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Diffstat (limited to 'src/crypto/x25519.c')
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diff --git a/src/crypto/x25519.c b/src/crypto/x25519.c new file mode 100644 index 000000000..750a2a71c --- /dev/null +++ b/src/crypto/x25519.c @@ -0,0 +1,808 @@ +/* + * Copyright (C) 2024 Michael Brown <mbrown@fensystems.co.uk>. + * + * This program is free software; you can redistribute it and/or + * modify it under the terms of the GNU General Public License as + * published by the Free Software Foundation; either version 2 of the + * License, or any later version. + * + * This program is distributed in the hope that it will be useful, but + * WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + * General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with this program; if not, write to the Free Software + * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA + * 02110-1301, USA. + * + * You can also choose to distribute this program under the terms of + * the Unmodified Binary Distribution Licence (as given in the file + * COPYING.UBDL), provided that you have satisfied its requirements. + */ + +FILE_LICENCE ( GPL2_OR_LATER_OR_UBDL ); + +/** @file + * + * X25519 key exchange + * + * This implementation is inspired by and partially based upon the + * paper "Implementing Curve25519/X25519: A Tutorial on Elliptic Curve + * Cryptography" by Martin Kleppmann, available for download from + * https://www.cl.cam.ac.uk/teaching/2122/Crypto/curve25519.pdf + * + * The underlying modular addition, subtraction, and multiplication + * operations are completely redesigned for substantially improved + * efficiency compared to the TweetNaCl implementation studied in that + * paper. + * + * TweetNaCl iPXE + * --------- ---- + * + * Storage size of each big integer 128 40 + * (in bytes) + * + * Stack usage for key exchange 1144 360 + * (in bytes, large objects only) + * + * Cost of big integer addition 16 5 + * (in number of 64-bit additions) + * + * Cost of big integer multiplication 273 31 + * (in number of 64-bit multiplications) + * + * The implementation is constant-time (provided that the underlying + * big integer operations are also constant-time). + */ + +#include <stdint.h> +#include <string.h> +#include <assert.h> +#include <ipxe/init.h> +#include <ipxe/x25519.h> + +/** X25519 reduction constant + * + * The X25519 field prime is p=2^255-19. This gives us: + * + * p = 2^255 - 19 + * 2^255 = p + 19 + * 2^255 = 19 (mod p) + * k * 2^255 = k * 19 (mod p) + * + * We can therefore reduce a value modulo p by taking the high-order + * bits of the value from bit 255 and above, multiplying by 19, and + * adding this to the low-order 255 bits of the value. + * + * This would be cumbersome to do in practice since it would require + * partitioning the value at a 255-bit boundary (and hence would + * require some shifting and masking operations). However, we can + * note that: + * + * k * 2^255 = k * 19 (mod p) + * k * 2 * 2^255 = k * 2 * 19 (mod p) + * k * 2^256 = k * 38 (mod p) + * + * We can therefore simplify the reduction to taking the high order + * bits of the value from bit 256 and above, multiplying by 38, and + * adding this to the low-order 256 bits of the value. + * + * Since 256 will inevitably be a multiple of the big integer element + * size (typically 32 or 64 bits), this avoids the need to perform any + * shifting or masking operations. + */ +#define X25519_REDUCE_256 38 + +/** X25519 multiplication step 1 result + * + * Step 1 of X25519 multiplication is to compute the product of two + * X25519 unsigned 258-bit integers. + * + * Both multiplication inputs