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hardwallet-bootloader/Src/uECC.c

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/* Copyright 2014, Kenneth MacKay. Licensed under the BSD 2-clause license. */
#include "uECC.h"
#include "uECC_types.h"
#define uECC_VLI_API static
#define CONCATX(a, ...) a ## __VA_ARGS__
#define CONCAT(a, ...) CONCATX(a, __VA_ARGS__)
#define STRX(a) #a
#define STR(a) STRX(a)
#define EVAL(...) EVAL1(EVAL1(EVAL1(EVAL1(__VA_ARGS__))))
#define EVAL1(...) EVAL2(EVAL2(EVAL2(EVAL2(__VA_ARGS__))))
#define EVAL2(...) EVAL3(EVAL3(EVAL3(EVAL3(__VA_ARGS__))))
#define EVAL3(...) EVAL4(EVAL4(EVAL4(EVAL4(__VA_ARGS__))))
#define EVAL4(...) __VA_ARGS__
#define DEC_1 0
#define DEC_2 1
#define DEC_3 2
#define DEC_4 3
#define DEC_5 4
#define DEC_6 5
#define DEC_7 6
#define DEC_8 7
#define DEC_9 8
#define DEC_10 9
#define DEC_11 10
#define DEC_12 11
#define DEC_13 12
#define DEC_14 13
#define DEC_15 14
#define DEC_16 15
#define DEC_17 16
#define DEC_18 17
#define DEC_19 18
#define DEC_20 19
#define DEC_21 20
#define DEC_22 21
#define DEC_23 22
#define DEC_24 23
#define DEC_25 24
#define DEC_26 25
#define DEC_27 26
#define DEC_28 27
#define DEC_29 28
#define DEC_30 29
#define DEC_31 30
#define DEC_32 31
#define DEC(N) CONCAT(DEC_, N)
#define SECOND_ARG(_, val, ...) val
#define SOME_CHECK_0 ~, 0
#define GET_SECOND_ARG(...) SECOND_ARG(__VA_ARGS__, SOME,)
#define SOME_OR_0(N) GET_SECOND_ARG(CONCAT(SOME_CHECK_, N))
#define EMPTY(...)
#define DEFER(...) __VA_ARGS__ EMPTY()
#define REPEAT_NAME_0() REPEAT_0
#define REPEAT_NAME_SOME() REPEAT_SOME
#define REPEAT_0(...)
#define REPEAT_SOME(N, stuff) DEFER(CONCAT(REPEAT_NAME_, SOME_OR_0(DEC(N))))()(DEC(N), stuff) stuff
#define REPEAT(N, stuff) EVAL(REPEAT_SOME(N, stuff))
#define REPEATM_NAME_0() REPEATM_0
#define REPEATM_NAME_SOME() REPEATM_SOME
#define REPEATM_0(...)
#define REPEATM_SOME(N, macro) macro(N) \
DEFER(CONCAT(REPEATM_NAME_, SOME_OR_0(DEC(N))))()(DEC(N), macro)
#define REPEATM(N, macro) EVAL(REPEATM_SOME(N, macro))
#define uECC_MAX_WORDS 8
#define BITS_TO_WORDS(num_bits) ((num_bits + ((uECC_WORD_SIZE * 8) - 1)) / (uECC_WORD_SIZE * 8))
#define BITS_TO_BYTES(num_bits) ((num_bits + 7) / 8)
struct uECC_Curve_t {
wordcount_t num_words;
wordcount_t num_bytes;
bitcount_t num_n_bits;
uECC_word_t p[uECC_MAX_WORDS];
uECC_word_t n[uECC_MAX_WORDS];
uECC_word_t G[uECC_MAX_WORDS * 2];
uECC_word_t b[uECC_MAX_WORDS];
void (*double_jacobian)(uECC_word_t * X1,
uECC_word_t * Y1,
uECC_word_t * Z1,
uECC_Curve curve);
void (*x_side)(uECC_word_t *result, const uECC_word_t *x, uECC_Curve curve);
void (*mmod_fast)(uECC_word_t *result, uECC_word_t *product);
};
static cmpresult_t uECC_vli_cmp_unsafe(const uECC_word_t *left,
const uECC_word_t *right,
wordcount_t num_words);
#if (uECC_PLATFORM == uECC_arm || uECC_PLATFORM == uECC_arm_thumb || \
uECC_PLATFORM == uECC_arm_thumb2)
#include "uECC_asm_arm.inc"
#endif
