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uECC.c
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/* Copyright 2014, Kenneth MacKay. Licensed under the BSD 2-clause license. */
#include "uECC.h"
#include "uECC_vli.h"
#ifdef KEYSTONE_BOOTLOADER
#include "string.h"
#elif defined(KEYSTONE_SM)
#include <sbi/sbi_string.h>
#define memcpy sbi_memcpy
#define memset sbi_memset
#else
#include <string.h>
#endif
#ifndef uECC_RNG_MAX_TRIES
#define uECC_RNG_MAX_TRIES 64
#endif
#if uECC_ENABLE_VLI_API
#define uECC_VLI_API
#else
#define uECC_VLI_API static
#endif
#if (uECC_PLATFORM == uECC_avr) || \
(uECC_PLATFORM == uECC_arm) || \
(uECC_PLATFORM == uECC_arm_thumb) || \
(uECC_PLATFORM == uECC_arm_thumb2)
#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))
#endif
#include "platform-specific.inc"
#if (uECC_WORD_SIZE == 1)
#if uECC_SUPPORTS_secp160r1
#define uECC_MAX_WORDS 21 /* Due to the size of curve_n. */
#endif
#if uECC_SUPPORTS_secp192r1
#undef uECC_MAX_WORDS
#define uECC_MAX_WORDS 24
#endif
#if uECC_SUPPORTS_secp224r1
#undef uECC_MAX_WORDS
#define uECC_MAX_WORDS 28
#endif
#if (uECC_SUPPORTS_secp256r1 || uECC_SUPPORTS_secp256k1)
#undef uECC_MAX_WORDS
#define uECC_MAX_WORDS 32
#endif
#elif (uECC_WORD_SIZE == 4)
#if uECC_SUPPORTS_secp160r1
#define uECC_MAX_WORDS 6 /* Due to the size of curve_n. */
#endif
#if uECC_SUPPORTS_secp192r1
#undef uECC_MAX_WORDS
#define uECC_MAX_WORDS 6
#endif
#if uECC_SUPPORTS_secp224r1
#undef uECC_MAX_WORDS
#define uECC_MAX_WORDS 7
#endif
#if (uECC_SUPPORTS_secp256r1 || uECC_SUPPORTS_secp256k1)
#undef uECC_MAX_WORDS
#define uECC_MAX_WORDS 8
#endif
#elif (uECC_WORD_SIZE == 8)
#if uECC_SUPPORTS_secp160r1
#define uECC_MAX_WORDS 3
#endif
#if uECC_SUPPORTS_secp192r1
#undef uECC_MAX_WORDS
#define uECC_MAX_WORDS 3
#endif
#if uECC_SUPPORTS_secp224r1
#undef uECC_MAX_WORDS
#define uECC_MAX_WORDS 4
#endif
#if (uECC_SUPPORTS_secp256r1 || uECC_SUPPORTS_secp256k1)
#undef uECC_MAX_WORDS
#define uECC_MAX_WORDS 4
#endif
#endif /* uECC_WORD_SIZE */
#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);
#if uECC_SUPPORT_COMPRESSED_POINT
void (*mod_sqrt)(uECC_word_t *a, uECC_Curve curve);
#endif
void (*x_side)(uECC_word_t *result, const uECC_word_t *x, uECC_Curve curve);
#if (uECC_OPTIMIZATION_LEVEL > 0)
void (*mmod_fast)(uECC_word_t *result, uECC_word_t *product);
#endif
};
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 "asm_arm.inc"
#endif
#if (uECC_PLATFORM == uECC_avr)
#include "asm_avr.inc"
#endif
#if default_RNG_defined
static uECC_RNG_Function g_rng_function = &default_RNG;
#else
static uECC_RNG_Function g_rng_function = 0;
#endif
void uECC_set_rng(uECC_RNG_Function rng_function) {
g_rng_function = rng_function;
}
uECC_RNG_Function uECC_get_rng(void) {
return g_rng_function;
}
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);