are limited to 258 bits, and so the + * product will have at most 516 bits. + */ +union x25519_multiply_step1 { + /** Raw product + * + * Big integer multiplication produces a result with a number + * of elements equal to the sum of the number of elements in + * each input. + */ + bigint_t ( X25519_SIZE + X25519_SIZE ) product; + /** Partition into low-order and high-order bits + * + * Reduction modulo p requires separating the low-order 256 + * bits from the remaining high-order bits. + * + * Since the value will never exceed 516 bits (see above), + * there will be at most 260 high-order bits. + */ + struct { + /** Low-order 256 bits */ + bigint_t ( bigint_required_size ( ( 256 /* bits */ + 7 ) / 8 ) ) + low_256bit; + /** High-order 260 bits */ + bigint_t ( bigint_required_size ( ( 260 /* bits */ + 7 ) / 8 ) ) + high_260bit; + } __attribute__ (( packed )) parts; +}; + +/** X25519 multiplication step 2 result + * + * Step 2 of X25519 multiplication is to multiply the high-order 260 + * bits from step 1 with the 6-bit reduction constant 38, and to add + * this to the low-order 256 bits from step 1. + * + * The multiplication inputs are limited to 260 and 6 bits + * respectively, and so the product will have at most 266 bits. After + * adding the low-order 256 bits from step 1, the result will have at + * most 267 bits. + */ +union x25519_multiply_step2 { + /** Raw product + * + * Big integer multiplication produces a result with a number + * of elements equal to the sum of the number of elements in + * each input. + */ + bigint_t ( bigint_required_size ( ( 260 /* bits */ + 7 ) / 8 ) + + bigint_required_size ( ( 6 /* bits */ + 7 ) / 8 ) ) product; + /** Big integer value + * + * The value will never exceed 267 bits (see above), and so + * may be consumed as a normal X25519 big integer. + */ + x25519_t value; + /** Partition into low-order and high-order bits + * + * Reduction modulo p requires separating the low-order 256 + * bits from the remaining high-order bits. + * + * Since the value will never exceed 267 bits (see above), + * there will be at most 11 high-order bits. + */ + struct { + /** Low-order 256 bits */ + bigint_t ( bigint_required_size ( ( 256 /* bits */ + 7 ) / 8 ) ) + low_256bit; + /** High-order 11 bits */ + bigint_t ( bigint_required_size ( ( 11 /* bits */ + 7 ) / 8 ) ) + high_11bit; + } __attribute__ (( packed )) parts; +}; + +/** X25519 multiplication step 3 result + * + * Step 3 of X25519 multiplication is to multiply the high-order 11 + * bits from step 2 with the 6-bit reduction constant 38, and to add + * this to the low-order 256 bits from step 2. + * + * The multiplication inputs are limited to 11 and 6 bits + * respectively, and so the product will have at most 17 bits. After + * adding the low-order 256 bits from step 2, the result will have at + * most 257 bits. + */ +union x25519_multiply_step3 { + /** Raw product + * + * Big integer multiplication produces a result with a number + * of elements equal to the sum of the number of elements in + * each input. + */ + bigint_t ( bigint_required_size ( ( 11 /* bits */ + 7 ) / 8 ) + + bigint_required_size ( ( 6 /* bits */ + 7 ) / 8 ) ) product; + /** Big integer value + * + * The value will never exceed 267 bits (see above), and so + * may be consumed as a normal X25519 big integer. + */ + x25519_t value; +}; + +/** X25519 multiplication temporary working space + * + * We overlap the buffers used by each step of the multiplication + * calculation to reduce the total stack space required: + * + * |--------------------------------------------------------| + * | <- pad -> | <------------ step 1 result -------------> | + * | | <- low 256 bits -> | <-- high 260 bits --> | + * | <------- step 2 result ------> | <-- step 3 result --> | + * |--------------------------------------------------------| + */ +union x25519_multiply_workspace { + /** Step 1 result */ + struct { + /** Padding to avoid collision between steps 1 and 2 + * + * The step 2 multiplication consumes the high 260 + * bits of step 1, and so the step 2 multiplication + * result must not overlap this portion of the step 1 + * result. + */ + uint8_t pad[ sizeof ( union x25519_multiply_step2 ) - + offsetof ( union x25519_multiply_step1, + parts.high_260bit ) ]; + /** Step 1 result */ + union x25519_multiply_step1 step1; + } __attribute__ (( packed )); + /** Steps 2 and 3 results */ + struct { + /** Step 2 result */ + union x25519_multiply_step2 step2; + /** Step 3 result */ + union x25519_multiply_step3 step3; + } __attribute__ (( packed )); +}; + +/** An X25519 elliptic curve point in projective coordinates + * + * A point (x,y) on the Montgomery curve used in X25519 is represented + * using projective coordinates (X/Z,Y/Z) so that intermediate + * calculations may be performed on both numerator and denominator + * separately, with the division step performed only once at the end + * of the calculation. + * + * The group operation calculation is performed using a Montgomery + * ladder as: + * + * X[2i] = ( X[i]^2 - Z[i]^2 )^2 + * X[2i+1] = ( X[i] * X[i+1] - Z[i] * Z[i+1] )^2 + * Z[2i] = 4 * X[i] * Z[i] * ( X[i]^2 + A * X[i] * Z[i] + Z[i]^2 ) + * Z[2i+1] = X[0] * ( X[i] * Z[i+1] - X[i+1] * Z[i] ) ^ 2 + * + * It is therefore not necessary to store (or use) the value of Y. + */ +struct x25519_projective { + /** X coordinate */ + union x25519_quad257 X; + /** Z coordinate */ + union x25519_quad257 Z; +}; + +/** An X25519 Montgomery ladder step */ +struct x25519_step { + /** X[n]/Z[n] */ + struct x25519_projective x_n; + /** X[n+1]/Z[n+1] */ + struct x25519_projective x_n1; +}; + +/** Constant p=2^255-19 (the finite field prime) */ +static const uint8_t x25519_p_raw[] = { + 0x7f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, + 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, + 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, + 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xed +}; + +/** Constant p=2^255-19 (the finite field prime) */ +static x25519_t x25519_p; + +/** Constant 2p=2^256-38 */ +static x25519_t x25519_2p; + +/** Constant 4p=2^257-76 */ +static x25519_t x25519_4p; + +/** Reduction constant (used during multiplication) */ +static const uint8_t x25519_reduce_256_raw[] = { X25519_REDUCE_256 }; + +/** Reduction constant (used during multiplication) */ +static bigint_t ( bigint_required_size ( sizeof ( x25519_reduce_256_raw ) ) ) + x25519_reduce_256; + +/** Constant 121665 (used in the Montgomery ladder) */ +static const uint8_t x25519_121665_raw[] = { 0x01, 0xdb, 0x41 }; + +/** Constant 121665 (used in the Montgomery ladder) */ +static union x25519_oct258 x25519_121665; + +/** + * Initialise constants + * + */ +static void x25519_init_constants ( void ) { + + /* Construct constant p */ + bigint_init ( &x25519_p, x25519_p_raw, sizeof ( x25519_p_raw ) ); + + /* Construct constant 2p */ + bigint_copy ( &x25519_p, &x25519_2p ); + bigint_add ( &x25519_p, &x25519_2p ); + + /* Construct constant 4p */ + bigint_copy ( &x25519_2p, &x25519_4p ); + bigint_add ( &x25519_2p, &x25519_4p ); + + /* Construct reduction constant */ + bigint_init ( &x25519_reduce_256, x25519_reduce_256_raw, + sizeof ( x25519_reduce_256_raw ) ); + + /* Construct constant 121665 */ + bigint_init ( &x25519_121665.value, x25519_121665_raw, + sizeof ( x25519_121665_raw ) ); +} + +/** Initialisation