int uECC_curve_private_key_size(uECC_Curve curve) {
return BITS_TO_BYTES(curve->num_n_bits);
}
int uECC_curve_public_key_size(uECC_Curve curve) {
return 2 * curve->num_bytes;
}
#if !asm_clear
uECC_VLI_API void uECC_vli_clear(uECC_word_t *vli, wordcount_t num_words) {
wordcount_t i;
for (i = 0; i < num_words; ++i) {
vli[i] = 0;
}
}
#endif /* !asm_clear */
/* Constant-time comparison to zero - secure way to compare long integers */
/* Returns 1 if vli == 0, 0 otherwise. */
uECC_VLI_API uECC_word_t uECC_vli_isZero(const uECC_word_t *vli, wordcount_t num_words) {
uECC_word_t bits = 0;
wordcount_t i;
for (i = 0; i < num_words; ++i) {
bits |= vli[i];
}
return (bits == 0);
}
/* Returns nonzero if bit 'bit' of vli is set. */
uECC_VLI_API uECC_word_t uECC_vli_testBit(const uECC_word_t *vli, bitcount_t bit) {
return (vli[bit >> uECC_WORD_BITS_SHIFT] & ((uECC_word_t)1 << (bit & uECC_WORD_BITS_MASK)));
}
/* Counts the number of words in vli. */
static wordcount_t vli_numDigits(const uECC_word_t *vli, const wordcount_t max_words) {
wordcount_t i;
/* Search from the end until we find a non-zero digit.
We do it in reverse because we expect that most digits will be nonzero. */
for (i = max_words - 1; i >= 0 && vli[i] == 0; --i) {
}
return (i + 1);
}
/* Counts the number of bits required to represent vli. */
uECC_VLI_API bitcount_t uECC_vli_numBits(const uECC_word_t *vli, const wordcount_t max_words) {
uECC_word_t i;
uECC_word_t digit;
wordcount_t num_digits = vli_numDigits(vli, max_words);
if (num_digits == 0) {
return 0;
}
digit = vli[num_digits - 1];
for (i = 0; digit; ++i) {
digit >>= 1;
}
return (((bitcount_t)(num_digits - 1) << uECC_WORD_BITS_SHIFT) + i);
}
/* Sets dest = src. */
#if !asm_set
uECC_VLI_API void uECC_vli_set(uECC_word_t *dest, const uECC_word_t *src, wordcount_t num_words) {
wordcount_t i;
for (i = 0; i < num_words; ++i) {
dest[i] = src[i];
}
}
#endif /* !asm_set */
/* Returns sign of left - right. */
static cmpresult_t uECC_vli_cmp_unsafe(const uECC_word_t *left,
const uECC_word_t *right,
wordcount_t num_words) {
wordcount_t i;
for (i = num_words - 1; i >= 0; --i) {
if (left[i] > right[i]) {
return 1;
} else if (left[i] < right[i]) {
return -1;
}
}
return 0;
}
/* Constant-time comparison function - secure way to compare long integers */
/* Returns one if left == right, zero otherwise. */
uECC_VLI_API uECC_word_t uECC_vli_equal(const uECC_word_t *left,
const uECC_word_t *right,
wordcount_t num_words) {
uECC_word_t diff = 0;
wordcount_t i;
for (i = num_words - 1; i >= 0; --i) {
diff |= (left[i] ^ right[i]);
}
return (diff == 0);
}
/* Computes vli = vli >> 1. */
#if !asm_rshift1
uECC_VLI_API void uECC_vli_rshift1(uECC_word_t *vli, wordcount_t num_words) {
uECC_word_t *end = vli;
uECC_word_t carry = 0;
vli += num_words;
while (vli-- > end) {
uECC_word_t temp = *vli;
*vli = (temp >> 1) | carry;
carry = temp << (uECC_WORD_BITS - 1);
}
}
#endif /* !asm_rshift1 */
/* Computes result = (left + right) % mod.
