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Detailed comments for secp256k1_scalar_split_lambda
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@roconnor-blockstream @sipa
roconnor-blockstream authored and sipa committed Oct 13, 2020
1 parent 76ed922 commit acab934
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Showing 4 changed files with 154 additions and 35 deletions.
9 changes: 5 additions & 4 deletions src/scalar.h
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Original file line number Diff line number Diff line change
Expand Up @@ -103,10 +103,11 @@ static void secp256k1_scalar_order_get_num(secp256k1_num *r);
static int secp256k1_scalar_eq(const secp256k1_scalar *a, const secp256k1_scalar *b);

#ifdef USE_ENDOMORPHISM
/** Find r1 and r2 such that r1+r2*2^128 = a. */
static void secp256k1_scalar_split_128(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *a);
/** Find r1 and r2 such that r1+r2*lambda = a, and r1 and r2 are maximum 128 bits long (see secp256k1_gej_mul_lambda). */
static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *a);
/** Find r1 and r2 such that r1+r2*2^128 = k. */
static void secp256k1_scalar_split_128(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *k);
/** Find r1 and r2 such that r1+r2*lambda = k,
* where r1 and r2 or their negations are maximum 128 bits long (see secp256k1_ge_mul_lambda). */
static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *k);
#endif

/** Multiply a and b (without taking the modulus!), divide by 2**shift, and round to the nearest integer. Shift must be at least 256. */
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10 changes: 5 additions & 5 deletions src/scalar_4x64_impl.h
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Expand Up @@ -913,13 +913,13 @@ static void secp256k1_scalar_sqr(secp256k1_scalar *r, const secp256k1_scalar *a)
}

#ifdef USE_ENDOMORPHISM
static void secp256k1_scalar_split_128(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *a) {
r1->d[0] = a->d[0];
r1->d[1] = a->d[1];
static void secp256k1_scalar_split_128(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *k) {
r1->d[0] = k->d[0];
r1->d[1] = k->d[1];
r1->d[2] = 0;
r1->d[3] = 0;
r2->d[0] = a->d[2];
r2->d[1] = a->d[3];
r2->d[0] = k->d[2];
r2->d[1] = k->d[3];
r2->d[2] = 0;
r2->d[3] = 0;
}
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18 changes: 9 additions & 9 deletions src/scalar_8x32_impl.h
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Expand Up @@ -673,19 +673,19 @@ static void secp256k1_scalar_sqr(secp256k1_scalar *r, const secp256k1_scalar *a)
}