}
uECC_VLI_API uECC_word_t uECC_vli_sub(uECC_word_t *result,
const uECC_word_t *left,
const uECC_word_t *right,
wordcount_t num_words);
/* Returns sign of left - right, in constant time. */
uECC_VLI_API cmpresult_t uECC_vli_cmp(const uECC_word_t *left,
const uECC_word_t *right,
wordcount_t num_words) {
uECC_word_t tmp[uECC_MAX_WORDS];
uECC_word_t neg = !!uECC_vli_sub(tmp, left, right, num_words);
uECC_word_t equal = uECC_vli_isZero(tmp, num_words);
return (!equal - 2 * neg);
}
/* 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, returning carry. Can modify in place. */
#if !asm_add
uECC_VLI_API uECC_word_t uECC_vli_add(uECC_word_t *result,
const uECC_word_t *left,
const uECC_word_t *right,
wordcount_t num_words) {
uECC_word_t carry = 0;
wordcount_t i;
for (i = 0; i < num_words; ++i) {
uECC_word_t sum = left[i] + right[i] + carry;
if (sum != left[i]) {
carry = (sum < left[i]);
}
result[i] = sum;
}
return carry;
}
#endif /* !asm_add */
/* Computes result = left - right, returning borrow. Can modify in place. */
#if !asm_sub
uECC_VLI_API uECC_word_t uECC_vli_sub(uECC_word_t *result,
const uECC_word_t *left,
const uECC_word_t *right,
wordcount_t num_words) {
uECC_word_t borrow = 0;
wordcount_t i;
for (i = 0; i < num_words; ++i) {
uECC_word_t diff = left[i] - right[i] - borrow;
if (diff != left[i]) {
borrow = (diff > left[i]);
}
result[i] = diff;
}
return borrow;
}
#endif /* !asm_sub */
#if !asm_mult || (uECC_SQUARE_FUNC && !asm_square) || \
(uECC_SUPPORTS_secp256k1 && (uECC_OPTIMIZATION_LEVEL > 0) && \
((uECC_WORD_SIZE == 1) || (uECC_WORD_SIZE == 8)))
static void muladd(uECC_word_t a,
uECC_word_t b,
uECC_word_t *r0,
uECC_word_t *r1,
uECC_word_t *r2) {
#if uECC_WORD_SIZE == 8 && !SUPPORTS_INT128
uint64_t a0 = a & 0xffffffffull;
uint64_t a1 = a >> 32;
uint64_t b0 = b & 0xffffffffull;
uint64_t b1 = b >> 32;
uint64_t i0 = a0 * b0;
uint64_t i1 = a0 * b1;
uint64_t i2 = a1 * b0;
uint64_t i3 = a1 * b1;
uint64_t p0, p1;
i2 += (i0 >> 32);
i2 += i1;
if (i2 < i1) { /* overflow */
i3 += 0x100000000ull;
}
p0 = (i0 & 0xffffffffull) | (i2 << 32);
p1 = i3 + (i2 >> 32);
*r0 += p0;
*r1 += (p1 + (*r0 < p0));
*r2 += ((*r1 < p1) || (*r1 == p1 && *r0 < p0));
#else
uECC_dword_t p = (uECC_dword_t)a * b;
uECC_dword_t r01 = ((uECC_dword_t)(*r1) << uECC_WORD_BITS) | *r0;
r01 += p;
*r2 += (r01 < p);
*r1 = r01 >> uECC_WORD_BITS;
*r0 = (uECC_word_t)r01;
#endif
}
#endif /* muladd needed */
#if !asm_mult
uECC_VLI_API void uECC_vli_mult(uECC_word_t *result,
const uECC_word_t *left,
const uECC_word_t *right,
wordcount_t num_words) {
uECC_word_t r0 = 0;
uECC_word_t r1 = 0;
uECC_word_t r2 = 0;
wordcount_t i, k;
/* Compute each digit of result in sequence, maintaining the carries. */
for (k = 0; k < num_words; ++k) {
for (i = 0; i <= k; ++i) {
muladd(left[i], right[k - i], &r0, &r1, &r2);
