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-rw-r--r--curve25519-donna.c294
1 files changed, 210 insertions, 84 deletions
diff --git a/curve25519-donna.c b/curve25519-donna.c
index 3309610..ef0b6d1 100644
--- a/curve25519-donna.c
+++ b/curve25519-donna.c
@@ -43,8 +43,7 @@
*
* This is, almost, a clean room reimplementation from the curve25519 paper. It
* uses many of the tricks described therein. Only the crecip function is taken
- * from the sample implementation.
- */
+ * from the sample implementation. */
#include <string.h>
#include <stdint.h>
@@ -63,25 +62,23 @@ typedef int64_t limb;
* significant first. The value of the field element is:
* x[0] + 2^26·x[1] + x^51·x[2] + 2^102·x[3] + ...
*
- * i.e. the limbs are 26, 25, 26, 25, ... bits wide.
- */
+ * i.e. the limbs are 26, 25, 26, 25, ... bits wide. */
/* Sum two numbers: output += in */
static void fsum(limb *output, const limb *in) {
unsigned i;
for (i = 0; i < 10; i += 2) {
- output[0+i] = (output[0+i] + in[0+i]);
- output[1+i] = (output[1+i] + in[1+i]);
+ output[0+i] = output[0+i] + in[0+i];
+ output[1+i] = output[1+i] + in[1+i];
}
}
/* Find the difference of two numbers: output = in - output
- * (note the order of the arguments!)
- */
+ * (note the order of the arguments!). */
static void fdifference(limb *output, const limb *in) {
unsigned i;
for (i = 0; i < 10; ++i) {
- output[i] = (in[i] - output[i]);
+ output[i] = in[i] - output[i];
}
}
@@ -97,7 +94,8 @@ static void fscalar_product(limb *output, const limb *in, const limb scalar) {
*
* output must be distinct to both inputs. The inputs are reduced coefficient
* form, the output is not.
- */
+ *
+ * output[x] <= 14 * the largest product of the input limbs. */
static void fproduct(limb *output, const limb *in2, const limb *in) {
output[0] = ((limb) ((s32) in2[0])) * ((s32) in[0]);
output[1] = ((limb) ((s32) in2[0])) * ((s32) in[1]) +
@@ -201,9 +199,15 @@ static void fproduct(limb *output, const limb *in2, const limb *in) {
output[18] = 2 * ((limb) ((s32) in2[9])) * ((s32) in[9]);
}
-/* Reduce a long form to a short form by taking the input mod 2^255 - 19. */
+/* Reduce a long form to a short form by taking the input mod 2^255 - 19.
+ *
+ * On entry: |output[i]| < 14*2^54
+ * On exit: |output[0..8]| < 280*2^54 */
static void freduce_degree(limb *output) {
- /* Each of these shifts and adds ends up multiplying the value by 19. */
+ /* Each of these shifts and adds ends up multiplying the value by 19.
+ *
+ * For output[0..8], the absolute entry value is < 14*2^54 and we add, at
+ * most, 19*14*2^54 thus, on exit, |output[0..8]| < 280*2^54. */
output[8] += output[18] << 4;
output[8] += output[18] << 1;
output[8] += output[18];
@@ -237,11 +241,13 @@ static void freduce_degree(limb *output) {
#error "This code only works on a two's complement system"
#endif
-/* return v / 2^26, using only shifts and adds. */
+/* return v / 2^26, using only shifts and adds.
+ *
+ * On entry: v can take any value. */
static inline limb
div_by_2_26(const limb v)
{
- /* High word of v; no shift needed*/
+ /* High word of v; no shift needed. */
const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32);
/* Set to all 1s if v was negative; else set to 0s. */
const int32_t sign = ((int32_t) highword) >> 31;
@@ -251,7 +257,9 @@ div_by_2_26(const limb v)
return (v + roundoff) >> 26;
}
-/* return v / (2^25), using only shifts and adds. */
+/* return v / (2^25), using only shifts and adds.
+ *
+ * On entry: v can take any value. */
static inline limb
div_by_2_25(const limb v)
{
@@ -265,17 +273,9 @@ div_by_2_25(const limb v)
return (v + roundoff) >> 25;
}
-static inline s32
-div_s32_by_2_25(const s32 v)
-{
- const s32 roundoff = ((uint32_t)(v >> 31)) >> 7;
- return (v + roundoff) >> 25;
-}
-
/* Reduce all coefficients of the short form input so that |x| < 2^26.
