/*
* 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 <errno.h>
#include <ipxe/init.h>
#include <ipxe/crypto.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;
/** Constant g=9 (the group generator) */
static struct x25519_value x25519_generator = {
.raw = { 9, }
};
/**
* 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)
* @ret rc Return status code
*/
int 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 );
/* Fail if result was all zeros (as required by RFC8422) */
return ( bigint_is_zero ( &point.value ) ? -EPERM : 0 );
}
/**
* Multiply scalar by curve point
*
* @v base Base point (or NULL to use generator)
* @v scalar Scalar multiple
* @v result Result point to fill in
* @ret rc Return status code
*/
static int x25519_curve_multiply ( const void *base, const void *scalar,
void *result ) {
/* Use base point if applicable */
if ( ! base )
base = &x25519_generator;
return x25519_key ( base, scalar, result );
}
/** X25519 elliptic curve */
struct elliptic_curve x25519_curve = {
.name = "x25519",
.keysize = sizeof ( struct x25519_value ),
.multiply = x25519_curve_multiply,
};