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https://github.com/trezor/trezor-firmware.git
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Extended comments, new function bn_add, a bug fix.
Describe normalized, partly reduced and reduced numbers. Comment which function expects which kind of input. Removed unused bn_bitlen. Add bn_add that does not reduce. Bug fix in ecdsa_validate_pubkey: bn_mod before bn_is_equal. Bug fix in hdnode_private_ckd: bn_mod after bn_addmod.
This commit is contained in:
Jochen Hoenicke
parent
53fa580b81
commit
f93b003cbc
150
bignum.c
150
bignum.c
@ -27,6 +27,31 @@
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#include "bignum.h"
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#include "macros.h"
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/* big number library */
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/* The structure bignum256 is an array of nine 32-bit values, which
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* are digits in base 2^30 representation. I.e. the number
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* bignum256 a;
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* represents the value
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* sum_{i=0}^8 a.val[i] * 2^{30 i}.
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*
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* The number is *normalized* iff every digit is < 2^30.
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*
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* As the name suggests, a bignum256 is intended to represent a 256
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* bit number, but it can represent 270 bits. Numbers are usually
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* reduced using a prime, either the group order or the field prime.
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* The reduction is often partly done by bn_fast_mod, and similarly
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* implicitly in bn_multiply. A *partly reduced number* is a
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* normalized number between 0 (inclusive) and 2*prime (exclusive).
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*
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* A partly reduced number can be fully reduced by calling bn_mod.
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* Only a fully reduced number is guaranteed to fit in 256 bit.
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*
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* All functions assume that the prime in question is slightly smaller
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* than 2^256. In particular it must be between 2^256-2^224 and
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* 2^256 and it must be a prime number.
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*/
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inline uint32_t read_be(const uint8_t *data)
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{
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return (((uint32_t)data[0]) << 24) |
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@ -43,7 +68,8 @@ inline void write_be(uint8_t *data, uint32_t x)
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data[3] = x;
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}
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// convert a raw bigendian 256 bit number to a normalized bignum
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// convert a raw bigendian 256 bit value into a normalized bignum.
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// out_number is partly reduced (since it fits in 256 bit).
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void bn_read_be(const uint8_t *in_number, bignum256 *out_number)
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{
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int i;
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@ -63,7 +89,7 @@ void bn_read_be(const uint8_t *in_number, bignum256 *out_number)
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}
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// convert a normalized bignum to a raw bigendian 256 bit number.
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// in_number must be normalized and < 2^256.
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// in_number must be fully reduced.
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void bn_write_be(const bignum256 *in_number, uint8_t *out_number)
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{
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int i;
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@ -77,6 +103,7 @@ void bn_write_be(const bignum256 *in_number, uint8_t *out_number)
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}
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}
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// sets a bignum to zero.
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void bn_zero(bignum256 *a)
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{
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int i;
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@ -85,6 +112,9 @@ void bn_zero(bignum256 *a)
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}
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}
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// checks that a bignum is zero.
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// a must be normalized
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// function is constant time (on some architectures, in particular ARM).
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int bn_is_zero(const bignum256 *a)
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{
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int i;
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@ -95,6 +125,9 @@ int bn_is_zero(const bignum256 *a)
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return !result;
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}
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// Check whether a < b
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// a and b must be normalized
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// function is constant time (on some architectures, in particular ARM).
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int bn_is_less(const bignum256 *a, const bignum256 *b)
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{
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int i;
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@ -107,6 +140,9 @@ int bn_is_less(const bignum256 *a, const bignum256 *b)
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return res1 > res2;
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}
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// Check whether a == b
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// a and b must be normalized
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// function is constant time (on some architectures, in particular ARM).
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int bn_is_equal(const bignum256 *a, const bignum256 *b) {
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int i;
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uint32_t result = 0;
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@ -116,6 +152,9 @@ int bn_is_equal(const bignum256 *a, const bignum256 *b) {
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return !result;
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}
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// Assigns res = cond ? truecase : falsecase
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// assumes that cond is either 0 or 1.
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// function is constant time.