*
- * On entry: |output[i]| < 2^62
- */
+ * On entry: |output[i]| < 280*2^54 */
static void freduce_coefficients(limb *output) {
unsigned i;
@@ -283,56 +283,65 @@ static void freduce_coefficients(limb *output) {
for (i = 0; i < 10; i += 2) {
limb over = div_by_2_26(output[i]);
+ /* The entry condition (that |output[i]| < 280*2^54) means that over is, at
+ * most, 280*2^28 in the first iteration of this loop. This is added to the
+ * next limb and we can approximate the resulting bound of that limb by
+ * 281*2^54. */
output[i] -= over << 26;
output[i+1] += over;
+ /* For the first iteration, |output[i+1]| < 281*2^54, thus |over| <
+ * 281*2^29. When this is added to the next limb, the resulting bound can
+ * be approximated as 281*2^54.
+ *
+ * For subsequent iterations of the loop, 281*2^54 remains a conservative
+ * bound and no overflow occurs. */
over = div_by_2_25(output[i+1]);
output[i+1] -= over << 25;
output[i+2] += over;
}
- /* Now |output[10]| < 2 ^ 38 and all other coefficients are reduced. */
+ /* Now |output[10]| < 281*2^29 and all other coefficients are reduced. */
output[0] += output[10] << 4;
output[0] += output[10] << 1;
output[0] += output[10];
output[10] = 0;
- /* Now output[1..9] are reduced, and |output[0]| < 2^26 + 19 * 2^38
- * So |over| will be no more than 77825 */
+ /* Now output[1..9] are reduced, and |output[0]| < 2^26 + 19*281*2^29
+ * So |over| will be no more than 2^16. */
{
limb over = div_by_2_26(output[0]);
output[0] -= over << 26;
output[1] += over;
}
- /* Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 77825
- * So |over| will be no more than 1. */
- {
- /* output[1] fits in 32 bits, so we can use div_s32_by_2_25 here. */
- s32 over32 = div_s32_by_2_25((s32) output[1]);
- output[1] -= over32 << 25;
- output[2] += over32;
- }
-
- /* Finally, output[0,1,3..9] are reduced, and output[2] is "nearly reduced":
- * we have |output[2]| <= 2^26. This is good enough for all of our math,
- * but it will require an extra freduce_coefficients before fcontract. */
+ /* Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 2^16 < 2^26. The
+ * bound on |output[1]| is sufficient to meet our needs. */
}
/* A helpful wrapper around fproduct: output = in * in2.
*
- * output must be distinct to both inputs. The output is reduced degree and
- * reduced coefficient.
- */
+ * On entry: |in[i]| < 2^27 and |in2[i]| < 2^27.
+ *
+ * output must be distinct to both inputs. The output is reduced degree
+ * (indeed, one need only provide storage for 10 limbs) and |output[i]| < 2^26. */
static void
fmul(limb *output, const limb *in, const limb *in2) {
limb t[19];
fproduct(t, in, in2);
+ /* |t[i]| < 14*2^54 */
freduce_degree(t);
freduce_coefficients(t);
+ /* |t[i]| < 2^26 */
memcpy(output, t, sizeof(limb) * 10);
}
+/* Square a number: output = in**2
+ *
+ * output must be distinct from the input. The inputs are reduced coefficient
+ * form, the output is not.
+ *
+ * output[x] <= 14 * the largest product of the input limbs. */
static void fsquare_inner(limb *output, const limb *in) {
output[0] = ((limb) ((s32) in[0])) * ((s32) in[0]);
output[1] = 2 * ((limb) ((s32) in[0])) * ((s32) in[1]);
@@ -391,12 +400,23 @@ static void fsquare_inner(limb *output, const limb *in) {
output[18] = 2 * ((limb) ((s32) in[9])) * ((s32) in[9]);
}
+/* fsquare sets output = in^2.
+ *
+ * On entry: The |in| argument is in reduced coefficients form and |in[i]| <
+ * 2^27.
