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Merge bitcoin-core/secp256k1#899: Reduce stratch space needed by ecmu…
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…lt_strauss_wnaf.

b797a50 Create a SECP256K1_ECMULT_TABLE_VERIFY macro. (Russell O'Connor)
a731200 Replace ECMULT_TABLE_GET_GE_STORAGE macro with a function. (Russell O'Connor)
fe34d9f Eliminate input_pos state field from ecmult_strauss_wnaf. (Russell O'Connor)
0397d00 Eliminate na_1 and na_lam state fields from ecmult_strauss_wnaf. (Russell O'Connor)
7ba3ffc Remove the unused pre_a_lam allocations. (Russell O'Connor)
b3b57ad Eliminate the pre_a_lam array from ecmult_strauss_wnaf. (Russell O'Connor)
ae7ba0f Remove the unused prej allocations. (Russell O'Connor)
e5c1889 Eliminate the prej array from ecmult_strauss_wnaf. (Russell O'Connor)
c9da1ba Move secp256k1_fe_one to field.h (Russell O'Connor)

Pull request description:

ACKs for top commit:
  sipa:
    ACK b797a50
  jonasnick:
    ACK b797a50

Tree-SHA512: 6742469979c306104a0861be76c2be86bf8ab14116b00afbd24f91b9e3ea843bf9b9a74552b367bd06ee617090019ad4df6be037d58937c8c869f8b37ddaa6cc
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jonasnick committed Jan 26, 2022
2 parents 0a40a48 + b797a50 commit d8a2463
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7 changes: 2 additions & 5 deletions src/ecmult_const_impl.h
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Original file line number Diff line number Diff line change
Expand Up @@ -19,13 +19,10 @@
* It only operates on tables sized for WINDOW_A wnaf multiples.
*/
static void secp256k1_ecmult_odd_multiples_table_globalz_windowa(secp256k1_ge *pre, secp256k1_fe *globalz, const secp256k1_gej *a) {
secp256k1_gej prej[ECMULT_TABLE_SIZE(WINDOW_A)];
secp256k1_fe zr[ECMULT_TABLE_SIZE(WINDOW_A)];

/* Compute the odd multiples in Jacobian form. */
secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), prej, zr, a);
/* Bring them to the same Z denominator. */
secp256k1_ge_globalz_set_table_gej(ECMULT_TABLE_SIZE(WINDOW_A), pre, globalz, prej, zr);
secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), pre, zr, globalz, a);
secp256k1_ge_table_set_globalz(ECMULT_TABLE_SIZE(WINDOW_A), pre, zr);
}

/* This is like `ECMULT_TABLE_GET_GE` but is constant time */
Expand Down
210 changes: 113 additions & 97 deletions src/ecmult_impl.h
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Original file line number Diff line number Diff line change
Expand Up @@ -47,7 +47,7 @@

/* The number of objects allocated on the scratch space for ecmult_multi algorithms */
#define PIPPENGER_SCRATCH_OBJECTS 6
#define STRAUSS_SCRATCH_OBJECTS 7
#define STRAUSS_SCRATCH_OBJECTS 5

#define PIPPENGER_MAX_BUCKET_WINDOW 12

Expand All @@ -56,71 +56,98 @@

#define ECMULT_MAX_POINTS_PER_BATCH 5000000

/** Fill a table 'prej' with precomputed odd multiples of a. Prej will contain
* the values [1*a,3*a,...,(2*n-1)*a], so it space for n values. zr[0] will
* contain prej[0].z / a.z. The other zr[i] values = prej[i].z / prej[i-1].z.
* Prej's Z values are undefined, except for the last value.
/** Fill a table 'pre_a' with precomputed odd multiples of a.
* pre_a will contain [1*a,3*a,...,(2*n-1)*a], so it needs space for n group elements.
* zr needs space for n field elements.
*
* Although pre_a is an array of _ge rather than _gej, it actually represents elements
* in Jacobian coordinates with their z coordinates omitted. The omitted z-coordinates
* can be recovered using z and zr. Using the notation z(b) to represent the omitted
* z coordinate of b:
* - z(pre_a[n-1]) = 'z'
* - z(pre_a[i-1]) = z(pre_a[i]) / zr[i] for n > i > 0
*
* Lastly the zr[0] value, which isn't used above, is set so that:
* - a.z = z(pre_a[0]) / zr[0]
*/
static void secp256k1_ecmult_odd_multiples_table(int n, secp256k1_gej *prej, secp256k1_fe *zr, const secp256k1_gej *a) {
secp256k1_gej d;
secp256k1_ge a_ge, d_ge;
static void secp256k1_ecmult_odd_multiples_table(int n, secp256k1_ge *pre_a, secp256k1_fe *zr, secp256k1_fe *z, const secp256k1_gej *a) {
secp256k1_gej d, ai;
secp256k1_ge d_ge;
int i;