Assumes that left < mod and right < mod, and that result does not overlap mod. */
uECC_VLI_API void uECC_vli_modAdd(uECC_word_t *result,
const uECC_word_t *left,
const uECC_word_t *right,
const uECC_word_t *mod,
wordcount_t num_words) {
uECC_word_t carry = uECC_vli_add(result, left, right, num_words);
if (carry || uECC_vli_cmp_unsafe(mod, result, num_words) != 1) {
/* result > mod (result = mod + remainder), so subtract mod to get remainder. */
uECC_vli_sub(result, result, mod, num_words);
}
}
/* Computes result = (left - right) % mod.
Assumes that left < mod and right < mod, and that result does not overlap mod. */
uECC_VLI_API void uECC_vli_modSub(uECC_word_t *result,
const uECC_word_t *left,
const uECC_word_t *right,
const uECC_word_t *mod,
wordcount_t num_words) {
uECC_word_t l_borrow = uECC_vli_sub(result, left, right, num_words);
if (l_borrow) {
/* In this case, result == -diff == (max int) - diff. Since -x % d == d - x,
we can get the correct result from result + mod (with overflow). */
uECC_vli_add(result, result, mod, num_words);
}
}
/* Computes result = product % mod, where product is 2N words long. */
/* Currently only designed to work for curve_p or curve_n. */
uECC_VLI_API void uECC_vli_mmod(uECC_word_t *result,
uECC_word_t *product,
const uECC_word_t *mod,
wordcount_t num_words) {
uECC_word_t mod_multiple[2 * uECC_MAX_WORDS];
uECC_word_t tmp[2 * uECC_MAX_WORDS];
uECC_word_t *v[2] = {tmp, product};
uECC_word_t index;
/* Shift mod so its highest set bit is at the maximum position. */
bitcount_t shift = (num_words * 2 * uECC_WORD_BITS) - uECC_vli_numBits(mod, num_words);
wordcount_t word_shift = shift / uECC_WORD_BITS;
wordcount_t bit_shift = shift % uECC_WORD_BITS;
uECC_word_t carry = 0;
uECC_vli_clear(mod_multiple, word_shift);
if (bit_shift > 0) {
for(index = 0; index < (uECC_word_t)num_words; ++index) {
mod_multiple[word_shift + index] = (mod[index] << bit_shift) | carry;
carry = mod[index] >> (uECC_WORD_BITS - bit_shift);
}
} else {
uECC_vli_set(mod_multiple + word_shift, mod, num_words);
}
for (index = 1; shift >= 0; --shift) {
uECC_word_t borrow = 0;
wordcount_t i;
for (i = 0; i < num_words * 2; ++i) {
uECC_word_t diff = v[index][i] - mod_multiple[i] - borrow;
if (diff != v[index][i]) {
borrow = (diff > v[index][i]);
}
v[1 - index][i] = diff;
}
index = !(index ^ borrow); /* Swap the index if there was no borrow */
uECC_vli_rshift1(mod_multiple, num_words);
mod_multiple[num_words - 1] |= mod_multiple[num_words] << (uECC_WORD_BITS - 1);
uECC_vli_rshift1(mod_multiple + num_words, num_words);
}
uECC_vli_set(result, v[index], num_words);
}
/* Computes result = (left * right) % mod. */
uECC_VLI_API void uECC_vli_modMult(uECC_word_t *result,
const uECC_word_t *left,
const uECC_word_t *right,
const uECC_word_t *mod,
wordcount_t num_words) {
uECC_word_t product[2 * uECC_MAX_WORDS];
uECC_vli_mult(product, left, right, num_words);
uECC_vli_mmod(result, product, mod, num_words);
}
uECC_VLI_API void uECC_vli_modMult_fast(uECC_word_t *result,
const uECC_word_t *left,
const uECC_word_t *right,
uECC_Curve curve) {
uECC_word_t product[2 * uECC_MAX_WORDS];
uECC_vli_mult(product, left, right, curve->num_words);
curve->mmod_fast(result, product);
}
uECC_VLI_API void uECC_vli_modSquare_fast(uECC_word_t *result,
const uECC_word_t *left,
uECC_Curve curve) {
uECC_vli_modMult_fast(result, left, left, curve);
}
#define EVEN(vli) (!(vli[0] & 1))
static void vli_modInv_update(uECC_word_t *uv,
const uECC_word_t *mod,
wordcount_t num_words) {
uECC_word_t carry = 0;
if (!EVEN(uv)) {
carry = uECC_vli_add(uv, uv, mod, num_words);
}
uECC_vli_rshift1(uv, num_words);
if (carry) {
uv[num_words - 1] |= HIGH_BIT_SET;
}
}
/* Computes result = (1 / input) % mod. All VLIs are the same size.