function */ +struct init_fn x25519_init_fn __init_fn ( INIT_NORMAL ) = { + .initialise = x25519_init_constants, +}; + +/** + * Add big integers modulo field prime + * + * @v augend Big integer to add + * @v addend Big integer to add + * @v result Big integer to hold result (may overlap augend) + */ +static inline __attribute__ (( always_inline )) void +x25519_add ( const union x25519_quad257 *augend, + const union x25519_quad257 *addend, + union x25519_oct258 *result ) { + int copy; + + /* Copy augend if necessary */ + copy = ( result != &augend->oct258 ); + build_assert ( __builtin_constant_p ( copy ) ); + if ( copy ) { + build_assert ( result != &addend->oct258 ); + bigint_copy ( &augend->oct258.value, &result->value ); + } + + /* Perform addition + * + * Both inputs are in the range [0,4p-1] and the resulting + * sum is therefore in the range [0,8p-2]. + * + * This range lies within the range [0,8p-1] and the result is + * therefore a valid X25519 unsigned 258-bit integer, as + * required. + */ + bigint_add ( &addend->value, &result->value ); +} + +/** + * Subtract big integers modulo field prime + * + * @v minuend Big integer from which to subtract + * @v subtrahend Big integer to subtract + * @v result Big integer to hold result (may overlap minuend) + */ +static inline __attribute__ (( always_inline )) void +x25519_subtract ( const union x25519_quad257 *minuend, + const union x25519_quad257 *subtrahend, + union x25519_oct258 *result ) { + int copy; + + /* Copy minuend if necessary */ + copy = ( result != &minuend->oct258 ); + build_assert ( __builtin_constant_p ( copy ) ); + if ( copy ) { + build_assert ( result != &subtrahend->oct258 ); + bigint_copy ( &minuend->oct258.value, &result->value ); + } + + /* Perform subtraction + * + * Both inputs are in the range [0,4p-1] and the resulting + * difference is therefore in the range [1-4p,4p-1]. + * + * This range lies partially outside the range [0,8p-1] and + * the result is therefore not yet a valid X25519 unsigned + * 258-bit integer. + */ + bigint_subtract ( &subtrahend->value, &result->value ); + + /* Add constant multiple of field prime p + * + * Add the constant 4p to the result. This brings the result + * within the range [1,8p-1] (without changing the value + * modulo p). + * + * This range lies within the range [0,8p-1] and the result is + * therefore now a valid X25519 unsigned 258-bit integer, as + * required. + */ + bigint_add ( &x25519_4p, &result->value ); +} + +/** + * Multiply big integers modulo field prime + * + * @v multiplicand Big integer to be multiplied + * @v multiplier Big integer to be multiplied + * @v result Big integer to hold result (may overlap either input) + */ +void x25519_multiply ( const union x25519_oct258 *multiplicand, + const union x25519_oct258 *multiplier, + union x25519_quad257 *result ) { + union x25519_multiply_workspace tmp; + union x25519_multiply_step1 *step1 = &tmp.step1; + union x25519_multiply_step2 *step2 = &tmp.step2; + union x25519_multiply_step3 *step3 = &tmp.step3; + + /* Step 1: perform raw multiplication + * + * step1 = multiplicand * multiplier + * + * Both inputs are 258-bit numbers and the step 1 result is + * therefore 258+258=516 bits. + */ + static_assert ( sizeof ( step1->product ) >= sizeof ( step1->parts ) ); + bigint_multiply ( &multiplicand->value, &multiplier->value, + &step1->product ); + + /* Step 2: reduce high-order 516-256=260 bits of step 1 result + * + * Use the identity 2^256=38 (mod p) to reduce the high-order + * bits of the step 1 result. We split the 516-bit result + * from step 1 into its low-order 256 bits and high-order 260 + * bits: + * + * step1 = step1(low 256 bits) + step1(high 