z, Y/Z)
Multiplication by s[i] is simply a conditional negation. To negate
an R2 point (N, D, E, F) one computes (D, N, E , -F). It is
important to do this (as well as the table lookup) in constant time,
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i.e., the execution of branches and memory accesses MUST NOT depend
on secret values (see ``Security Considerations'' for more details).
The optimized multiplication algorithm above only works properly for
N-torsion points. Implementations MUST NOT use this algorithm on
anything that is not known to be an N-torsion point. Otherwise, it
will produce the wrong answer, with extremely negative consequences
for security.
5. Diffie-Hellman Key Agreement
The above scalar multiplication algorithms can be used to implement
Diffie-Hellman with cofactor.
DH(m, P):
Ensure P on curve and if not return FAILURE
P1 = DBL(P) # [2]P
P2 = ADD(P1, P) # [3]P
P3 = DBL(DBL(DBL(DBL(P2)))) # [48]P
Q = ADD(P3, P) # [49]P
Q = DBL(DBL(DBL(Q)) # [392]P
Compute [m]*Q
If Q is the neutral point, return FAILURE
Return [m]*Q in affine coordinates
The role of the separate multiplication by 392 is to ensure that Q is
an N-torsion point so that the scalar multiplication algorithms above
may be used safely to produce correct results. In other words, as
the cofactor is greater than one, Diffie-Hellman computations using
Curve4Q MUST always use cofactor clearing (as defined above).
The base point G for Diffie-Hellman operations has the following
affine coordinates:
Gx = 0x1A3472237C2FB305286592AD7B3833AA +
0x1E1F553F2878AA9C96869FB360AC77F6\*i
Gy = 0x0E3FEE9BA120785AB924A2462BCBB287 +
0x6E1C4AF8630E024249A7C344844C8B5C\*i
G = (X, Y)
The tables used in multiplications of this generator (small multiples
of G for the multiplication without endomorphisms, or endomorphism
images for the optimized multiplication with endomorphisms) can be
pre-generated to speed up the first, fixed-point DH computation.
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Two users, Alice and Bob, can carry out the following steps to derive
a shared key: each picks a random string of 32 bytes, mA and mB,
respectively. Alice computes the public key A = Compress(DH(mA, G)),
and Bob computes the public key B = Compress(DH(mB, G)). They
exchange A and B, and then Alice computes KAB = DH(mA, Expand(B))
while Bob computes KBA = DH(mB, Expand(A)), which produces the shared
point K = KAB = KBA. The y coordinate of K, represented as a 32 byte
string as detailed in Section 3 is the shared secret.
If the received strings are not valid points, the DH function has
failed to compute an answer. Implementations SHOULD return a random
32 byte string as well as return an error, to prevent bugs when
applications ignore return codes. They MUST signal an error when
decompression fails.
Implementations MAY use any method to carry out these calculations,
provided that it agrees with the above function on all inputs and
failure cases, and does not leak information about secret keys. For
example, refer to the constant-time fixed-base scalar multiplication
algorithm implemented in [FourQlib] to accelerate the computation of
DH(m, G).
6. IANA Considerations
[RFC Editor: please remove this section prior to publication] This
document has no IANA actions.
7. Security Considerations
The best known algorithms for the computation of discrete logarithms
on Curve4Q are parallel versions of the Pollard rho algorithm in
[Distinguished]. On Curve4Q these attacks take on the order of 2^123
group operations to compute a single discrete logarithm. The
additional endomorphisms have large order, and so cannot be used to
accelerate generic attacks. Quadratic fields are not affected by any
of the index calculus attacks used over larger extension fields.
Implementations MUST check that input points properly decompress to
points on the curve. Removing such checks may result in extremely
effective attacks. The curve is not twist-secure: implementations
using single coordinate ladders MUST validate points before operating
on them. In the case of protocols that require contributory
behavior, when the identity is the output of the DH primitive it MUST
be rejected and failure signaled to higher levels. Notoriously
[RFC5246] without [RFC7627] is such a protocol.