VERIFY_CHECK(!a->infinity);

secp256k1_gej_double_var(&d, a, NULL);

/*
* Perform the additions on an isomorphism where 'd' is affine: drop the z coordinate
* of 'd', and scale the 1P starting value's x/y coordinates without changing its z.
* Perform the additions using an isomorphic curve Y^2 = X^3 + 7*C^6 where C := d.z.
* The isomorphism, phi, maps a secp256k1 point (x, y) to the point (x*C^2, y*C^3) on the other curve.
* In Jacobian coordinates phi maps (x, y, z) to (x*C^2, y*C^3, z) or, equivalently to (x, y, z/C).
*
* phi(x, y, z) = (x*C^2, y*C^3, z) = (x, y, z/C)
* d_ge := phi(d) = (d.x, d.y, 1)
* ai := phi(a) = (a.x*C^2, a.y*C^3, a.z)
*
* The group addition functions work correctly on these isomorphic curves.
* In particular phi(d) is easy to represent in affine coordinates under this isomorphism.
* This lets us use the faster secp256k1_gej_add_ge_var group addition function that we wouldn't be able to use otherwise.
*/
d_ge.x = d.x;
d_ge.y = d.y;
d_ge.infinity = 0;

secp256k1_ge_set_gej_zinv(&a_ge, a, &d.z);
prej[0].x = a_ge.x;
prej[0].y = a_ge.y;
prej[0].z = a->z;
prej[0].infinity = 0;
secp256k1_ge_set_xy(&d_ge, &d.x, &d.y);
secp256k1_ge_set_gej_zinv(&pre_a[0], a, &d.z);
secp256k1_gej_set_ge(&ai, &pre_a[0]);
ai.z = a->z;

/* pre_a[0] is the point (a.x*C^2, a.y*C^3, a.z*C) which is equvalent to a.
* Set zr[0] to C, which is the ratio between the omitted z(pre_a[0]) value and a.z.
*/
zr[0] = d.z;

for (i = 1; i < n; i++) {
secp256k1_gej_add_ge_var(&prej[i], &prej[i-1], &d_ge, &zr[i]);
secp256k1_gej_add_ge_var(&ai, &ai, &d_ge, &zr[i]);
secp256k1_ge_set_xy(&pre_a[i], &ai.x, &ai.y);
}

/*
* Each point in 'prej' has a z coordinate too small by a factor of 'd.z'. Only
* the final point's z coordinate is actually used though, so just update that.
/* Multiply the last z-coordinate by C to undo the isomorphism.
* Since the z-coordinates of the pre_a values are implied by the zr array of z-coordinate ratios,
* undoing the isomorphism here undoes the isomorphism for all pre_a values.
*/
secp256k1_fe_mul(&prej[n-1].z, &prej[n-1].z, &d.z);
secp256k1_fe_mul(z, &ai.z, &d.z);
}

/** The following two macro retrieves a particular odd multiple from a table
* of precomputed multiples. */
#define ECMULT_TABLE_GET_GE(r,pre,n,w) do { \
#define SECP256K1_ECMULT_TABLE_VERIFY(n,w) \
VERIFY_CHECK(((n) & 1) == 1); \
VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
VERIFY_CHECK((n) <= ((1 << ((w)-1)) - 1)); \
if ((n) > 0) { \
*(r) = (pre)[((n)-1)/2]; \
} else { \
*(r) = (pre)[(-(n)-1)/2]; \
secp256k1_fe_negate(&((r)->y), &((r)->y), 1); \
} \
} while(0)