}
result[k] = r0;
r0 = r1;
r1 = r2;
r2 = 0;
}
for (k = num_words; k < num_words * 2 - 1; ++k) {
for (i = (k + 1) - num_words; i < num_words; ++i) {
muladd(left[i], right[k - i], &r0, &r1, &r2);
}
result[k] = r0;
r0 = r1;
r1 = r2;
r2 = 0;
}
result[num_words * 2 - 1] = r0;
}
#endif /* !asm_mult */
#if uECC_SQUARE_FUNC
#if !asm_square
static void mul2add(uECC_word_t a,
uECC_word_t b,
uECC_word_t *r0,
uECC_word_t *r1,
uECC_word_t *r2) {
#if uECC_WORD_SIZE == 8 && !SUPPORTS_INT128
uint64_t a0 = a & 0xffffffffull;
uint64_t a1 = a >> 32;
uint64_t b0 = b & 0xffffffffull;
uint64_t b1 = b >> 32;
uint64_t i0 = a0 * b0;
uint64_t i1 = a0 * b1;
uint64_t i2 = a1 * b0;
uint64_t i3 = a1 * b1;
uint64_t p0, p1;
i2 += (i0 >> 32);
i2 += i1;
if (i2 < i1)
{ /* overflow */
i3 += 0x100000000ull;
}
p0 = (i0 & 0xffffffffull) | (i2 << 32);
p1 = i3 + (i2 >> 32);
*r2 += (p1 >> 63);
p1 = (p1 << 1) | (p0 >> 63);
p0 <<= 1;
*r0 += p0;
*r1 += (p1 + (*r0 < p0));
*r2 += ((*r1 < p1) || (*r1 == p1 && *r0 < p0));
#else
uECC_dword_t p = (uECC_dword_t)a * b;
uECC_dword_t r01 = ((uECC_dword_t)(*r1) << uECC_WORD_BITS) | *r0;
*r2 += (p >> (uECC_WORD_BITS * 2 - 1));
p *= 2;
r01 += p;
*r2 += (r01 < p);
*r1 = r01 >> uECC_WORD_BITS;
*r0 = (uECC_word_t)r01;
#endif
}
uECC_VLI_API void uECC_vli_square(uECC_word_t *result,
const uECC_word_t *left,
wordcount_t num_words) {
uECC_word_t r0 = 0;
uECC_word_t r1 = 0;
uECC_word_t r2 = 0;
wordcount_t i, k;
for (k = 0; k < num_words * 2 - 1; ++k) {
uECC_word_t min = (k < num_words ? 0 : (k + 1) - num_words);
for (i = min; i <= k && i <= k - i; ++i) {
if (i < k-i) {
mul2add(left[i], left[k - i], &r0, &r1, &r2);
} else {
muladd(left[i], left[k - i], &r0, &r1, &r2);
}
}
result[k] = r0;
r0 = r1;
r1 = r2;
r2 = 0;
}
result[num_words * 2 - 1] = r0;
}
#endif /* !asm_square */
#else /* uECC_SQUARE_FUNC */
#if uECC_ENABLE_VLI_API
uECC_VLI_API void uECC_vli_square(uECC_word_t *result,
const uECC_word_t *left,
wordcount_t num_words) {
uECC_vli_mult(result, left, left, num_words);
}
#endif /* uECC_ENABLE_VLI_API */
#endif /* uECC_SQUARE_FUNC */
/* 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);
#if (uECC_OPTIMIZATION_LEVEL > 0)
curve->mmod_fast(result, product);
#else
uECC_vli_mmod(result, product, curve->p, curve->num_words);
#endif
}
#if uECC_SQUARE_FUNC
#if uECC_ENABLE_VLI_API
/* Computes result = left^2 % mod. */
uECC_VLI_API void uECC_vli_modSquare(uECC_word_t *result,
const uECC_word_t *left,
const uECC_word_t *mod,
wordcount_t num_words) {
uECC_word_t product[2 * uECC_MAX_WORDS];
uECC_vli_square(product, left, num_words);
uECC_vli_mmod(result, product, mod, num_words);
}
#endif /* uECC_ENABLE_VLI_API */
uECC_VLI_API void uECC_vli_modSquare_fast(uECC_word_t *result,
const uECC_word_t *left,
uECC_Curve curve) {
uECC_word_t product[2 * uECC_MAX_WORDS];
