/***********************************************************************

 * Copyright (c) 2013, 2014 Pieter Wuille                              *

 * Distributed under the MIT software license, see the accompanying    *

 * file COPYING or https://www.opensource.org/licenses/mit-license.php.*

 ***********************************************************************/

#ifndef SECP256K1_FIELD_H

#define SECP256K1_FIELD_H

#include "util.h"

/* This file defines the generic interface for working with secp256k1_fe

 * objects, which represent field elements (integers modulo 2^256 - 2^32 - 977).

 *

 * The actual definition of the secp256k1_fe type depends on the chosen field

 * implementation; see the field_5x52.h and field_10x26.h files for details.

 *

 * All secp256k1_fe objects have implicit properties that determine what

 * operations are permitted on it. These are purely a function of what

 * secp256k1_fe_ operations are applied on it, generally (implicitly) fixed at

 * compile time, and do not depend on the chosen field implementation. Despite

 * that, what these properties actually entail for the field representation

 * values depends on the chosen field implementation. These properties are:

 * - magnitude: an integer in [0,32]

 * - normalized: 0 or 1; normalized=1 implies magnitude <= 1.

 *

 * In VERIFY mode, they are materialized explicitly as fields in the struct,

 * allowing run-time verification of these properties. In that case, the field

 * implementation also provides a secp256k1_fe_verify routine to verify that

 * these fields match the run-time value and perform internal consistency

 * checks. */

#ifdef VERIFY

#  define SECP256K1_FE_VERIFY_FIELDS \

    int magnitude; \

    int normalized;

#else

#  define SECP256K1_FE_VERIFY_FIELDS

#endif

#if defined(SECP256K1_WIDEMUL_INT128)

#include "field_5x52.h"

#elif defined(SECP256K1_WIDEMUL_INT64)

#include "field_10x26.h"

#else

#error "Please select wide multiplication implementation"

#endif

#ifdef VERIFY

/* Magnitude and normalized value for constants. */

#define SECP256K1_FE_VERIFY_CONST(d7, d6, d5, d4, d3, d2, d1, d0) \

    /* Magnitude is 0 for constant 0; 1 otherwise. */ \

    , (((d7) | (d6) | (d5) | (d4) | (d3) | (d2) | (d1) | (d0)) != 0) \

    /* Normalized is 1 unless sum(d_i<<(32*i) for i=0..7) exceeds field modulus. */ \

    , (!(((d7) & (d6) & (d5) & (d4) & (d3) & (d2)) == 0xfffffffful && ((d1) == 0xfffffffful || ((d1) == 0xfffffffe && (d0 >= 0xfffffc2f)))))

#else

#define SECP256K1_FE_VERIFY_CONST(d7, d6, d5, d4, d3, d2, d1, d0)

#endif

/** This expands to an initializer for a secp256k1_fe valued sum((i*32) * d_i, i=0..7) mod p.

 *

 * It has magnitude 1, unless d_i are all 0, in which case the magnitude is 0.

 * It is normalized, unless sum(2^(i*32) * d_i, i=0..7) >= p.

 *

 * SECP256K1_FE_CONST_INNER is provided by the implementation.

 */

#define SECP256K1_FE_CONST(d7, d6, d5, d4, d3, d2, d1, d0) {SECP256K1_FE_CONST_INNER((d7), (d6), (d5), (d4), (d3), (d2), (d1), (d0)) SECP256K1_FE_VERIFY_CONST((d7), (d6), (d5), (d4), (d3), (d2), (d1), (d0)) }

static const secp256k1_fe secp256k1_fe_one = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1);

static const secp256k1_fe secp256k1_const_beta = SECP256K1_FE_CONST(

    0x7ae96a2bul, 0x657c0710ul, 0x6e64479eul, 0xac3434e9ul,

    0x9cf04975ul, 0x12f58995ul, 0xc1396c28ul, 0x719501eeul

);