#define ECMULT_TABLE_GET_GE_STORAGE(r,pre,n,w) do { \
VERIFY_CHECK(((n) & 1) == 1); \
VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
VERIFY_CHECK((n) <= ((1 << ((w)-1)) - 1)); \
if ((n) > 0) { \
secp256k1_ge_from_storage((r), &(pre)[((n)-1)/2]); \
} else { \
secp256k1_ge_from_storage((r), &(pre)[(-(n)-1)/2]); \
secp256k1_fe_negate(&((r)->y), &((r)->y), 1); \
} \
} while(0)
VERIFY_CHECK((n) <= ((1 << ((w)-1)) - 1));

SECP256K1_INLINE static void secp256k1_ecmult_table_get_ge(secp256k1_ge *r, const secp256k1_ge *pre, int n, int w) {
SECP256K1_ECMULT_TABLE_VERIFY(n,w)
if (n > 0) {
*r = pre[(n-1)/2];
} else {
*r = pre[(-n-1)/2];
secp256k1_fe_negate(&(r->y), &(r->y), 1);
}
}

SECP256K1_INLINE static void secp256k1_ecmult_table_get_ge_lambda(secp256k1_ge *r, const secp256k1_ge *pre, const secp256k1_fe *x, int n, int w) {
SECP256K1_ECMULT_TABLE_VERIFY(n,w)
if (n > 0) {
secp256k1_ge_set_xy(r, &x[(n-1)/2], &pre[(n-1)/2].y);
} else {
secp256k1_ge_set_xy(r, &x[(-n-1)/2], &pre[(-n-1)/2].y);
secp256k1_fe_negate(&(r->y), &(r->y), 1);
}
}

SECP256K1_INLINE static void secp256k1_ecmult_table_get_ge_storage(secp256k1_ge *r, const secp256k1_ge_storage *pre, int n, int w) {
SECP256K1_ECMULT_TABLE_VERIFY(n,w)
if (n > 0) {
secp256k1_ge_from_storage(r, &pre[(n-1)/2]);
} else {
secp256k1_ge_from_storage(r, &pre[(-n-1)/2]);
secp256k1_fe_negate(&(r->y), &(r->y), 1);
}
}

/** Convert a number to WNAF notation. The number becomes represented by sum(2^i * wnaf[i], i=0..bits),
* with the following guarantees:
Expand Down Expand Up @@ -182,19 +209,16 @@ static int secp256k1_ecmult_wnaf(int *wnaf, int len, const secp256k1_scalar *a,
}

struct secp256k1_strauss_point_state {
secp256k1_scalar na_1, na_lam;
int wnaf_na_1[129];
int wnaf_na_lam[129];
int bits_na_1;
int bits_na_lam;
size_t input_pos;
};

struct secp256k1_strauss_state {
secp256k1_gej* prej;
secp256k1_fe* zr;
/* aux is used to hold z-ratios, and then used to hold pre_a[i].x * BETA values. */
secp256k1_fe* aux;
secp256k1_ge* pre_a;
secp256k1_ge* pre_a_lam;
struct secp256k1_strauss_point_state* ps;
};

Expand All @@ -212,17 +236,19 @@ static void secp256k1_ecmult_strauss_wnaf(const struct secp256k1_strauss_state *
size_t np;
size_t no = 0;

secp256k1_fe_set_int(&Z, 1);
for (np = 0; np < num; ++np) {
secp256k1_gej tmp;
secp256k1_scalar na_1, na_lam;
if (secp256k1_scalar_is_zero(&na[np]) || secp256k1_gej_is_infinity(&a[np])) {
continue;
}
state->ps[no].input_pos = np;
/* split na into na_1 and na_lam (where na = na_1 + na_lam*lambda, and na_1 and na_lam are ~128 bit) */
secp256k1_scalar_split_lambda(&state->ps[no].na_1, &state->ps[no].na_lam, &na[np]);
secp256k1_scalar_split_lambda(&na_1, &na_lam, &na[np]);