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void bn_cmov(bignum256 *res, int cond, const bignum256 *truecase, const bignum256 *falsecase)
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{
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int i;
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@ -129,15 +168,8 @@ void bn_cmov(bignum256 *res, int cond, const bignum256 *truecase, const bignum25
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}
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}
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int bn_bitlen(const bignum256 *a) {
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int i = 8, j;
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while (i >= 0 && a->val[i] == 0) i--;
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if (i == -1) return 0;
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j = 29;
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while ((a->val[i] & (1 << j)) == 0) j--;
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return i * 30 + j + 1;
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}
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// shift number to the left, i.e multiply it by 2.
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// a must be normalized. The result is normalized but not reduced.
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void bn_lshift(bignum256 *a)
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{
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int i;
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@ -147,6 +179,8 @@ void bn_lshift(bignum256 *a)
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a->val[0] = (a->val[0] << 1) & 0x3FFFFFFF;
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}
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// shift number to the right, i.e divide by 2 while rounding down.
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// a must be normalized. The result is normalized.
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void bn_rshift(bignum256 *a)
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{
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int i;
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@ -157,8 +191,10 @@ void bn_rshift(bignum256 *a)
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}
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// multiply x by 1/2 modulo prime.
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// assumes x < 2*prime,
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// guarantees x < 4*prime on exit.
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// it computes x = (x & 1) ? (x + prime) >> 1 : x >> 1.
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// assumes x is normalized.
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// if x was partly reduced, it is also partly reduced on exit.
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// function is constant time.
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void bn_mult_half(bignum256 * x, const bignum256 *prime)
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{
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int j;
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@ -177,8 +213,8 @@ void bn_mult_half(bignum256 * x, const bignum256 *prime)
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}
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// multiply x by k modulo prime.
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// assumes x < prime,
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// guarantees x < prime on exit.
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// assumes x is normalized, 0 <= k <= 4.
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// guarantees x is partly reduced.
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void bn_mult_k(bignum256 *x, uint8_t k, const bignum256 *prime)
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{
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int j;
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@ -188,7 +224,8 @@ void bn_mult_k(bignum256 *x, uint8_t k, const bignum256 *prime)
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bn_fast_mod(x, prime);
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}
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// assumes x < 2*prime, result < prime
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// compute x = x mod prime by computing x >= prime ? x - prime : x.
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// assumes x partly reduced, guarantees x fully reduced.
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void bn_mod(bignum256 *x, const bignum256 *prime)
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{
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int i = 8;
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@ -214,6 +251,9 @@ void bn_mod(bignum256 *x, const bignum256 *prime)
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}
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}
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// auxiliary function for multiplication.
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// compute k * x as a 540 bit number in base 2^30 (normalized).
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// assumes that k and x are normalized.
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void bn_multiply_long(const bignum256 *k, const bignum256 *x, uint32_t res[18])
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{
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int i, j;
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@ -223,6 +263,7 @@ void bn_multiply_long(const bignum256 *k, const bignum256 *x, uint32_t res[18])
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for (i = 0; i < 9; i++)
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{
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for (j = 0; j <= i; j++) {
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// no overflow, since 9*2^60 < 2^64
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temp += k->val[j] * (uint64_t)x->val[i - j];
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}
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res[i] = temp & 0x3FFFFFFFu;
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@ -232,6 +273,7 @@ void bn_multiply_long(const bignum256 *k, const bignum256 *x, uint32_t res[18])
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for (; i < 17; i++)
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{
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for (j = i - 8; j < 9 ; j++) {
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// no overflow, since 9*2^60 < 2^64
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temp += k->val[j] * (uint64_t)x->val[i - j];
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}
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res[i] = temp & 0x3FFFFFFFu;
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@ -240,13 +282,16 @@ void bn_multiply_long(const bignum256 *k, const bignum256 *x, uint32_t res[18])
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res[17] = temp;
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}
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// auxiliary function for multiplication.
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// reduces res modulo prime.