#ifndef VERIFY

/* In non-VERIFY mode, we #define the fe operations to be identical to their

 * internal field implementation, to avoid the potential overhead of a

 * function call (even though presumably inlinable). */

#  define secp256k1_fe_normalize secp256k1_fe_impl_normalize

#  define secp256k1_fe_normalize_weak secp256k1_fe_impl_normalize_weak

#  define secp256k1_fe_normalize_var secp256k1_fe_impl_normalize_var

#  define secp256k1_fe_normalizes_to_zero secp256k1_fe_impl_normalizes_to_zero

#  define secp256k1_fe_normalizes_to_zero_var secp256k1_fe_impl_normalizes_to_zero_var

#  define secp256k1_fe_set_int secp256k1_fe_impl_set_int

#  define secp256k1_fe_is_zero secp256k1_fe_impl_is_zero

#  define secp256k1_fe_is_odd secp256k1_fe_impl_is_odd

#  define secp256k1_fe_cmp_var secp256k1_fe_impl_cmp_var

#  define secp256k1_fe_set_b32_mod secp256k1_fe_impl_set_b32_mod

#  define secp256k1_fe_set_b32_limit secp256k1_fe_impl_set_b32_limit

#  define secp256k1_fe_get_b32 secp256k1_fe_impl_get_b32

#  define secp256k1_fe_negate_unchecked secp256k1_fe_impl_negate_unchecked

#  define secp256k1_fe_mul_int_unchecked secp256k1_fe_impl_mul_int_unchecked

#  define secp256k1_fe_add secp256k1_fe_impl_add

#  define secp256k1_fe_mul secp256k1_fe_impl_mul

#  define secp256k1_fe_sqr secp256k1_fe_impl_sqr

#  define secp256k1_fe_cmov secp256k1_fe_impl_cmov

#  define secp256k1_fe_to_storage secp256k1_fe_impl_to_storage

#  define secp256k1_fe_from_storage secp256k1_fe_impl_from_storage

#  define secp256k1_fe_inv secp256k1_fe_impl_inv

#  define secp256k1_fe_inv_var secp256k1_fe_impl_inv_var

#  define secp256k1_fe_get_bounds secp256k1_fe_impl_get_bounds

#  define secp256k1_fe_half secp256k1_fe_impl_half

#  define secp256k1_fe_add_int secp256k1_fe_impl_add_int

#  define secp256k1_fe_is_square_var secp256k1_fe_impl_is_square_var

#endif /* !defined(VERIFY) */

/** Normalize a field element.

 *

 * On input, r must be a valid field element.

 * On output, r represents the same value but has normalized=1 and magnitude=1.

 */

static void secp256k1_fe_normalize(secp256k1_fe *r);

/** Give a field element magnitude 1.

 *

 * On input, r must be a valid field element.

 * On output, r represents the same value but has magnitude=1. Normalized is unchanged.

 */

static void secp256k1_fe_normalize_weak(secp256k1_fe *r);

/** Normalize a field element, without constant-time guarantee.

 *

 * Identical in behavior to secp256k1_fe_normalize, but not constant time in r.

 */

static void secp256k1_fe_normalize_var(secp256k1_fe *r);

/** Determine whether r represents field element 0.

 *

 * On input, r must be a valid field element.

 * Returns whether r = 0 (mod p).

 */

static int secp256k1_fe_normalizes_to_zero(const secp256k1_fe *r);

/** Determine whether r represents field element 0, without constant-time guarantee.

 *

 * Identical in behavior to secp256k1_normalizes_to_zero, but not constant time in r.

 */

static int secp256k1_fe_normalizes_to_zero_var(const secp256k1_fe *r);

/** Set a field element to an integer in range [0,0x7FFF].

 *

 * On input, r does not need to be initialized, a must be in [0,0x7FFF].

 * On output, r represents value a, is normalized and has magnitude (a!=0).

 */

static void secp256k1_fe_set_int(secp256k1_fe *r, int a);

/** Clear a field element to prevent leaking sensitive information. */

static void secp256k1_fe_clear(secp256k1_fe *a);

/** Determine whether a represents field element 0.

 *

 * On input, a must be a valid normalized field element.

 * Returns whether a = 0 (mod p).

 *

 * This behaves identical to secp256k1_normalizes_to_zero{,_var}, but requires

 * normalized input (and is much faster).

 */

static int secp256k1_fe_is_zero(const secp256k1_fe *a);

/** Determine whether a (mod p) is odd.

 *

 * On input, a must be a valid normalized field element.

 * Returns (int(a) mod p) & 1.