/* build wnaf representation for na_1 and na_lam. */
state->ps[no].bits_na_1 = secp256k1_ecmult_wnaf(state->ps[no].wnaf_na_1, 129, &state->ps[no].na_1, WINDOW_A);
state->ps[no].bits_na_lam = secp256k1_ecmult_wnaf(state->ps[no].wnaf_na_lam, 129, &state->ps[no].na_lam, WINDOW_A);
state->ps[no].bits_na_1 = secp256k1_ecmult_wnaf(state->ps[no].wnaf_na_1, 129, &na_1, WINDOW_A);
state->ps[no].bits_na_lam = secp256k1_ecmult_wnaf(state->ps[no].wnaf_na_lam, 129, &na_lam, WINDOW_A);
VERIFY_CHECK(state->ps[no].bits_na_1 <= 129);
VERIFY_CHECK(state->ps[no].bits_na_lam <= 129);
if (state->ps[no].bits_na_1 > bits) {
Expand All @@ -231,40 +257,36 @@ static void secp256k1_ecmult_strauss_wnaf(const struct secp256k1_strauss_state *
if (state->ps[no].bits_na_lam > bits) {
bits = state->ps[no].bits_na_lam;
}
++no;
}

/* Calculate odd multiples of a.
* All multiples are brought to the same Z 'denominator', which is stored
* in Z. Due to secp256k1' isomorphism we can do all operations pretending
* that the Z coordinate was 1, use affine addition formulae, and correct
* the Z coordinate of the result once at the end.
* The exception is the precomputed G table points, which are actually
* affine. Compared to the base used for other points, they have a Z ratio
* of 1/Z, so we can use secp256k1_gej_add_zinv_var, which uses the same
* isomorphism to efficiently add with a known Z inverse.
*/
if (no > 0) {
/* Compute the odd multiples in Jacobian form. */
secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), state->prej, state->zr, &a[state->ps[0].input_pos]);
for (np = 1; np < no; ++np) {
secp256k1_gej tmp = a[state->ps[np].input_pos];
/* Calculate odd multiples of a.
* All multiples are brought to the same Z 'denominator', which is stored
* in Z. Due to secp256k1' isomorphism we can do all operations pretending
* that the Z coordinate was 1, use affine addition formulae, and correct
* the Z coordinate of the result once at the end.
* The exception is the precomputed G table points, which are actually
* affine. Compared to the base used for other points, they have a Z ratio
* of 1/Z, so we can use secp256k1_gej_add_zinv_var, which uses the same
* isomorphism to efficiently add with a known Z inverse.
*/
tmp = a[np];
if (no) {
#ifdef VERIFY
secp256k1_fe_normalize_var(&(state->prej[(np - 1) * ECMULT_TABLE_SIZE(WINDOW_A) + ECMULT_TABLE_SIZE(WINDOW_A) - 1].z));
secp256k1_fe_normalize_var(&Z);
#endif
secp256k1_gej_rescale(&tmp, &(state->prej[(np - 1) * ECMULT_TABLE_SIZE(WINDOW_A) + ECMULT_TABLE_SIZE(WINDOW_A) - 1].z));
secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), state->prej + np * ECMULT_TABLE_SIZE(WINDOW_A), state->zr + np * ECMULT_TABLE_SIZE(WINDOW_A), &tmp);
secp256k1_fe_mul(state->zr + np * ECMULT_TABLE_SIZE(WINDOW_A), state->zr + np * ECMULT_TABLE_SIZE(WINDOW_A), &(a[state->ps[np].input_pos].z));
secp256k1_gej_rescale(&tmp, &Z);
}
/* Bring them to the same Z denominator. */
secp256k1_ge_globalz_set_table_gej(ECMULT_TABLE_SIZE(WINDOW_A) * no, state->pre_a, &Z, state->prej, state->zr);
} else {
secp256k1_fe_set_int(&Z, 1);
secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), state->pre_a + no * ECMULT_TABLE_SIZE(WINDOW_A), state->aux + no * ECMULT_TABLE_SIZE(WINDOW_A), &Z, &tmp);
if (no) secp256k1_fe_mul(state->aux + no * ECMULT_TABLE_SIZE(WINDOW_A), state->aux + no * ECMULT_TABLE_SIZE(WINDOW_A), &(a[np].z));