See "From Euclid's GCD to Montgomery Multiplication to the Great Divide" */
uECC_VLI_API void uECC_vli_modInv(uECC_word_t *result,
const uECC_word_t *input,
const uECC_word_t *mod,
wordcount_t num_words) {
uECC_word_t a[uECC_MAX_WORDS], b[uECC_MAX_WORDS], u[uECC_MAX_WORDS], v[uECC_MAX_WORDS];
cmpresult_t cmpResult;
if (uECC_vli_isZero(input, num_words)) {
uECC_vli_clear(result, num_words);
return;
}
uECC_vli_set(a, input, num_words);
uECC_vli_set(b, mod, num_words);
uECC_vli_clear(u, num_words);
u[0] = 1;
uECC_vli_clear(v, num_words);
while ((cmpResult = uECC_vli_cmp_unsafe(a, b, num_words)) != 0) {
if (EVEN(a)) {
uECC_vli_rshift1(a, num_words);
vli_modInv_update(u, mod, num_words);
} else if (EVEN(b)) {
uECC_vli_rshift1(b, num_words);
vli_modInv_update(v, mod, num_words);
} else if (cmpResult > 0) {
uECC_vli_sub(a, a, b, num_words);
uECC_vli_rshift1(a, num_words);
if (uECC_vli_cmp_unsafe(u, v, num_words) < 0) {
uECC_vli_add(u, u, mod, num_words);
}
uECC_vli_sub(u, u, v, num_words);
vli_modInv_update(u, mod, num_words);
} else {
uECC_vli_sub(b, b, a, num_words);
uECC_vli_rshift1(b, num_words);
if (uECC_vli_cmp_unsafe(v, u, num_words) < 0) {
uECC_vli_add(v, v, mod, num_words);
}
uECC_vli_sub(v, v, u, num_words);
vli_modInv_update(v, mod, num_words);
}
}
uECC_vli_set(result, u, num_words);
}
/* ------ Point operations ------ */
#include "uECC_curve-specific.inc"
/* Returns 1 if 'point' is the point at infinity, 0 otherwise. */
#define EccPoint_isZero(point, curve) uECC_vli_isZero((point), (curve)->num_words * 2)
/* Point multiplication algorithm using Montgomery's ladder with co-Z coordinates.
From http://eprint.iacr.org/2011/338.pdf
*/
/* Modify (x1, y1) => (x1 * z^2, y1 * z^3) */
static void apply_z(uECC_word_t * X1,
uECC_word_t * Y1,
const uECC_word_t * const Z,
uECC_Curve curve) {
uECC_word_t t1[uECC_MAX_WORDS];
uECC_vli_modSquare_fast(t1, Z, curve); /* z^2 */
uECC_vli_modMult_fast(X1, X1, t1, curve); /* x1 * z^2 */
uECC_vli_modMult_fast(t1, t1, Z, curve); /* z^3 */
uECC_vli_modMult_fast(Y1, Y1, t1, curve); /* y1 * z^3 */
}
/* Input P = (x1, y1, Z), Q = (x2, y2, Z)
Output P' = (x1', y1', Z3), P + Q = (x3, y3, Z3)
or P => P', Q => P + Q
*/
static void XYcZ_add(uECC_word_t * X1,
uECC_word_t * Y1,
uECC_word_t * X2,
uECC_word_t * Y2,
uECC_Curve curve) {
/* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */
uECC_word_t t5[uECC_MAX_WORDS];
wordcount_t num_words = curve->num_words;
uECC_vli_modSub(t5, X2, X1, curve->p, num_words); /* t5 = x2 - x1 */
uECC_vli_modSquare_fast(t5, t5, curve); /* t5 = (x2 - x1)^2 = A */
uECC_vli_modMult_fast(X1, X1, t5, curve); /* t1 = x1*A = B */
uECC_vli_modMult_fast(X2, X2, t5, curve); /* t3 = x2*A = C */
uECC_vli_modSub(Y2, Y2, Y1, curve->p, num_words); /* t4 = y2 - y1 */
uECC_vli_modSquare_fast(t5, Y2, curve); /* t5 = (y2 - y1)^2 = D */
uECC_vli_modSub(t5, t5, X1, curve->p, num_words); /* t5 = D - B */