+ *
+ * On exit: The |output| argument is in reduced coefficients form (indeed, one
+ * need only provide storage for 10 limbs) and |out[i]| < 2^26. */
static void
fsquare(limb *output, const limb *in) {
limb t[19];
fsquare_inner(t, in);
+ /* |t[i]| < 14*2^54 because the largest product of two limbs will be <
+ * 2^(27+27) and fsquare_inner adds together, at most, 14 of those
+ * products. */
freduce_degree(t);
freduce_coefficients(t);
+ /* |t[i]| < 2^26 */
memcpy(output, t, sizeof(limb) * 10);
}
@@ -417,7 +437,7 @@ fexpand(limb *output, const u8 *input) {
F(6, 19, 1, 0x3ffffff);
F(7, 22, 3, 0x1ffffff);
F(8, 25, 4, 0x3ffffff);
- F(9, 28, 6, 0x3ffffff);
+ F(9, 28, 6, 0x1ffffff);
#undef F
}
@@ -425,60 +445,143 @@ fexpand(limb *output, const u8 *input) {
#error "This code only works when >> does sign-extension on negative numbers"
#endif
+/* s32_eq returns 0xffffffff iff a == b and zero otherwise. */
+static s32 s32_eq(s32 a, s32 b) {
+ a = ~(a ^ b);
+ a &= a << 16;
+ a &= a << 8;
+ a &= a << 4;
+ a &= a << 2;
+ a &= a << 1;
+ return a >> 31;
+}
+
+/* s32_gte returns 0xffffffff if a >= b and zero otherwise, where a and b are
+ * both non-negative. */
+static s32 s32_gte(s32 a, s32 b) {
+ a -= b;
+ /* a >= 0 iff a >= b. */
+ return ~(a >> 31);
+}
+
/* Take a fully reduced polynomial form number and contract it into a
- * little-endian, 32-byte array
- */
+ * little-endian, 32-byte array.
+ *
+ * On entry: |input_limbs[i]| < 2^26 */
static void
-fcontract(u8 *output, limb *input) {
+fcontract(u8 *output, limb *input_limbs) {
int i;
int j;
+ s32 input[10];
+ s32 mask;
+
+ /* |input_limbs[i]| < 2^26, so it's valid to convert to an s32. */
+ for (i = 0; i < 10; i++) {
+ input[i] = input_limbs[i];
+ }
for (j = 0; j < 2; ++j) {
for (i = 0; i < 9; ++i) {
if ((i & 1) == 1) {
- /* This calculation is a time-invariant way to make input[i] positive
- by borrowing from the next-larger limb.
- */
- const s32 mask = (s32)(input[i]) >> 31;
- const s32 carry = -(((s32)(input[i]) & mask) >> 25);
- input[i] = (s32)(input[i]) + (carry << 25);
- input[i+1] = (s32)(input[i+1]) - carry;
+ /* This calculation is a time-invariant way to make input[i]
+ * non-negative by borrowing from the next-larger limb. */
+ const s32 mask = input[i] >> 31;
+ const s32 carry = -((input[i] & mask) >> 25);
+ input[i] = input[i] + (carry << 25);
+ input[i+1] = input[i+1] - carry;
} else {
- const s32 mask = (s32)(input[i]) >> 31;
- const s32 carry = -(((s32)(input[i]) & mask) >> 26);
- input[i] = (s32)(input[i]) + (carry << 26);
- input[i+1] = (s32)(input[i+1]) - carry;
+ const s32 mask = input[i] >> 31;
+ const s32 carry = -((input[i] & mask) >> 26);
+ input[i] = input[i] + (carry << 26);
+ input[i+1] = input[i+1] - carry;
}
}
+
+ /* There's no greater limb for input[9] to borrow from, but we can multiply
+ * by 19 and borrow from input[0], which is valid mod 2^255-19. */
{
- const s32 mask = (s32)(input[9]) >> 31;
- const s32 carry = -(((s32)(input[9]) & mask) >> 25);
- input[9] = (s32)(input[9]) + (carry << 25);
- input[0] = (s32)(input[0]) - (carry * 19);
+ const s32 mask = input[9] >> 31;
+ const s32 carry = -((input[9] & mask) >> 25);
+ input[9] = input[9] + (carry << 25);
+ input[0] = input[0] - (carry * 19);
}
+
+ /* After the first iteration, input[1..9] are non-negative and fit within
+ * 25 or 26 bits, depending on position. However, input[0] may be
+ * negative. */
}
/* The first borrow-propagation pass above ended with every limb
except (possibly) input[0] non-negative.
- Since each input limb except input[0] is decreased by at most 1
- by a borrow-propagation pass, the second borrow-propagation pass
- could only have wrapped around to decrease input[0] again if the
- first pass left input[0] negative *and* input[1] through input[9]
- were all zero. In that case, input[1] is now 2^25 - 1, and this
- last borrow-propagation step will leave input[1] non-negative.