 */

static int secp256k1_fe_is_odd(const secp256k1_fe *a);

/** Determine whether two field elements are equal.

 *

 * On input, a and b must be valid field elements with magnitudes not exceeding

 * 1 and 31, respectively.

 * Returns a = b (mod p).

 */

static int secp256k1_fe_equal(const secp256k1_fe *a, const secp256k1_fe *b);

/** Compare the values represented by 2 field elements, without constant-time guarantee.

 *

 * On input, a and b must be valid normalized field elements.

 * Returns 1 if a > b, -1 if a < b, and 0 if a = b (comparisons are done as integers

 * in range 0..p-1).

 */

static int secp256k1_fe_cmp_var(const secp256k1_fe *a, const secp256k1_fe *b);

/** Set a field element equal to the element represented by a provided 32-byte big endian value

 * interpreted modulo p.

 *

 * On input, r does not need to be initialized. a must be a pointer to an initialized 32-byte array.

 * On output, r = a (mod p). It will have magnitude 1, and not be normalized.

 */

static void secp256k1_fe_set_b32_mod(secp256k1_fe *r, const unsigned char *a);

/** Set a field element equal to a provided 32-byte big endian value, checking for overflow.

 *

 * On input, r does not need to be initialized. a must be a pointer to an initialized 32-byte array.

 * On output, r = a if (a < p), it will be normalized with magnitude 1, and 1 is returned.

 * If a >= p, 0 is returned, and r will be made invalid (and must not be used without overwriting).

 */

static int secp256k1_fe_set_b32_limit(secp256k1_fe *r, const unsigned char *a);

/** Convert a field element to 32-byte big endian byte array.

 * On input, a must be a valid normalized field element, and r a pointer to a 32-byte array.

 * On output, r = a (mod p).

 */

static void secp256k1_fe_get_b32(unsigned char *r, const secp256k1_fe *a);

/** Negate a field element.

 *

 * On input, r does not need to be initialized. a must be a valid field element with

 * magnitude not exceeding m. m must be an integer constant expression in [0,31].

 * Performs {r = -a}.

 * On output, r will not be normalized, and will have magnitude m+1.

 */

#define secp256k1_fe_negate(r, a, m) ASSERT_INT_CONST_AND_DO(m, secp256k1_fe_negate_unchecked(r, a, m))

/** Like secp256k1_fe_negate_unchecked but m is not checked to be an integer constant expression.

 *

 * Should not be called directly outside of tests.

 */

static void secp256k1_fe_negate_unchecked(secp256k1_fe *r, const secp256k1_fe *a, int m);

/** Add a small integer to a field element.

 *

 * Performs {r += a}. The magnitude of r increases by 1, and normalized is cleared.

 * a must be in range [0,0x7FFF].

 */

static void secp256k1_fe_add_int(secp256k1_fe *r, int a);

/** Multiply a field element with a small integer.

 *

 * On input, r must be a valid field element. a must be an integer constant expression in [0,32].

 * The magnitude of r times a must not exceed 32.

 * Performs {r *= a}.

 * On output, r's magnitude is multiplied by a, and r will not be normalized.

 */

#define secp256k1_fe_mul_int(r, a) ASSERT_INT_CONST_AND_DO(a, secp256k1_fe_mul_int_unchecked(r, a))

/** Like secp256k1_fe_mul_int but a is not checked to be an integer constant expression.

 * 

 * Should not be called directly outside of tests.

 */

static void secp256k1_fe_mul_int_unchecked(secp256k1_fe *r, int a);

/** Increment a field element by another.

 *

 * On input, r and a must be valid field elements, not necessarily normalized.

 * The sum of their magnitudes must not exceed 32.

 * Performs {r += a}.

 * On output, r will not be normalized, and will have magnitude incremented by a's.

 */

static void secp256k1_fe_add(secp256k1_fe *r, const secp256k1_fe *a);

/** Multiply two field elements.

 *

 * On input, a and b must be valid field elements; r does not need to be initialized.

 * r and a may point to the same object, but neither may point to the object pointed

 * to by b. The magnitudes of a and b must not exceed 8.

 * Performs {r = a * b}

 * On output, r will have magnitude 1, but won't be normalized.