#ifdef USE_ENDOMORPHISM
static void secp256k1_scalar_split_128(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *a) {
r1->d[0] = a->d[0];
r1->d[1] = a->d[1];
r1->d[2] = a->d[2];
r1->d[3] = a->d[3];
static void secp256k1_scalar_split_128(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *k) {
r1->d[0] = k->d[0];
r1->d[1] = k->d[1];
r1->d[2] = k->d[2];
r1->d[3] = k->d[3];
r1->d[4] = 0;
r1->d[5] = 0;
r1->d[6] = 0;
r1->d[7] = 0;
r2->d[0] = a->d[4];
r2->d[1] = a->d[5];
r2->d[2] = a->d[6];
r2->d[3] = a->d[7];
r2->d[0] = k->d[4];
r2->d[1] = k->d[5];
r2->d[2] = k->d[6];
r2->d[3] = k->d[7];
r2->d[4] = 0;
r2->d[5] = 0;
r2->d[6] = 0;
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152 changes: 135 additions & 17 deletions src/scalar_impl.h
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Original file line number Diff line number Diff line change
Expand Up @@ -279,19 +279,31 @@ static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar
* lambda is {0x53,0x63,0xad,0x4c,0xc0,0x5c,0x30,0xe0,0xa5,0x26,0x1c,0x02,0x88,0x12,0x64,0x5a,
* 0x12,0x2e,0x22,0xea,0x20,0x81,0x66,0x78,0xdf,0x02,0x96,0x7c,0x1b,0x23,0xbd,0x72}
*
* "Guide to Elliptic Curve Cryptography" (Hankerson, Menezes, Vanstone) gives an algorithm
* (algorithm 3.74) to find k1 and k2 given k, such that k1 + k2 * lambda == k mod n, and k1
* and k2 have a small size.
* It relies on constants a1, b1, a2, b2. These constants for the value of lambda above are:
* Both lambda and beta are primitive cube roots of unity. That is lamba^3 == 1 mod n and
* beta^3 == 1 mod p, where n is the curve order and p is the field order.
*
* Futhermore, because (X^3 - 1) = (X - 1)(X^2 + X + 1), the primitive cube roots of unity are
* roots of X^2 + X + 1. Therefore lambda^2 + lamba == -1 mod n and beta^2 + beta == -1 mod p.
* (The other primitive cube roots of unity are lambda^2 and beta^2 respectively.)
*
* Let l = -1/2 + i*sqrt(3)/2, the complex root of X^2 + X + 1. We can define a ring
* homomorphism phi : Z[l] -> Z_n where phi(a + b*l) == a + b*lambda mod n. The kernel of phi
* is a lattice over Z[l] (considering Z[l] as a Z-module). This lattice is generated by a
* reduced basis {a1 + b1*l, a2 + b2*l} where
*
* - a1 = {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15}
* - b1 = -{0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28,0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3}
* - a2 = {0x01,0x14,0xca,0x50,0xf7,0xa8,0xe2,0xf3,0xf6,0x57,0xc1,0x10,0x8d,0x9d,0x44,0xcf,0xd8}
* - b2 = {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15}
*
* The algorithm then computes c1 = round(b1 * k / n) and c2 = round(b2 * k / n), and gives
* "Guide to Elliptic Curve Cryptography" (Hankerson, Menezes, Vanstone) gives an algorithm
* (algorithm 3.74) to find k1 and k2 given k, such that k1 + k2 * lambda == k mod n, and k1
* and k2 have a small size.
*
* The algorithm computes c1 = round(b2 * k / n) and c2 = round((-b1) * k / n), and gives
* k1 = k - (c1*a1 + c2*a2) and k2 = -(c1*b1 + c2*b2). Instead, we use modular arithmetic, and
* compute k1 as k - k2 * lambda, avoiding the need for constants a1 and a2.
* compute k - k2 * lambda (mod n) which is equivalent to k1 (mod n), avoiding the need for
* the constants a1 and a2.
*
* g1, g2 are precomputed constants used to replace division with a rounded multiplication
* when decomposing the scalar for an endomorphism-based point multiplication.
Expand All @@ -303,16 +315,122 @@ static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar
* Cryptography on Sensor Networks Using the MSP430X Microcontroller" (Gouvea, Oliveira, Lopez),
* Section 4.3 (here we use a somewhat higher-precision estimate):
* d = a1*b2 - b1*a2
* g1 = round((2^384)*b2/d)
* g2 = round((2^384)*(-b1)/d)
* g1 = round(2^384 * b2/d)
* g2 = round(2^384 * (-b1)/d)
*
* (Note that d is also equal to the curve order, n, here because [a1,b1] and [a2,b2]
* can be found as outputs of the Extended Euclidean Algorithm on inputs n and lambda).
*
* The function below splits k into r1 and r2, such that
* - r1 + lambda * r2 == k (mod n)
* - either r1 < 2^128 or -r1 mod n < 2^128
* - either r2 < 2^128 or -r2 mod n < 2^128
*
* Proof.
*
* Let
* - epsilon1 = 2^256 * |g1/2^384 - b2/d|
* - epsilon2 = 2^256 * |g2/2^384 - (-b1)/d|
* - c1 = round(k*g1/2^384)
* - c2 = round(k*g2/2^384)
*
* Lemma 1: |c1 - k*b2/d| < 2^-1 + epsilon1
*
* (Note that 'd' is also equal to the curve order here because [a1,b1] and [a2,b2] are found
* as outputs of the Extended Euclidean Algorithm on inputs 'order' and 'lambda').
* |c1 - k*b2/d|
* =
* |c1 - k*g1/2^384 + k*g1/2^384 - k*b2/d|
* <= {triangle inequality}
* |c1 - k*g1/2^384| + |k*g1/2^384 - k*b2/d|
* =
* |c1 - k*g1/2^384| + k*|g1/2^384 - b2/d|
* < {rounding in c1 and 0 <= k < 2^256}
* 2^-1 + 2^256 * |g1/2^384 - b2/d|
* = {definition of epsilon1}
* 2^-1 + epsilon1
*
* The function below splits a in r1 and r2, such that r1 + lambda * r2 == a (mod order).
* Lemma 2: |c2 - k*(-b1)/d| < 2^-1 + epsilon2
*
* |c2 - k*(-b1)/d|
* =
* |c2 - k*g2/2^384 + k*g2/2^384 - k*(-b1)/d|
* <= {triangle inequality}
* |c2 - k*g2/2^384| + |k*g2/2^384 - k*(-b1)/d|
* =
* |c2 - k*g2/2^384| + k*|g2/2^384 - (-b1)/d|
* < {rounding in c2 and 0 <= k < 2^256}
* 2^-1 + 2^256 * |g2/2^384 - (-b1)/d|
* = {definition of epsilon2}
* 2^-1 + epsilon2
*
* Let
* - k1 = k - c1*a1 - c2*a2
* - k2 = - c1*b1 - c2*b2
*
* Lemma 3: |k1| < (a1 + a2 + 1)/2 < 2^128
*
* |k1|
* = {definition of k1}
* |k - c1*a1 - c2*a2|
* = {(a1*b2 - b1*a2)/n = 1}
* |k*(a1*b2 - b1*a2)/n - c1*a1 - c2*a2|
* =
* |a1*(k*b2/n - c1) + a2*(k*(-b1)/n - c2)|
* <= {triangle inequality}
* a1*|k*b2/n - c1| + a2*|k*(-b1)/n - c2|
* < {Lemma 1 and Lemma 2}
* a1*(2^-1 + epslion1) + a2*(2^-1 + epsilon2)
* < {rounding up to an integer}
* (a1 + a2 + 1)/2
* < {rounding up to a power of 2}
* 2^128
*
* Lemma 4: |k2| < (-b1 + b2)/2 + 1 < 2^128
*
* |k2|
* = {definition of k2}
* |- c1*a1 - c2*a2|
* = {(b1*b2 - b1*b2)/n = 0}
* |k*(b1*b2 - b1*b2)/n - c1*b1 - c2*b2|
* =
* |b1*(k*b2/n - c1) + b2*(k*(-b1)/n - c2)|
* <= {triangle inequality}
* (-b1)*|k*b2/n - c1| + b2*|k*(-b1)/n - c2|
* < {Lemma 1 and Lemma 2}
* (-b1)*(2^-1 + epslion1) + b2*(2^-1 + epsilon2)
* < {rounding up to an integer}
* (-b1 + b2)/2 + 1
* < {rounding up to a power of 2}
* 2^128
*
* Let
* - r2 = k2 mod n
* - r1 = k - r2*lambda mod n.
*
* Notice that r1 is defined such that r1 + r2 * lambda == k (mod n).
*
* Lemma 5: r1 == k1 mod n.
*
* r1
* == {definition of r1 and r2}
* k - k2*lambda
* == {definition of k2}
* k - (- c1*b1 - c2*b2)*lambda
* ==
* k + c1*b1*lambda + c2*b2*lambda
* == {a1 + b1*lambda == 0 mod n and a2 + b2*lambda == 0 mod n}
* k - c1*a1 - c2*a2
* == {definition of k1}
* k1
*
* From Lemma 3, Lemma 4, Lemma 5 and the definition of r2, we can conclude that
*
* - either r1 < 2^128 or -r1 mod n < 2^128
* - either r2 < 2^128 or -r2 mod n < 2^128.
*
* Q.E.D.
*/