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// assumes res normalized, res < 2^(30(i-7)) * 2 * prime
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// guarantees res normalized, res < 2^(30(i-8)) * 2 * prime
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void bn_multiply_reduce_step(uint32_t res[18], const bignum256 *prime, uint32_t i) {
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// let k = i-8.
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// invariants:
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// res[0..(i+1)] = k * x (mod prime)
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// 0 <= res < 2^(30k + 256) * (2^31)
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// estimate (res / prime)
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// coef = res / 2^(30k + 256) rounded down
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// on entry:
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// 0 <= res < 2^(30k + 31) * prime
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// estimate coef = (res / prime / 2^30k)
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// by coef = res / 2^(30k + 256) rounded down
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// 0 <= coef < 2^31
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// subtract (coef * 2^(30k) * prime) from res
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// note that we unrolled the first iteration
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@ -257,11 +302,11 @@ void bn_multiply_reduce_step(uint32_t res[18], const bignum256 *prime, uint32_t
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res[i - 8] = temp & 0x3FFFFFFF;
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for (j = 1; j < 9; j++) {
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temp >>= 30;
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// Note: coeff * prime->val <= (2^31-1) * (2^30-1)
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// Note: coeff * prime->val[j] <= (2^31-1) * (2^30-1)
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// Hence, this addition will not underflow.
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temp += 0x1FFFFFFF80000000ull + res[i - 8 + j] - prime->val[j] * (uint64_t)coef;
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res[i - 8 + j] = temp & 0x3FFFFFFF;
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// 0 <= temp < 2^61
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// 0 <= temp < 2^61 + 2^30
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}
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temp >>= 30;
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temp += 0x1FFFFFFF80000000ull + res[i - 8 + j];
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@ -271,18 +316,20 @@ void bn_multiply_reduce_step(uint32_t res[18], const bignum256 *prime, uint32_t
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// and
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// coef * 2^(30k + 256) <= oldres < (coef+1) * 2^(30k + 256)
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// Hence, 0 <= res < 2^30k (2^256 + coef * (2^256 - prime))
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// Since coef * (2^256 - prime) < 2^256, we get
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// 0 <= res < 2^(30k + 226) (2^31)
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// Thus the invariant holds again.
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// < 2^30k (2^256 + 2^31 * 2^224)
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// < 2^30k (2 * prime)
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}
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// auxiliary function for multiplication.
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// reduces x = res modulo prime.
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// assumes res normalized, res < 2^270 * 2 * prime
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// guarantees x partly reduced, i.e., x < 2 * prime
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void bn_multiply_reduce(bignum256 *x, uint32_t res[18], const bignum256 *prime)
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{
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int i;
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// res = k * x is a normalized number (every limb < 2^30)
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// 0 <= res < 2^526.
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// compute modulo p division is only estimated so this may give result greater than prime but not bigger than 2 * prime
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// 0 <= res < 2^270 * 2 * prime.
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for (i = 16; i >= 8; i--) {
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bn_multiply_reduce_step(res, prime, i);
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assert(res[i + 1] == 0);
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@ -294,11 +341,9 @@ void bn_multiply_reduce(bignum256 *x, uint32_t res[18], const bignum256 *prime)
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}
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// Compute x := k * x (mod prime)
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// both inputs must be smaller than 2 * prime.
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// result is reduced to 0 <= x < 2 * prime
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// This only works for primes between 2^256-2^196 and 2^256.
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// this particular implementation accepts inputs up to 2^263 or 128*prime.
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// both inputs must be smaller than 180 * prime.
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// result is partly reduced (0 <= x < 2 * prime)
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// This only works for primes between 2^256-2^224 and 2^256.
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void bn_multiply(const bignum256 *k, bignum256 *x, const bignum256 *prime)
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{
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uint32_t res[18] = {0};
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@ -307,9 +352,11 @@ void bn_multiply(const bignum256 *k, bignum256 *x, const bignum256 *prime)
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MEMSET_BZERO(res, sizeof(res));
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}
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// input x can be any normalized number that fits (0 <= x < 2^270).
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// partly reduce x modulo prime
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// input x does not have to be normalized.
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// x can be any number that fits.