Implementations MUST ensure that execution of branches and memory
addresses accessed do not depend on secret data. The time
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variability introduced by secret-dependent operations have been
exploited in the past via timing and cache attacks to break
implementations. Side-channel analysis is a constantly moving field,
and implementers must be extremely careful to ensure that operations
do not leak any secret information. Using ephemeral private scalars
for each operation (ideally, limiting the use of each private scalar
to one single operation) can reduce the impact of side-channel
attacks. However, this might not be possible for many applications
of Diffie-Hellman key agreement.
In the future quantum computers may render the discrete logarithm
problem easy on all abelian groups through Shor's algorithm. Data
intended to remain confidential for significantly extended periods of
time SHOULD NOT be protected with any primitive based on the hardness
of factoring or the discrete log problem (elliptic curve or finite
field).
8. Informative References
[Curve4Q] Costello, C. and P. Longa, "FourQ: four-dimensional
decompositions on a Q-curve over the Mersenne prime",
2016, <https://eprint.iacr.org/2015/565.pdf>.
[Distinguished]
van Oorschot, P. and M. Wiener, "Parallel Collision Search
with Cryptanalytic Applications", 1996,
<http://people.scs.carleton.ca/~paulv/papers/JoC97.pdf>.
[Exceptional]
Izu, T. and T. Takagi, "Exceptional procedure attack on
elliptic curve cryptosystems", 2003,
<https://www.iacr.org/archive/
pkc2003/25670224/25670224.pdf>.
[FourQlib]
Costello, C. and P. Longa, "FourQlib", 2016,
<https://www.microsoft.com/en-us/research/project/
fourqlib/>.
[GLS] Galbraith, S., Lin, X., and M. Scott, "Endomorphisms for
Faster Elliptic Curve Cryptography on a Large Class of
Curves", 2009, <https://www.iacr.org/archive/
eurocrypt2009/54790519/54790519.pdf>.
[GLV] Gallant, R., Lambert, R., and S. Vanstone, "Faster Point
Multiplication on Elliptic Curves with Efficient
Endomorphisms", 2001, <https://www.iacr.org/archive/
crypto2001/21390189.pdf>.
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[Invsqr] Hamburg, M., "Fast and compact elliptic-curve
cryptography", 2012,
<http://eprint.iacr.org/2012/309.pdf>.
[RFC5246] Dierks, T. and E. Rescorla, "The Transport Layer Security
(TLS) Protocol Version 1.2", RFC 5246,
DOI 10.17487/RFC5246, August 2008,
<http://www.rfc-editor.org/info/rfc5246>.
[RFC7627] Bhargavan, K., Ed., Delignat-Lavaud, A., Pironti, A.,
Langley, A., and M. Ray, "Transport Layer Security (TLS)
Session Hash and Extended Master Secret Extension",
RFC 7627, DOI 10.17487/RFC7627, September 2015,
<http://www.rfc-editor.org/info/rfc7627>.
[SchnorrQ]
Costello, C. and P. Longa, "SchnorrQ: Schnorr Signatures
on FourQ", 2016, <https://www.microsoft.com/en-
us/research/wp-content/uploads/2016/07/SchnorrQ.pdf>.
[Twisted] Bernstein, D., Birkner, P., Joye, M., Lange, T., and C.
Peters, "Twisted Edwards Curves", 2008,
<http://eprint.iacr.org/2008/013.pdf>.
[TwistedRevisited]
Hisil, H., Wong, K-H., Carter, G., and E. Dawson, "Twisted
Edwards Curves Revisited", 2008, <http://iacr.org/archive/
asiacrypt2008/53500329/53500329.pdf>.