static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *a) {
static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *k) {
secp256k1_scalar c1, c2;
static const secp256k1_scalar minus_lambda = SECP256K1_SCALAR_CONST(
0xAC9C52B3UL, 0x3FA3CF1FUL, 0x5AD9E3FDUL, 0x77ED9BA4UL,
Expand All @@ -334,16 +452,16 @@ static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar
0xE4437ED6UL, 0x010E8828UL, 0x6F547FA9UL, 0x0ABFE4C4UL,
0x221208ACUL, 0x9DF506C6UL, 0x1571B4AEUL, 0x8AC47F71UL
);
VERIFY_CHECK(r1 != a);
VERIFY_CHECK(r2 != a);
VERIFY_CHECK(r1 != k);
VERIFY_CHECK(r2 != k);
/* these _var calls are constant time since the shift amount is constant */
secp256k1_scalar_mul_shift_var(&c1, a, &g1, 384);
secp256k1_scalar_mul_shift_var(&c2, a, &g2, 384);
secp256k1_scalar_mul_shift_var(&c1, k, &g1, 384);
secp256k1_scalar_mul_shift_var(&c2, k, &g2, 384);
secp256k1_scalar_mul(&c1, &c1, &minus_b1);
secp256k1_scalar_mul(&c2, &c2, &minus_b2);
secp256k1_scalar_add(r2, &c1, &c2);
secp256k1_scalar_mul(r1, r2, &minus_lambda);
secp256k1_scalar_add(r1, r1, a);
secp256k1_scalar_add(r1, r1, k);
}
#endif
#endif
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