uECC_vli_modSub(t5, t5, X2, curve->p, num_words); /* t5 = D - B - C = x3 */
uECC_vli_modSub(X2, X2, X1, curve->p, num_words); /* t3 = C - B */
uECC_vli_modMult_fast(Y1, Y1, X2, curve); /* t2 = y1*(C - B) */
uECC_vli_modSub(X2, X1, t5, curve->p, num_words); /* t3 = B - x3 */
uECC_vli_modMult_fast(Y2, Y2, X2, curve); /* t4 = (y2 - y1)*(B - x3) */
uECC_vli_modSub(Y2, Y2, Y1, curve->p, num_words); /* t4 = y3 */
uECC_vli_set(X2, t5, num_words);
}
uECC_VLI_API void uECC_vli_bytesToNative(uECC_word_t *native,
const uint8_t *bytes,
int num_bytes) {
wordcount_t i;
uECC_vli_clear(native, (num_bytes + (uECC_WORD_SIZE - 1)) / uECC_WORD_SIZE);
for (i = 0; i < num_bytes; ++i) {
unsigned b = num_bytes - 1 - i;
native[b / uECC_WORD_SIZE] |=
(uECC_word_t)bytes[i] << (8 * (b % uECC_WORD_SIZE));
}
}
int uECC_valid_point(const uECC_word_t *point, uECC_Curve curve) {
uECC_word_t tmp1[uECC_MAX_WORDS];
uECC_word_t tmp2[uECC_MAX_WORDS];
wordcount_t num_words = curve->num_words;
/* The point at infinity is invalid. */
if (EccPoint_isZero(point, curve)) {
return 0;
}
/* x and y must be smaller than p. */
if (uECC_vli_cmp_unsafe(curve->p, point, num_words) != 1 ||
uECC_vli_cmp_unsafe(curve->p, point + num_words, num_words) != 1) {
return 0;
}
uECC_vli_modSquare_fast(tmp1, point + num_words, curve);
curve->x_side(tmp2, point, curve); /* tmp2 = x^3 + ax + b */
/* Make sure that y^2 == x^3 + ax + b */
return (int)(uECC_vli_equal(tmp1, tmp2, num_words));
}
/* -------- ECDSA code -------- */
static void bits2int(uECC_word_t *native,
const uint8_t *bits,
unsigned bits_size,
uECC_Curve curve) {
unsigned num_n_bytes = BITS_TO_BYTES(curve->num_n_bits);
unsigned num_n_words = BITS_TO_WORDS(curve->num_n_bits);
int shift;
uECC_word_t carry;
uECC_word_t *ptr;
if (bits_size > num_n_bytes) {
bits_size = num_n_bytes;
}
uECC_vli_clear(native, num_n_words);
uECC_vli_bytesToNative(native, bits, bits_size);
if (bits_size * 8 <= (unsigned)curve->num_n_bits) {
return;
}
shift = bits_size * 8 - curve->num_n_bits;
carry = 0;
ptr = native + num_n_words;
while (ptr-- > native) {
uECC_word_t temp = *ptr;
*ptr = (temp >> shift) | carry;
carry = temp << (uECC_WORD_BITS - shift);
}
/* Reduce mod curve_n */
if (uECC_vli_cmp_unsafe(curve->n, native, num_n_words) != 1) {
uECC_vli_sub(native, native, curve->n, num_n_words);
}
}
static bitcount_t smax(bitcount_t a, bitcount_t b) {
return (a > b ? a : b);
}
int uECC_verify(const uint8_t *public_key,
const uint8_t *message_hash,
unsigned hash_size,
const uint8_t *signature,
uECC_Curve curve) {
uECC_word_t u1[uECC_MAX_WORDS], u2[uECC_MAX_WORDS];
uECC_word_t z[uECC_MAX_WORDS];
uECC_word_t sum[uECC_MAX_WORDS * 2];
uECC_word_t rx[uECC_MAX_WORDS];
uECC_word_t ry[uECC_MAX_WORDS];
uECC_word_t tx[uECC_MAX_WORDS];
uECC_word_t ty[uECC_MAX_WORDS];
uECC_word_t tz[uECC_MAX_WORDS];
const uECC_word_t *points[4];