++no;
}

/* Bring them to the same Z denominator. */
secp256k1_ge_table_set_globalz(ECMULT_TABLE_SIZE(WINDOW_A) * no, state->pre_a, state->aux);

for (np = 0; np < no; ++np) {
for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
secp256k1_ge_mul_lambda(&state->pre_a_lam[np * ECMULT_TABLE_SIZE(WINDOW_A) + i], &state->pre_a[np * ECMULT_TABLE_SIZE(WINDOW_A) + i]);
secp256k1_fe_mul(&state->aux[np * ECMULT_TABLE_SIZE(WINDOW_A) + i], &state->pre_a[np * ECMULT_TABLE_SIZE(WINDOW_A) + i].x, &secp256k1_const_beta);
}
}

Expand All @@ -290,20 +312,20 @@ static void secp256k1_ecmult_strauss_wnaf(const struct secp256k1_strauss_state *
secp256k1_gej_double_var(r, r, NULL);
for (np = 0; np < no; ++np) {
if (i < state->ps[np].bits_na_1 && (n = state->ps[np].wnaf_na_1[i])) {
ECMULT_TABLE_GET_GE(&tmpa, state->pre_a + np * ECMULT_TABLE_SIZE(WINDOW_A), n, WINDOW_A);
secp256k1_ecmult_table_get_ge(&tmpa, state->pre_a + np * ECMULT_TABLE_SIZE(WINDOW_A), n, WINDOW_A);
secp256k1_gej_add_ge_var(r, r, &tmpa, NULL);
}
if (i < state->ps[np].bits_na_lam && (n = state->ps[np].wnaf_na_lam[i])) {
ECMULT_TABLE_GET_GE(&tmpa, state->pre_a_lam + np * ECMULT_TABLE_SIZE(WINDOW_A), n, WINDOW_A);
secp256k1_ecmult_table_get_ge_lambda(&tmpa, state->pre_a + np * ECMULT_TABLE_SIZE(WINDOW_A), state->aux + np * ECMULT_TABLE_SIZE(WINDOW_A), n, WINDOW_A);
secp256k1_gej_add_ge_var(r, r, &tmpa, NULL);
}
}
if (i < bits_ng_1 && (n = wnaf_ng_1[i])) {
ECMULT_TABLE_GET_GE_STORAGE(&tmpa, secp256k1_pre_g, n, WINDOW_G);
secp256k1_ecmult_table_get_ge_storage(&tmpa, secp256k1_pre_g, n, WINDOW_G);
secp256k1_gej_add_zinv_var(r, r, &tmpa, &Z);
}
if (i < bits_ng_128 && (n = wnaf_ng_128[i])) {
ECMULT_TABLE_GET_GE_STORAGE(&tmpa, secp256k1_pre_g_128, n, WINDOW_G);
secp256k1_ecmult_table_get_ge_storage(&tmpa, secp256k1_pre_g_128, n, WINDOW_G);
secp256k1_gej_add_zinv_var(r, r, &tmpa, &Z);
}
}
Expand All @@ -314,23 +336,19 @@ static void secp256k1_ecmult_strauss_wnaf(const struct secp256k1_strauss_state *
}

static void secp256k1_ecmult(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_scalar *na, const secp256k1_scalar *ng) {
secp256k1_gej prej[ECMULT_TABLE_SIZE(WINDOW_A)];
secp256k1_fe zr[ECMULT_TABLE_SIZE(WINDOW_A)];
secp256k1_fe aux[ECMULT_TABLE_SIZE(WINDOW_A)];
secp256k1_ge pre_a[ECMULT_TABLE_SIZE(WINDOW_A)];
struct secp256k1_strauss_point_state ps[1];
secp256k1_ge pre_a_lam[ECMULT_TABLE_SIZE(WINDOW_A)];
struct secp256k1_strauss_state state;

state.prej = prej;
state.zr = zr;
state.aux = aux;
state.pre_a = pre_a;
state.pre_a_lam = pre_a_lam;
state.ps = ps;
secp256k1_ecmult_strauss_wnaf(&state, r, 1, a, na, ng);
}