260 bits) * 2^256 + * + * and then perform the calculation: + * + * step2 = step1 (mod p) + * = step1(low 256 bits) + step1(high 260 bits) * 2^256 (mod p) + * = step1(low 256 bits) + step1(high 260 bits) * 38 (mod p) + * + * There are 6 bits in the constant value 38. The step 2 + * multiplication product will therefore have 260+6=266 bits, + * and the step 2 result (after the addition) will therefore + * have 267 bits. + */ + static_assert ( sizeof ( step2->product ) >= sizeof ( step2->value ) ); + static_assert ( sizeof ( step2->product ) >= sizeof ( step2->parts ) ); + bigint_grow ( &step1->parts.low_256bit, &result->value ); + bigint_multiply ( &step1->parts.high_260bit, &x25519_reduce_256, + &step2->product ); + bigint_add ( &result->value, &step2->value ); + + /* Step 3: reduce high-order 267-256=11 bits of step 2 result + * + * Use the identity 2^256=38 (mod p) again to reduce the + * high-order bits of the step 2 result. As before, we split + * the 267-bit result from step 2 into its low-order 256 bits + * and high-order 11 bits: + * + * step2 = step2(low 256 bits) + step2(high 11 bits) * 2^256 + * + * and then perform the calculation: + * + * step3 = step2 (mod p) + * = step2(low 256 bits) + step2(high 11 bits) * 2^256 (mod p) + * = step2(low 256 bits) + step2(high 11 bits) * 38 (mod p) + * + * There are 6 bits in the constant value 38. The step 3 + * multiplication product will therefore have 11+6=19 bits, + * and the step 3 result (after the addition) will therefore + * have 257 bits. + * + * A loose upper bound for the step 3 result (after the + * addition) is given by: + * + * step3 < ( 2^256 - 1 ) + ( 2^19 - 1 ) + * < ( 2^257 - 2^256 - 1 ) + ( 2^19 - 1 ) + * < ( 2^257 - 76 ) - 2^256 + 2^19 + 74 + * < 4 * ( 2^255 - 19 ) - 2^256 + 2^19 + 74 + * < 4p - 2^256 + 2^19 + 74 + * + * and so the step 3 result is strictly less than 4p, and + * therefore lies within the range [0,4p-1]. + */ + memset ( &step3->value, 0, sizeof ( step3->value ) ); + bigint_grow ( &step2->parts.low_256bit, &result->value ); + bigint_multiply ( &step2->parts.high_11bit, &x25519_reduce_256, + &step3->product ); + bigint_add ( &step3->value, &result->value ); + + /* Step 1 calculates the product of the input operands, and + * each subsequent step reduces the number of bits in the + * result while preserving this value (modulo p). The final + * result is therefore equal to the product of the input + * operands (modulo p), as required. + * + * The step 3 result lies within the range [0,4p-1] and the + * final result is therefore a valid X25519 unsigned 257-bit + * integer, as required. + */ +} + +/** + * Compute multiplicative inverse + * + * @v invertend Big integer to be inverted + * @v result Big integer to hold result (may not overlap input) + */ +void x25519_invert ( const union x25519_oct258 *invertend, + union x25519_quad257 *result ) { + int i; + + /* Sanity check */ + assert ( invertend != &result->oct258 ); + + /* Calculate inverse as x^(-1)=x^(p-2) where p is the field prime + * + * The field prime is p=2^255-19 and so: + * + * p - 2 = 2^255 - 21 + * = (2^255 - 1) - 2^4 - 2^2 + * + * i.e. p-2 is a 254-bit number in which all bits are set + * apart from bit 2 and bit 4. + * + * We use the square-and-multiply method to compute x^(p-2). + */ + bigint_copy ( &invertend->value, &result->value ); + for ( i = 253 ; i >= 0 ; i-- ) { + + /* Square running total */ + x25519_multiply ( &result->oct258, &result->oct258, result ); + + /* For each set bit in the exponent, multiply by invertend */ + if ( ( i != 2 ) && ( i != 4 ) ) { + x25519_multiply ( invertend, &result->oct258, result ); + } + } +} + +/** + * Reduce big integer via