- */
+ If input[0] was negative after the first pass, then it was because of a
+ carry from input[9]. On entry, input[9] < 2^26 so the carry was, at most,
+ one, since (2**26-1) >> 25 = 1. Thus input[0] >= -19.
+
+ In the second pass, each limb is decreased by at most one. Thus the second
+ borrow-propagation pass could only have wrapped around to decrease
+ input[0] again if the first pass left input[0] negative *and* input[1]
+ through input[9] were all zero. In that case, input[1] is now 2^25 - 1,
+ and this last borrow-propagation step will leave input[1] non-negative. */
{
- const s32 mask = (s32)(input[0]) >> 31;
- const s32 carry = -(((s32)(input[0]) & mask) >> 26);
- input[0] = (s32)(input[0]) + (carry << 26);
- input[1] = (s32)(input[1]) - carry;
+ const s32 mask = input[0] >> 31;
+ const s32 carry = -((input[0] & mask) >> 26);
+ input[0] = input[0] + (carry << 26);
+ input[1] = input[1] - carry;
+ }
+
+ /* All input[i] are now non-negative. However, there might be values between
+ * 2^25 and 2^26 in a limb which is, nominally, 25 bits wide. */
+ for (j = 0; j < 2; j++) {
+ for (i = 0; i < 9; i++) {
+ if ((i & 1) == 1) {
+ const s32 carry = input[i] >> 25;
+ input[i] &= 0x1ffffff;
+ input[i+1] += carry;
+ } else {
+ const s32 carry = input[i] >> 26;
+ input[i] &= 0x3ffffff;
+ input[i+1] += carry;
+ }
+ }
+
+ {
+ const s32 carry = input[9] >> 25;
+ input[9] &= 0x1ffffff;
+ input[0] += 19*carry;
+ }
+ }
+
+ /* If the first carry-chain pass, just above, ended up with a carry from
+ * input[9], and that caused input[0] to be out-of-bounds, then input[0] was
+ * < 2^26 + 2*19, because the carry was, at most, two.
+ *
+ * If the second pass carried from input[9] again then input[0] is < 2*19 and
+ * the input[9] -> input[0] carry didn't push input[0] out of bounds. */
+
+ /* It still remains the case that input might be between 2^255-19 and 2^255.
+ * In this case, input[1..9] must take their maximum value and input[0] must
+ * be >= (2^255-19) & 0x3ffffff, which is 0x3ffffed. */
+ mask = s32_gte(input[0], 0x3ffffed);
+ for (i = 1; i < 10; i++) {
+ if ((i & 1) == 1) {
+ mask &= s32_eq(input[i], 0x1ffffff);
+ } else {
+ mask &= s32_eq(input[i], 0x3ffffff);
+ }
}
- /* Both passes through the above loop, plus the last 0-to-1 step, are
- necessary: if input[9] is -1 and input[0] through input[8] are 0,
- negative values will remain in the array until the end.
- */
+ /* mask is either 0xffffffff (if input >= 2^255-19) and zero otherwise. Thus
+ * this conditionally subtracts 2^255-19. */
+ input[0] -= mask & 0x3ffffed;
+
+ for (i = 1; i < 10; i++) {
+ if ((i & 1) == 1) {
+ input[i] -= mask & 0x1ffffff;
+ } else {
+ input[i] -= mask & 0x3ffffff;
+ }
+ }
input[1] <<= 2;
input[2] <<= 3;
@@ -516,7 +619,9 @@ fcontract(u8 *output, limb *input) {
* x z: short form, destroyed
* xprime zprime: short form, destroyed
* qmqp: short form, preserved
- */
+ *
+ * On entry and exit, the absolute value of the limbs of all inputs and outputs
+ * are < 2^26. */
static void fmonty(limb *x2, limb *z2, /* output 2Q */
limb *x3, limb *z3, /* output Q + Q' */
limb *x, limb *z, /* input Q */
@@ -527,43 +632,69 @@ static void fmonty(limb *x2, limb *z2, /* output 2Q */
memcpy(origx, x, 10 * sizeof(limb));
fsum(x, z);
+ /* |x[i]| < 2^27 */
fdifference(z, origx); /* does x - z */
+ /* |z[i]| < 2^27 */
memcpy(origxprime, xprime, sizeof(limb) * 10);
fsum(xprime, zprime);
+ /* |xprime[i]| < 2^27 */
fdifference(zprime, origxprime);
+ /* |zprime[i]| < 2^27 */
fproduct(xxprime, xprime, z);
+ /* |xxprime[i]| < 14*2^54: the largest product of two limbs will be <
+ * 2^(27+27) and fproduct adds together, at most, 14 of those products.