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// prime must be between (2^256 - 2^224) and 2^256
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// result is smaller than 2*prime
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// result is partly reduced, smaller than 2*prime
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void bn_fast_mod(bignum256 *x, const bignum256 *prime)
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{
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int j;
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@ -330,6 +377,8 @@ void bn_fast_mod(bignum256 *x, const bignum256 *prime)
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// square root of x = x^((p+1)/4)
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// http://en.wikipedia.org/wiki/Quadratic_residue#Prime_or_prime_power_modulus
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// assumes x is normalized but not necessarily reduced.
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// guarantees x is reduced
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void bn_sqrt(bignum256 *x, const bignum256 *prime)
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{
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// this method compute x^1/2 = x^(prime+1)/4
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@ -678,10 +727,18 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
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#endif
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void bn_normalize(bignum256 *a) {
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bn_addi(a, 0);
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}
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// add two numbers a = a + b
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// assumes that a, b are normalized
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// guarantees that a is normalized
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void bn_add(bignum256 *a, const bignum256 *b)
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{
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int i;
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uint32_t tmp = 0;
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for (i = 0; i < 9; i++) {
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tmp += a->val[i];
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tmp += a->val[i] + b->val[i];
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a->val[i] = tmp & 0x3FFFFFFF;
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tmp >>= 30;
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}
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@ -697,22 +754,25 @@ void bn_addmod(bignum256 *a, const bignum256 *b, const bignum256 *prime)
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}
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void bn_addi(bignum256 *a, uint32_t b) {
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a->val[0] += b;
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bn_normalize(a);
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int i;
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uint32_t tmp = b;
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for (i = 0; i < 9; i++) {
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tmp += a->val[i];
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a->val[i] = tmp & 0x3FFFFFFF;
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tmp >>= 30;
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}
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}
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void bn_subi(bignum256 *a, uint32_t b, const bignum256 *prime) {
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int i;
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for (i = 0; i < 9; i++) {
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a->val[i] += prime->val[i];
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}
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assert (b <= prime->val[0]);
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// the possible underflow will be taken care of when adding the prime
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a->val[0] -= b;
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bn_fast_mod(a, prime);
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bn_add(a, prime);
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}
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// res = a - b mod prime. More exactly res = a + (2*prime - b).
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// precondition: 0 <= b < 2*prime, 0 <= a < prime
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// res < 3*prime
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// b must be a partly reduced number
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// result is normalized but not reduced.
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void bn_subtractmod(const bignum256 *a, const bignum256 *b, bignum256 *res, const bignum256 *prime)
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{
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int i;
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4
bignum.h
4
bignum.h
@ -53,8 +53,6 @@ int bn_is_equal(const bignum256 *a, const bignum256 *b);