static size_t secp256k1_strauss_scratch_size(size_t n_points) {
static const size_t point_size = (2 * sizeof(secp256k1_ge) + sizeof(secp256k1_gej) + sizeof(secp256k1_fe)) * ECMULT_TABLE_SIZE(WINDOW_A) + sizeof(struct secp256k1_strauss_point_state) + sizeof(secp256k1_gej) + sizeof(secp256k1_scalar);
static const size_t point_size = (sizeof(secp256k1_ge) + sizeof(secp256k1_fe)) * ECMULT_TABLE_SIZE(WINDOW_A) + sizeof(struct secp256k1_strauss_point_state) + sizeof(secp256k1_gej) + sizeof(secp256k1_scalar);
return n_points*point_size;
}

Expand All @@ -351,13 +369,11 @@ static int secp256k1_ecmult_strauss_batch(const secp256k1_callback* error_callba
* constant and strauss_scratch_size accordingly. */
points = (secp256k1_gej*)secp256k1_scratch_alloc(error_callback, scratch, n_points * sizeof(secp256k1_gej));
scalars = (secp256k1_scalar*)secp256k1_scratch_alloc(error_callback, scratch, n_points * sizeof(secp256k1_scalar));
state.prej = (secp256k1_gej*)secp256k1_scratch_alloc(error_callback, scratch, n_points * ECMULT_TABLE_SIZE(WINDOW_A) * sizeof(secp256k1_gej));
state.zr = (secp256k1_fe*)secp256k1_scratch_alloc(error_callback, scratch, n_points * ECMULT_TABLE_SIZE(WINDOW_A) * sizeof(secp256k1_fe));
state.aux = (secp256k1_fe*)secp256k1_scratch_alloc(error_callback, scratch, n_points * ECMULT_TABLE_SIZE(WINDOW_A) * sizeof(secp256k1_fe));
state.pre_a = (secp256k1_ge*)secp256k1_scratch_alloc(error_callback, scratch, n_points * ECMULT_TABLE_SIZE(WINDOW_A) * sizeof(secp256k1_ge));
state.pre_a_lam = (secp256k1_ge*)secp256k1_scratch_alloc(error_callback, scratch, n_points * ECMULT_TABLE_SIZE(WINDOW_A) * sizeof(secp256k1_ge));
state.ps = (struct secp256k1_strauss_point_state*)secp256k1_scratch_alloc(error_callback, scratch, n_points * sizeof(struct secp256k1_strauss_point_state));

if (points == NULL || scalars == NULL || state.prej == NULL || state.zr == NULL || state.pre_a == NULL || state.pre_a_lam == NULL || state.ps == NULL) {
if (points == NULL || scalars == NULL || state.aux == NULL || state.pre_a == NULL || state.ps == NULL) {
secp256k1_scratch_apply_checkpoint(error_callback, scratch, scratch_checkpoint);
return 0;
}
Expand Down
6 changes: 6 additions & 0 deletions src/field.h
View file
Original file line number Diff line number Diff line change
Expand Up @@ -32,6 +32,12 @@
#error "Please select wide multiplication implementation"
#endif

static const secp256k1_fe secp256k1_fe_one = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1);
static const secp256k1_fe secp256k1_const_beta = SECP256K1_FE_CONST(
0x7ae96a2bul, 0x657c0710ul, 0x6e64479eul, 0xac3434e9ul,
0x9cf04975ul, 0x12f58995ul, 0xc1396c28ul, 0x719501eeul
);

/** Normalize a field element. This brings the field element to a canonical representation, reduces
* its magnitude to 1, and reduces it modulo field size `p`.
*/
Expand Down
2 changes: 0 additions & 2 deletions src/field_impl.h
View file
Original file line number Diff line number Diff line change
Expand Up @@ -135,6 +135,4 @@ static int secp256k1_fe_sqrt(secp256k1_fe *r, const secp256k1_fe *a) {
return secp256k1_fe_equal(&t1, a);
}

static const secp256k1_fe secp256k1_fe_one = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1);

#endif /* SECP256K1_FIELD_IMPL_H */
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