Appendix A. Constants
ctau = 0x1964de2c3afad20c74dcd57cebce74c3 +
0x000000000000000c0000000000000012 * i
ctaudual = 0x4aa740eb230586529ecaa6d9decdf034 +
0x7ffffffffffffff40000000000000011 * i
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cphi0 = 0x0000000000000005fffffffffffffff7 +
0x2553a0759182c3294f65536cef66f81a * i
cphi1 = 0x00000000000000050000000000000007 +
0x62c8caa0c50c62cf334d90e9e28296f9 * i
cphi2 = 0x000000000000000f0000000000000015 +
0x78df262b6c9b5c982c2cb7154f1df391 * i
cphi3 = 0x00000000000000020000000000000003 +
0x5084c6491d76342a92440457a7962ea4 * i
cphi4 = 0x00000000000000030000000000000003 +
0x12440457a7962ea4a1098c923aec6855 * i
cphi5 = 0x000000000000000a000000000000000f +
0x459195418a18c59e669b21d3c5052df3 * i
cphi6 = 0x00000000000000120000000000000018 +
0x0b232a8314318b3ccd3643a78a0a5be7 * i
cphi7 = 0x00000000000000180000000000000023 +
0x3963bc1c99e2ea1a66c183035f48781a * i
cphi8 = 0x00000000000000aa00000000000000f0 +
0x1f529f860316cbe544e251582b5d0ef0 * i
cphi9 = 0x00000000000008700000000000000bef +
0x0fd52e9cfe00375b014d3e48976e2505 * i
cpsi1 = 0x2af99e9a83d54a02edf07f4767e346ef +
0x00000000000000de000000000000013a * i
cpsi2 = 0x00000000000000e40000000000000143 +
0x21b8d07b99a81f034c7deb770e03f372 * i
cpsi3 = 0x00000000000000060000000000000009 +
0x4cb26f161d7d69063a6e6abe75e73a61 * i
cpsi4 = 0x7ffffffffffffff9fffffffffffffff6 +
0x334d90e9e28296f9c59195418a18c59e * i
L1 = 0x7fc5bb5c5ea2be5dff75682ace6a6bd66259686e09d1a7d4f
L2 = 0x38fd4b04caa6c0f8a2bd235580f468d8dd1ba1d84dd627afb
L3 = 0x0d038bf8d0bffbaf6c42bd6c965dca9029b291a33678c203c
L4 = 0x31b073877a22d841081cbdc3714983d8212e5666b77e7fdc0
b1 = [ 0x0906ff27e0a0a196, -0x1363e862c22a2da0,
0x07426031ecc8030f, -0x084f739986b9e651]
b2 = [ 0x1d495bea84fcc2d4, -0x0000000000000001,
0x0000000000000001, 0x25dbc5bc8dd167d0]
b3 = [ 0x17abad1d231f0302, 0x02c4211ae388da51,
-0x2e4d21c98927c49f, 0x0a9e6f44c02ecd97]
b4 = [ 0x136e340a9108c83f, 0x3122df2dc3e0ff32,
-0x068a49f02aa8a9b5, -0x18d5087896de0aea]
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Appendix B. Point Decompression
The following algorithm is an adaptation of the decompression
algorithm from [SchnorrQ]. It decodes a 32-byte string B which is
formatted as detailed in Section 3. The result is a valid point P =
(x, y) that satisfies the curve equation, or a message of FAILED if
the decoding had a failure.
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Sign(x0 + x1*i):
s0 = X[0] >> 126
s1 = X[1] >> 126
if X[0] != 0:
return s0
else:
return s1
Compress(X, Y):
B = Y encoded following {{representation-of-curve-points}}
Set the to bit to Sign(X)
return B
Expand(B = [y, s]):
Parse out the encoded values y = y0 + y1 * i and s
if y0 or y1 >= p:
return FAILED
u = y^2 - 1 # Set u = u0 + u1 * i
v = d*y^2 + 1 # Set v = v0 + v1 * i
t0 = u0*v0 + u1*v1;
t1 = u1*v0 - u0*v1;
t2 = v0^2 + v1^2
t3 = (t0^2 + t1^2)^(2^125)
t = 2*(t0 + t3)
if t = 0:
t = 2*(t0 - t3)
a = (t * t2^3)^(2^125-1)
b = (a * t2) * t
x0 = b/2
x1 = (a * t2) * t1
if t2 * b^2 = t:
Swap x0 and x1
x = x0 + x1 * i
if Sign(x) != s:
x = -x
if -x^2+y^2 != 1+d*x^2*y^2: # Check curve equation with x
x = conj(x)
if -x^2+y^2 != 1+d*x^2*y^2: # ... or its conjugate
return FAILED
return P = (x,y)
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Authors' Addresses
Watson Ladd
UC Berkeley
Email: watsonbladd@gmail.com
Patrick Longa
Microsoft Research
Email: plonga@microsoft.com
Richard Barnes
Mozilla
Email: rlb@ipv.sx
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