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void bn_cmov(bignum256 *res, int cond, const bignum256 *truecase, const bignum256 *falsecase);
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int bn_bitlen(const bignum256 *a);
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void bn_lshift(bignum256 *a);
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void bn_rshift(bignum256 *a);
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@ -75,6 +73,8 @@ void bn_inverse(bignum256 *x, const bignum256 *prime);
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void bn_normalize(bignum256 *a);
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void bn_add(bignum256 *a, const bignum256 *b);
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void bn_addmod(bignum256 *a, const bignum256 *b, const bignum256 *prime);
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void bn_addi(bignum256 *a, uint32_t b);
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1
bip32.c
1
bip32.c
@ -143,6 +143,7 @@ int hdnode_private_ckd(HDNode *inout, uint32_t i)
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}
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if (!failed) {
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bn_addmod(&a, &b, &default_curve->order);
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bn_mod(&a, &default_curve->order);
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if (bn_is_zero(&a)) {
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failed = true;
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}
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22
ecdsa.c
22
ecdsa.c
@ -287,7 +287,7 @@ void point_jacobian_add(const curve_point *p1, jacobian_curve_point *p2, const e
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bn_fast_mod(&h, prime);
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// h = x1' - x2;
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bn_addmod(&xz, &p2->x, prime);
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bn_add(&xz, &p2->x);
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// xz = x1' + x2
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is_doubling = bn_is_zero(&h) | bn_is_equal(&h, prime);
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@ -296,7 +296,7 @@ void point_jacobian_add(const curve_point *p1, jacobian_curve_point *p2, const e
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bn_subtractmod(&yz, &p2->y, &r, prime);
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// r = y1' - y2;
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bn_addmod(&yz, &p2->y, prime);
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bn_add(&yz, &p2->y);
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// yz = y1' + y2
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r2 = p2->x;
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@ -347,6 +347,7 @@ void point_jacobian_double(jacobian_curve_point *p, const ecdsa_curve *curve) {
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bignum256 az4, m, msq, ysq, xysq;
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const bignum256 *prime = &curve->prime;
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assert (-3 <= curve->a && curve->a <= 0);
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||||
/* usual algorithm:
|
||||
*
|
||||
* lambda = (3((x/z^2)^2 + a) / 2y/z^3) = (3x^2 + az^4)/2yz
|
||||
@ -861,7 +862,7 @@ void uncompress_coords(const ecdsa_curve *curve, uint8_t odd, const bignum256 *x
|
||||
bn_multiply(x, y, &curve->prime); // y is x^2
|
||||
bn_subi(y, -curve->a, &curve->prime); // y is x^2 + a
|
||||
bn_multiply(x, y, &curve->prime); // y is x^3 + ax
|
||||
bn_addmod(y, &curve->b, &curve->prime); // y is x^3 + ax + b
|
||||
bn_add(y, &curve->b); // y is x^3 + ax + b
|
||||
bn_sqrt(y, &curve->prime); // y = sqrt(y)
|
||||
if ((odd & 0x01) != (y->val[0] & 1)) {
|
||||
bn_subtract(&curve->prime, y, y); // y = -y
|
||||
@ -891,7 +892,7 @@ int ecdsa_read_pubkey(const ecdsa_curve *curve, const uint8_t *pub_key, curve_po
|
||||
|
||||
int ecdsa_validate_pubkey(const ecdsa_curve *curve, const curve_point *pub)
|
||||
{
|
||||
bignum256 y_2, x_3_b;
|
||||
bignum256 y_2, x3_ax_b;
|
||||
|
||||
if (point_is_infinity(pub)) {
|
||||
return 0;
|
||||
@ -902,19 +903,20 @@ int ecdsa_validate_pubkey(const ecdsa_curve *curve, const curve_point *pub)
|
||||
}
|
||||
|
||||
memcpy(&y_2, &(pub->y), sizeof(bignum256));
|
||||
memcpy(&x_3_b, &(pub->x), sizeof(bignum256));
|
||||
memcpy(&x3_ax_b, &(pub->x), sizeof(bignum256));
|
||||
|
||||
// y^2
|
||||
bn_multiply(&(pub->y), &y_2, &curve->prime);
|
||||
bn_mod(&y_2, &curve->prime);
|
||||
|
||||
// x^3 + ax + b
|
||||
bn_multiply(&(pub->x), &x_3_b, &curve->prime); // x^2
|
||||
bn_subi(&x_3_b, -curve->a, &curve->prime); // x^2 + a
|
||||
bn_multiply(&(pub->x), &x_3_b, &curve->prime); // x^3 + ax
|
||||
bn_addmod(&x_3_b, &curve->b, &curve->prime); // x^3 + ax + b
|
||||
bn_multiply(&(pub->x), &x3_ax_b, &curve->prime); // x^2
|
||||
bn_subi(&x3_ax_b, -curve->a, &curve->prime); // x^2 + a
|
||||
bn_multiply(&(pub->x), &x3_ax_b, &curve->prime); // x^3 + ax
|
||||
bn_addmod(&x3_ax_b, &curve->b, &curve->prime); // x^3 + ax + b
|
||||
bn_mod(&x3_ax_b, &curve->prime);
|
||||
|
||||
if (!bn_is_equal(&x_3_b, &y_2)) {
|
||||
if (!bn_is_equal(&x3_ax_b, &y_2)) {
|
||||
return 0;
|
||||
}
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user