uECC_vli_square(product, left, curve->num_words);
#if (uECC_OPTIMIZATION_LEVEL > 0)
curve->mmod_fast(result, product);
#else
uECC_vli_mmod(result, product, curve->p, curve->num_words);
#endif
}
#else /* uECC_SQUARE_FUNC */
#if uECC_ENABLE_VLI_API
uECC_VLI_API void uECC_vli_modSquare(uECC_word_t *result,
const uECC_word_t *left,
const uECC_word_t *mod,
wordcount_t num_words) {
uECC_vli_modMult(result, left, left, mod, num_words);
}
#endif /* uECC_ENABLE_VLI_API */
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);
}
#endif /* uECC_SQUARE_FUNC */
#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 "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 */
}
/* P = (x1, y1) => 2P, (x2, y2) => P' */
static void XYcZ_initial_double(uECC_word_t * X1,
uECC_word_t * Y1,
uECC_word_t * X2,
uECC_word_t * Y2,
const uECC_word_t * const initial_Z,
uECC_Curve curve) {
uECC_word_t z[uECC_MAX_WORDS];
wordcount_t num_words = curve->num_words;
if (initial_Z) {
uECC_vli_set(z, initial_Z, num_words);
} else {
uECC_vli_clear(z, num_words);
z[0] = 1;
}
uECC_vli_set(X2, X1, num_words);
uECC_vli_set(Y2, Y1, num_words);
apply_z(X1, Y1, z, curve);
curve->double_jacobian(X1, Y1, z, curve);
apply_z(X2, Y2, z, curve);
}
/* 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
sub = x1' - x3 (used for subsequent call to XYcZ_addC()).
*/
static void XYcZ_add(uECC_word_t * X1,
uECC_word_t * Y1,
uECC_word_t * X2,
uECC_word_t * Y2,
uECC_word_t * sub,
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); /* x1' = 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); /* y1' = y1*(C - B) */
uECC_vli_modSub(sub, X1, t5, curve->p, num_words); /* s = B - x3 */
uECC_vli_modMult_fast(Y2, Y2, sub, 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); /* move x3 to output */
}
/* Input P = (x1, y1, Z), Q = (x2, y2, Z), sub = x1 - x2
Output P - Q = (x3', y3', Z3), P + Q = (x3, y3, Z3)
or P => P - Q, Q => P + Q
*/
static void XYcZ_addC(uECC_word_t * X1,
uECC_word_t * Y1,
uECC_word_t * X2,
uECC_word_t * Y2,
uECC_word_t * sub,
uECC_Curve curve) {
/* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */
uECC_word_t t5[uECC_MAX_WORDS];
uECC_word_t t6[uECC_MAX_WORDS];
uECC_word_t t7[uECC_MAX_WORDS];
wordcount_t num_words = curve->num_words;
uECC_vli_modSquare_fast(t5, sub, 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_modAdd(t5, Y2, Y1, curve->p, num_words); /* t5 = y2 + y1 */
uECC_vli_modSub(Y2, Y2, Y1, curve->p, num_words); /* t4 = y2 - y1 */
uECC_vli_modSub(t6, X2, X1, curve->p, num_words); /* t6 = C - B */
uECC_vli_modMult_fast(Y1, Y1, t6, curve); /* t2 = y1 * (C - B) = E */
uECC_vli_modAdd(t6, X1, X2, curve->p, num_words); /* t6 = B + C */
uECC_vli_modSquare_fast(X2, Y2, curve); /* t3 = (y2 - y1)^2 = D */
uECC_vli_modSub(X2, X2, t6, curve->p, num_words); /* t3 = D - (B + C) = x3 */
