commit 8cc086059253347c82ebb1ff072abde56cd1da1a Author: Nick Mathewson <nickm@torproject.org> Date: Tue Jul 15 15:42:20 2014 +0200 Update to latest curve25519-donna32 --- changes/curve25519-donna32-bug | 10 + src/ext/curve25519_donna/curve25519-donna.c | 290 ++++++++++++++++++++------- 2 files changed, 223 insertions(+), 77 deletions(-) diff --git a/changes/curve25519-donna32-bug b/changes/curve25519-donna32-bug new file mode 100644 index 0000000..54892d7 --- /dev/null +++ b/changes/curve25519-donna32-bug @@ -0,0 +1,10 @@ + o Major bugfixes: + + - Fix a bug in the bounds-checking in the 32-bit curve25519-donna + implementation that caused incorrect results on 32-bit + implementations when certain malformed inputs were used along with + a small class of private ntor keys. This bug does not currently + appear to allow an attacker to learn private keys or impersonate a + Tor server, but it could provide a means to distinguish 32-bit Tor + implementations from 64-bit Tor implementations. + diff --git a/src/ext/curve25519_donna/curve25519-donna.c b/src/ext/curve25519_donna/curve25519-donna.c index 5c6821c..75c9f91 100644 --- a/src/ext/curve25519_donna/curve25519-donna.c +++ b/src/ext/curve25519_donna/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 "orconfig.h" @@ -61,25 +60,23 @@ 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]; } } @@ -95,7 +92,8 @@ 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]) + @@ -199,9 +197,15 @@ 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]; @@ -235,11 +239,13 @@ 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; @@ -249,7 +255,9 @@ 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) { @@ -263,6 +271,9 @@ div_by_2_25(const limb v) return (v + roundoff) >> 25; } +/* return v / (2^25), using only shifts and adds. + * + * On entry: v can take any value. */ static inline s32 div_s32_by_2_25(const s32 v) { @@ -272,8 +283,7 @@ div_s32_by_2_25(const s32 v) /* 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; @@ -281,56 +291,65 @@ 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]); @@ -389,12 +408,23 @@ 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); } @@ -423,60 +453,143 @@ 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; } - /* 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. - */ + /* 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); + } + } + + /* 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; @@ -514,7 +627,9 @@ 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 */ @@ -525,43 +640,69 @@ static void fmonty(limb *x2, limb *z2, /* output 2Q */ memcpy(origx, x, 10 * sizeof(limb)); fsum(x, z); - fdifference(z, origx); // does 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 @@ -572,8 +713,7 @@ 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; @@ -590,8 +730,7 @@ 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}; @@ -709,8 +848,6 @@ 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]; @@ -726,7 +863,6 @@ 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; }