 */

static void secp256k1_fe_mul(secp256k1_fe *r, const secp256k1_fe *a, const secp256k1_fe * SECP256K1_RESTRICT b);

/** Square a field element.

 *

 * On input, a must be a valid field element; r does not need to be initialized. The magnitude

 * of a must not exceed 8.

 * Performs {r = a**2}

 * On output, r will have magnitude 1, but won't be normalized.

 */

static void secp256k1_fe_sqr(secp256k1_fe *r, const secp256k1_fe *a);

/** Compute a square root of a field element.

 *

 * On input, a must be a valid field element with magnitude<=8; r need not be initialized.

 * If sqrt(a) exists, performs {r = sqrt(a)} and returns 1.

 * Otherwise, sqrt(-a) exists. The function performs {r = sqrt(-a)} and returns 0.

 * The resulting value represented by r will be a square itself.

 * Variables r and a must not point to the same object.

 * On output, r will have magnitude 1 but will not be normalized.

 */

static int secp256k1_fe_sqrt(secp256k1_fe * SECP256K1_RESTRICT r, const secp256k1_fe * SECP256K1_RESTRICT a);

/** Compute the modular inverse of a field element.

 *

 * On input, a must be a valid field element; r need not be initialized.

 * Performs {r = a**(p-2)} (which maps 0 to 0, and every other element to its

 * inverse).

 * On output, r will have magnitude (a.magnitude != 0) and be normalized.

 */

static void secp256k1_fe_inv(secp256k1_fe *r, const secp256k1_fe *a);

/** Compute the modular inverse of a field element, without constant-time guarantee.

 *

 * Behaves identically to secp256k1_fe_inv, but is not constant-time in a.

 */

static void secp256k1_fe_inv_var(secp256k1_fe *r, const secp256k1_fe *a);

/** Convert a field element to secp256k1_fe_storage.

 *

 * On input, a must be a valid normalized field element.

 * Performs {r = a}.

 */

static void secp256k1_fe_to_storage(secp256k1_fe_storage *r, const secp256k1_fe *a);

/** Convert a field element back from secp256k1_fe_storage.

 *

 * On input, r need not be initialized.

 * Performs {r = a}.

 * On output, r will be normalized and will have magnitude 1.

 */

static void secp256k1_fe_from_storage(secp256k1_fe *r, const secp256k1_fe_storage *a);

/** If flag is true, set *r equal to *a; otherwise leave it. Constant-time.  Both *r and *a must be initialized.*/

static void secp256k1_fe_storage_cmov(secp256k1_fe_storage *r, const secp256k1_fe_storage *a, int flag);

/** Conditionally move a field element in constant time.

 *

 * On input, both r and a must be valid field elements. Flag must be 0 or 1.

 * Performs {r = flag ? a : r}.

 *

 * On output, r's magnitude will be the maximum of both input magnitudes.

 * It will be normalized if and only if both inputs were normalized.

 */

static void secp256k1_fe_cmov(secp256k1_fe *r, const secp256k1_fe *a, int flag);

/** Halve the value of a field element modulo the field prime in constant-time.

 *

 * On input, r must be a valid field element.

 * On output, r will be normalized and have magnitude floor(m/2) + 1 where m is

 * the magnitude of r on input.

 */

static void secp256k1_fe_half(secp256k1_fe *r);

/** Sets r to a field element with magnitude m, normalized if (and only if) m==0.

 *  The value is chosen so that it is likely to trigger edge cases related to

 *  internal overflows. */

static void secp256k1_fe_get_bounds(secp256k1_fe *r, int m);

/** Determine whether a is a square (modulo p).

 *

 * On input, a must be a valid field element.

 */

static int secp256k1_fe_is_square_var(const secp256k1_fe *a);

/** Check invariants on a field element (no-op unless VERIFY is enabled). */

static void secp256k1_fe_verify(const secp256k1_fe *a);

#define SECP256K1_FE_VERIFY(a) secp256k1_fe_verify(a)

/** Check that magnitude of a is at most m (no-op unless VERIFY is enabled). */

static void secp256k1_fe_verify_magnitude(const secp256k1_fe *a, int m);

#define SECP256K1_FE_VERIFY_MAGNITUDE(a, m) secp256k1_fe_verify_magnitude(a, m)

#endif /* SECP256K1_FIELD_H */