uECC_vli_modSub(t7, X1, X2, curve->p, num_words); /* t7 = B - x3 */
uECC_vli_modMult_fast(Y2, Y2, t7, curve); /* t4 = (y2 - y1)*(B - x3) */
uECC_vli_modSub(Y2, Y2, Y1, curve->p, num_words); /* t4 = (y2 - y1)*(B - x3) - E = y3 */
uECC_vli_modSquare_fast(t7, t5, curve); /* t7 = (y2 + y1)^2 = F */
uECC_vli_modSub(t7, t7, t6, curve->p, num_words); /* t7 = F - (B + C) = x3' */
uECC_vli_modSub(t6, t7, X1, curve->p, num_words); /* t6 = x3' - B */
uECC_vli_modMult_fast(t6, t6, t5, curve); /* t6 = (y2+y1)*(x3' - B) */
uECC_vli_modSub(Y1, t6, Y1, curve->p, num_words); /* t2 = (y2+y1)*(x3' - B) - E = y3' */
uECC_vli_set(X1, t7, num_words); /* move x3' to output */
}
/* result may overlap point. */
static void EccPoint_mult(uECC_word_t * result,
const uECC_word_t * point,
const uECC_word_t * scalar,
const uECC_word_t * initial_Z,
bitcount_t num_bits,
uECC_Curve curve) {
/* R0 and R1 */
uECC_word_t Rx[2][uECC_MAX_WORDS];
uECC_word_t Ry[2][uECC_MAX_WORDS];
uECC_word_t z[uECC_MAX_WORDS];
uECC_word_t sub[uECC_MAX_WORDS];
bitcount_t i;
uECC_word_t nb;
wordcount_t num_words = curve->num_words;
uECC_vli_set(Rx[1], point, num_words);
uECC_vli_set(Ry[1], point + num_words, num_words);
XYcZ_initial_double(Rx[1], Ry[1], Rx[0], Ry[0], initial_Z, curve);
uECC_vli_modSub(sub, Rx[0], Rx[1], curve->p, num_words);
for (i = num_bits - 2; i > 0; --i) {
nb = !uECC_vli_testBit(scalar, i);
XYcZ_addC(Rx[1 - nb], Ry[1 - nb], Rx[nb], Ry[nb], sub, curve);
XYcZ_add(Rx[nb], Ry[nb], Rx[1 - nb], Ry[1 - nb], sub, curve);
}
nb = !uECC_vli_testBit(scalar, 0);
XYcZ_addC(Rx[1 - nb], Ry[1 - nb], Rx[nb], Ry[nb], sub, curve);
/* Find final 1/Z value. */
uECC_vli_modSub(z, Rx[1], Rx[0], curve->p, num_words); /* X1 - X0 */
uECC_vli_modMult_fast(z, z, Ry[1 - nb], curve); /* Yb * (X1 - X0) */
uECC_vli_modMult_fast(z, z, point, curve); /* xP * Yb * (X1 - X0) */
uECC_vli_modInv(z, z, curve->p, num_words); /* 1 / (xP * Yb * (X1 - X0)) */
uECC_vli_modMult_fast(z, z, point + num_words, curve); /* yP / (xP * Yb * (X1 - X0)) */
uECC_vli_modMult_fast(z, z, Rx[1 - nb], curve); /* Xb * yP / (xP * Yb * (X1 - X0)) */
/* End 1/Z calculation */
XYcZ_add(Rx[nb], Ry[nb], Rx[1 - nb], Ry[1 - nb], sub, curve);
apply_z(Rx[0], Ry[0], z, curve);
uECC_vli_set(result, Rx[0], num_words);
uECC_vli_set(result + num_words, Ry[0], num_words);
}
static uECC_word_t regularize_k(const uECC_word_t * const k,
uECC_word_t *k0,
uECC_word_t *k1,
uECC_Curve curve) {
wordcount_t num_n_words = BITS_TO_WORDS(curve->num_n_bits);
bitcount_t num_n_bits = curve->num_n_bits;
uECC_word_t carry = uECC_vli_add(k0, k, curve->n, num_n_words) ||
(num_n_bits < ((bitcount_t)num_n_words * uECC_WORD_SIZE * 8) &&
uECC_vli_testBit(k0, num_n_bits));
uECC_vli_add(k1, k0, curve->n, num_n_words);
return carry;
}
/* Generates a random integer in the range 0 < random < top.