const uECC_word_t *point;
bitcount_t num_bits;
bitcount_t i;
uECC_word_t _public[uECC_MAX_WORDS * 2];
uECC_word_t r[uECC_MAX_WORDS], s[uECC_MAX_WORDS];
wordcount_t num_words = curve->num_words;
wordcount_t num_n_words = BITS_TO_WORDS(curve->num_n_bits);
rx[num_n_words - 1] = 0;
r[num_n_words - 1] = 0;
s[num_n_words - 1] = 0;
uECC_vli_bytesToNative(_public, public_key, curve->num_bytes);
uECC_vli_bytesToNative(_public + num_words, public_key + curve->num_bytes, curve->num_bytes);
uECC_vli_bytesToNative(r, signature, curve->num_bytes);
uECC_vli_bytesToNative(s, signature + curve->num_bytes, curve->num_bytes);
/* r, s must not be 0. */
if (uECC_vli_isZero(r, num_words) || uECC_vli_isZero(s, num_words)) {
return 0;
}
/* r, s must be < n. */
if (uECC_vli_cmp_unsafe(curve->n, r, num_n_words) != 1 ||
uECC_vli_cmp_unsafe(curve->n, s, num_n_words) != 1) {
return 0;
}
/* Calculate u1 and u2. */
uECC_vli_modInv(z, s, curve->n, num_n_words); /* z = 1/s */
u1[num_n_words - 1] = 0;
bits2int(u1, message_hash, hash_size, curve);
uECC_vli_modMult(u1, u1, z, curve->n, num_n_words); /* u1 = e/s */
uECC_vli_modMult(u2, r, z, curve->n, num_n_words); /* u2 = r/s */
/* Calculate sum = G + Q. */
uECC_vli_set(sum, _public, num_words);
uECC_vli_set(sum + num_words, _public + num_words, num_words);
uECC_vli_set(tx, curve->G, num_words);
uECC_vli_set(ty, curve->G + num_words, num_words);
uECC_vli_modSub(z, sum, tx, curve->p, num_words); /* z = x2 - x1 */
XYcZ_add(tx, ty, sum, sum + num_words, curve);
uECC_vli_modInv(z, z, curve->p, num_words); /* z = 1/z */
apply_z(sum, sum + num_words, z, curve);
/* Use Shamir's trick to calculate u1*G + u2*Q */
points[0] = 0;
points[1] = curve->G;
points[2] = _public;
points[3] = sum;
num_bits = smax(uECC_vli_numBits(u1, num_n_words),
uECC_vli_numBits(u2, num_n_words));
point = points[(!!uECC_vli_testBit(u1, num_bits - 1)) |
((!!uECC_vli_testBit(u2, num_bits - 1)) << 1)];
uECC_vli_set(rx, point, num_words);
uECC_vli_set(ry, point + num_words, num_words);
uECC_vli_clear(z, num_words);
z[0] = 1;
for (i = num_bits - 2; i >= 0; --i) {
uECC_word_t index;
curve->double_jacobian(rx, ry, z, curve);
index = (!!uECC_vli_testBit(u1, i)) | ((!!uECC_vli_testBit(u2, i)) << 1);
point = points[index];
if (point) {
uECC_vli_set(tx, point, num_words);
uECC_vli_set(ty, point + num_words, num_words);
apply_z(tx, ty, z, curve);
uECC_vli_modSub(tz, rx, tx, curve->p, num_words); /* Z = x2 - x1 */
XYcZ_add(tx, ty, rx, ry, curve);
uECC_vli_modMult_fast(z, z, tz, curve);
}
}
uECC_vli_modInv(z, z, curve->p, num_words); /* Z = 1/Z */
apply_z(rx, ry, z, curve);
/* v = x1 (mod n) */
if (uECC_vli_cmp_unsafe(curve->n, rx, num_n_words) != 1) {
uECC_vli_sub(rx, rx, curve->n, num_n_words);
}
/* Accept only if v == r. */
return (int)(uECC_vli_equal(rx, r, num_words));
}
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