+ * (Approximating that to 2^58 doesn't work out.) */
fproduct(zzprime, x, zprime);
+ /* |zzprime[i]| < 14*2^54 */
freduce_degree(xxprime);
freduce_coefficients(xxprime);
+ /* |xxprime[i]| < 2^26 */
freduce_degree(zzprime);
freduce_coefficients(zzprime);
+ /* |zzprime[i]| < 2^26 */
memcpy(origxprime, xxprime, sizeof(limb) * 10);
fsum(xxprime, zzprime);
+ /* |xxprime[i]| < 2^27 */
fdifference(zzprime, origxprime);
+ /* |zzprime[i]| < 2^27 */
fsquare(xxxprime, xxprime);
+ /* |xxxprime[i]| < 2^26 */
fsquare(zzzprime, zzprime);
+ /* |zzzprime[i]| < 2^26 */
fproduct(zzprime, zzzprime, qmqp);
+ /* |zzprime[i]| < 14*2^52 */
freduce_degree(zzprime);
freduce_coefficients(zzprime);
+ /* |zzprime[i]| < 2^26 */
memcpy(x3, xxxprime, sizeof(limb) * 10);
memcpy(z3, zzprime, sizeof(limb) * 10);
fsquare(xx, x);
+ /* |xx[i]| < 2^26 */
fsquare(zz, z);
+ /* |zz[i]| < 2^26 */
fproduct(x2, xx, zz);
+ /* |x2[i]| < 14*2^52 */
freduce_degree(x2);
freduce_coefficients(x2);
+ /* |x2[i]| < 2^26 */
fdifference(zz, xx); /* does zz = xx - zz */
+ /* |zz[i]| < 2^27 */
memset(zzz + 10, 0, sizeof(limb) * 9);
fscalar_product(zzz, zz, 121665);
+ /* |zzz[i]| < 2^(27+17) */
/* No need to call freduce_degree here:
fscalar_product doesn't increase the degree of its input. */
freduce_coefficients(zzz);
+ /* |zzz[i]| < 2^26 */
fsum(zzz, xx);
+ /* |zzz[i]| < 2^27 */
fproduct(z2, zz, zzz);
+ /* |z2[i]| < 14*2^(26+27) */
freduce_degree(z2);
freduce_coefficients(z2);
+ /* |z2|i| < 2^26 */
}
/* Conditionally swap two reduced-form limb arrays if 'iswap' is 1, but leave
@@ -574,8 +705,7 @@ static void fmonty(limb *x2, limb *z2, /* output 2Q */
* wrong results. Also, the two limb arrays must be in reduced-coefficient,
* reduced-degree form: the values in a[10..19] or b[10..19] aren't swapped,
* and all all values in a[0..9],b[0..9] must have magnitude less than
- * INT32_MAX.
- */
+ * INT32_MAX. */
static void
swap_conditional(limb a[19], limb b[19], limb iswap) {
unsigned i;
@@ -592,8 +722,7 @@ swap_conditional(limb a[19], limb b[19], limb iswap) {
*
* resultx/resultz: the x coordinate of the resulting curve point (short form)
* n: a little endian, 32-byte number
- * q: a point of the curve (short form)
- */
+ * q: a point of the curve (short form) */
static void
cmult(limb *resultx, limb *resultz, const u8 *n, const limb *q) {
limb a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0};
@@ -711,8 +840,6 @@ crecip(limb *out, const limb *z) {
/* 2^255 - 21 */ fmul(out,t1,z11);
}
-int curve25519_donna(u8 *, const u8 *, const u8 *);
-
int
curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) {
limb bp[10], x[10], z[11], zmone[10];
@@ -728,7 +855,6 @@ curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) {
cmult(x, z, e, bp);
crecip(zmone, z);
fmul(z, x, zmone);
- freduce_coefficients(z);
fcontract(mypublic, z);
return 0;
}
generated by cgit v1.2.3 (git 2.39.1) at 2024-11-27 10:57:33 +0000