Both random and top have num_words words. */
uECC_VLI_API int uECC_generate_random_int(uECC_word_t *random,
const uECC_word_t *top,
wordcount_t num_words) {
uECC_word_t mask = (uECC_word_t)-1;
uECC_word_t tries;
bitcount_t num_bits = uECC_vli_numBits(top, num_words);
if (!g_rng_function) {
return 0;
}
for (tries = 0; tries < uECC_RNG_MAX_TRIES; ++tries) {
if (!g_rng_function((uint8_t *)random, num_words * uECC_WORD_SIZE)) {
return 0;
}
random[num_words - 1] &= mask >> ((bitcount_t)(num_words * uECC_WORD_SIZE * 8 - num_bits));
if (!uECC_vli_isZero(random, num_words) &&
uECC_vli_cmp(top, random, num_words) == 1) {
return 1;
}
}
return 0;
}
static uECC_word_t EccPoint_compute_public_key(uECC_word_t *result,
uECC_word_t *private_key,
uECC_Curve curve) {
uECC_word_t tmp1[uECC_MAX_WORDS];
uECC_word_t tmp2[uECC_MAX_WORDS];
uECC_word_t *p2[2] = {tmp1, tmp2};
uECC_word_t *initial_Z = 0;
uECC_word_t carry;
/* Regularize the bitcount for the private key so that attackers cannot use a side channel
attack to learn the number of leading zeros. */
carry = regularize_k(private_key, tmp1, tmp2, curve);
/* If an RNG function was specified, try to get a random initial Z value to improve
protection against side-channel attacks. */
if (g_rng_function) {
if (!uECC_generate_random_int(p2[carry], curve->p, curve->num_words)) {
return 0;
}
initial_Z = p2[carry];
}
EccPoint_mult(result, curve->G, p2[!carry], initial_Z, curve->num_n_bits + 1, curve);
if (EccPoint_isZero(result, curve)) {
return 0;
}
return 1;
}
#if uECC_VLI_NATIVE_LITTLE_ENDIAN
#define uECC_vli_nativeToBytes(bytes, num_bytes, native) memcpy((bytes), (native), (num_bytes))
#define uECC_vli_bytesToNative(native, bytes, num_bytes) memcpy((native), (bytes), (num_bytes))
#else /* uECC_VLI_NATIVE_LITTLE_ENDIAN */
#if uECC_WORD_SIZE == 1
uECC_VLI_API void uECC_vli_nativeToBytes(uint8_t *bytes,
int num_bytes,
const uint8_t *native) {
wordcount_t i;
for (i = 0; i < num_bytes; ++i) {
bytes[i] = native[(num_bytes - 1) - i];
}
}
uECC_VLI_API void uECC_vli_bytesToNative(uint8_t *native,
const uint8_t *bytes,
int num_bytes) {
uECC_vli_nativeToBytes(native, num_bytes, bytes);
}
#else