conditional subtraction + * + * @v subtrahend Big integer to subtract + * @v value Big integer to be subtracted from, if possible + */ +static void x25519_reduce_by ( const x25519_t *subtrahend, x25519_t *value ) { + unsigned int max_bit = ( ( 8 * sizeof ( *value ) ) - 1 ); + x25519_t tmp; + + /* Conditionally subtract subtrahend + * + * Subtract the subtrahend, discarding the result (in constant + * time) if the subtraction underflows. + */ + bigint_copy ( value, &tmp ); + bigint_subtract ( subtrahend, value ); + bigint_swap ( value, &tmp, bigint_bit_is_set ( value, max_bit ) ); +} + +/** + * Reduce big integer to canonical range + * + * @v value Big integer to be reduced + */ +void x25519_reduce ( union x25519_quad257 *value ) { + + /* Conditionally subtract 2p + * + * Subtract twice the field prime, discarding the result (in + * constant time) if the subtraction underflows. + * + * The input value is in the range [0,4p-1]. After this + * conditional subtraction, the value is in the range + * [0,2p-1]. + */ + x25519_reduce_by ( &x25519_2p, &value->value ); + + /* Conditionally subtract p + * + * Subtract the field prime, discarding the result (in + * constant time) if the subtraction underflows. + * + * The value is already in the range [0,2p-1]. After this + * conditional subtraction, the value is in the range [0,p-1] + * and is therefore the canonical representation. + */ + x25519_reduce_by ( &x25519_p, &value->value ); +} + +/** + * Compute next step of the Montgomery ladder + * + * @v base Base point + * @v bit Bit value + * @v step Ladder step + */ +static void x25519_step ( const union x25519_quad257 *base, int bit, + struct x25519_step *step ) { + union x25519_quad257 *a = &step->x_n.X; + union x25519_quad257 *b = &step->x_n1.X; + union x25519_quad257 *c = &step->x_n.Z; + union x25519_quad257 *d = &step->x_n1.Z; + union x25519_oct258 e; + union x25519_quad257 f; + union x25519_oct258 *v1_e; + union x25519_oct258 *v2_a; + union x25519_oct258 *v3_c; + union x25519_oct258 *v4_b; + union x25519_quad257 *v5_d; + union x25519_quad257 *v6_f; + union x25519_quad257 *v7_a; + union x25519_quad257 *v8_c; + union x25519_oct258 *v9_e; + union x25519_oct258 *v10_a; + union x25519_quad257 *v11_b; + union x25519_oct258 *v12_c; + union x25519_quad257 *v13_a; + union x25519_oct258 *v14_a; + union x25519_quad257 *v15_c; + union x25519_quad257 *v16_a; + union x25519_quad257 *v17_d; + union x25519_quad257 *v18_b; + + /* See the referenced paper "Implementing Curve25519/X25519: A + * Tutorial on Elliptic Curve Cryptography" for the reasoning + * behind this calculation. + */ + + /* Reuse storage locations for intermediate results where possible */ + v1_e = &e; + v2_a = container_of ( &a->value, union x25519_oct258, value ); + v3_c = container_of ( &c->value, union x25519_oct258, value ); + v4_b = container_of ( &b->value, union x25519_oct258, value ); + v5_d = d; + v6_f = &f; + v7_a = a; + v8_c = c; + v9_e = &e; + v10_a = container_of ( &a->value, union x25519_oct258, value ); + v11_b = b; + v12_c = container_of ( &c->value, union x25519_oct258, value ); + v13_a = a; + v14_a = container_of ( &a->value, union x25519_oct258, value ); + v15_c = c; + v16_a = a; + v17_d = d; + v18_b = b; + + /* Select inputs */ + bigint_swap ( &a->value, &b->value, bit ); + bigint_swap ( &c->value, &d->value, bit ); + + /* v1 = a + c */ + x25519_add ( a, c, v1_e ); + + /* v2 = a - c */ + x25519_subtract ( a, c, v2_a ); + + /* v3 = b + d */ + x25519_add ( b, d, v3_c ); + + /* v4 = b - d */ + x25519_subtract ( b, d, v4_b ); + + /* v5 = v1^2 = (a + c)^2 = a^2 + 2ac + c^2 */ + x25519_multiply ( v1_e, v1_e, v5_d ); + + /* v6 = v2^2 = (a - c)^2 = a^2 - 2ac + c^2 */ + x25519_multiply ( v2_a, v2_a, v6_f ); + + /* v7 = v3 * v2 = (b + d) * (a - c) = ab - bc + ad - cd */ + x25519_multiply ( v3_c, v2_a, v7_a ); + + /* v8 = v4 * v1 = (b - d) * (a + c) = ab + bc - ad - cd */ + x25519_multiply ( v4_b, v1_e, v8_c ); + + /* v9 = v7 + v8 = 2 * (ab - cd) */ + x25519_add ( v7_a, v8_c, v9_e ); + + /* v10 = v7 - v8 = 2 * (ad - bc) */ + x25519_subtract ( v7_a, v8_c, v10_a ); + + /* v11 = v10^2 = 4 * (ad - bc)^2 */ + x25519_multiply ( v10_a, v10_a, v11_b ); + + /* v12 = v5 - v6 = (a + c)^2 - (a - c)^2 = 4ac */ + x25519_subtract ( v5_d, v6_f, v12_c ); + + /* v13 = v12 * 121665 = 486660ac = (A-2) * ac */ + x25519_multiply ( v12_c, &x25519_121665, v13_a ); + + /* v14 = v13 + v5 = (A-2) * ac + a^2 + 2ac + c^2 = a^2 + A * ac + c^2 */ + x25519_add ( v13_a, v5_d, v14_a ); + + /* v15 = v12 * v14 = 4ac * (a^2 + A * ac + c^2) */ + x25519_multiply ( v12_c, v14_a, v15_c ); + + /* v16 = v5 * v6 = (a + c)^2 * (a - c)^2 = (a^2 - c^2)^2 */ + x25519_multiply ( &v5_d->oct258, &v6_f->oct258, v16_a ); + + /* v17 = v11 * base = 4 * base * (ad - bc)^2 */ + x25519_multiply ( &v11_b->oct258, &base->oct258, v17_d ); + + /* v18 = v9^2 = 4 * (ab - cd)^2 */ + x25519_multiply ( v9_e, v9_e, v18_b ); + + /* Select outputs */ + bigint_swap ( &a->value, &b->value, bit ); + bigint_swap ( &c->value, &d->value, bit ); +} + +/** + * Multiply X25519 elliptic curve point + * + * @v base Base point + * @v scalar Scalar multiple + * @v result Point to hold result (may overlap base point) + */ +static void x25519_ladder ( const union x25519_quad257 *base, + struct x25519_value *scalar, + union x25519_quad257 *result ) { + static const uint8_t zero[] = { 0 }; + static const uint8_t one[] = { 1 }; + struct x25519_step step; + union x25519_quad257 *tmp; + int bit; + int i; + + /* Initialise ladder */ + bigint_init ( &step.x_n.X.value, one, sizeof ( one ) ); + bigint_init ( &step.x_n.Z.value, zero, sizeof ( zero ) ); + bigint_copy ( &base->value, &step.x_n1.X.value ); + bigint_init ( &step.x_n1.Z.value, one, sizeof ( one ) ); + + /* Use ladder */ + for ( i = 254 ; i >= 0 ; i-- ) { + bit = ( ( scalar->raw[ i / 8 ] >> ( i % 8 ) ) & 1 ); + x25519_step ( base, bit, &step ); + } + + /* Convert back to affine coordinate */ + tmp = &step.x_n1.X; + x25519_invert ( &step.x_n.Z.oct258, tmp ); + x25519_multiply ( &step.x_n.X.oct258, &tmp->oct258, result ); + x25519_reduce ( result ); +} + +/** + * Reverse X25519 value endianness + * + * @v value Value to reverse + */ +static void x25519_reverse ( struct x25519_value *value ) { + uint8_t *low = value->raw; + uint8_t *high = &value->raw[ sizeof ( value->raw ) - 1 ]; + uint8_t tmp; + + /* Reverse bytes */ + do { + tmp = *low; + *low = *high; + *high = tmp; + } while ( ++low < --high ); +} + +/** + * Calculate X25519 key + * + * @v base Base point + * @v scalar Scalar multiple + * @v result Point to hold result (may overlap base point) + */ +void x25519_key ( const struct x25519_value *base, + const struct x25519_value *scalar, + struct x25519_value *result ) { + struct x25519_value *tmp = result; + union x25519_quad257 point; + + /* Reverse base point and clear high bit as required by RFC7748 */ + memcpy ( tmp, base, sizeof ( *tmp ) ); + x25519_reverse ( tmp ); + tmp->raw[0] &= 0x7f; + bigint_init ( &point.value, tmp->raw, sizeof ( tmp->raw ) ); + + /* Clamp scalar as required by RFC7748 */ + memcpy ( tmp, scalar, sizeof ( *tmp ) ); + tmp->raw[0] &= 0xf8; + tmp->raw[31] |= 0x40; + + /* Multiply elliptic curve point */ + x25519_ladder ( &point, tmp, &point ); + + /* Reverse result */ + bigint_done ( &point.value, result->raw, sizeof ( result->raw ) ); + x25519_reverse ( result ); +} |