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								// Copyright 2015 Jeffrey Wilcke, Felix Lange, Gustav Simonsson. All rights reserved.
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								// Use of this source code is governed by a BSD-style license that can be found in
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								// the LICENSE file.
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								// secp256k1_context_create_sign_verify creates a context for signing and signature verification.
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								static secp256k1_context* secp256k1_context_create_sign_verify() {
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									return secp256k1_context_create(SECP256K1_CONTEXT_SIGN | SECP256K1_CONTEXT_VERIFY);
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								}
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								// secp256k1_ext_ecdsa_recover recovers the public key of an encoded compact signature.
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								//
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								// Returns: 1: recovery was successful
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								//          0: recovery was not successful
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								// Args:    ctx:        pointer to a context object (cannot be NULL)
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								//  Out:    pubkey_out: the serialized 65-byte public key of the signer (cannot be NULL)
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								//  In:     sigdata:    pointer to a 65-byte signature with the recovery id at the end (cannot be NULL)
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								//          msgdata:    pointer to a 32-byte message (cannot be NULL)
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								static int secp256k1_ext_ecdsa_recover(
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									const secp256k1_context* ctx,
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									unsigned char *pubkey_out,
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									const unsigned char *sigdata,
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									const unsigned char *msgdata
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								) {
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									secp256k1_ecdsa_recoverable_signature sig;
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									secp256k1_pubkey pubkey;
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									if (!secp256k1_ecdsa_recoverable_signature_parse_compact(ctx, &sig, sigdata, (int)sigdata[64])) {
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										return 0;
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									}
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									if (!secp256k1_ecdsa_recover(ctx, &pubkey, &sig, msgdata)) {
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										return 0;
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									}
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									size_t outputlen = 65;
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									return secp256k1_ec_pubkey_serialize(ctx, pubkey_out, &outputlen, &pubkey, SECP256K1_EC_UNCOMPRESSED);
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								}
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								// secp256k1_ext_ecdsa_verify verifies an encoded compact signature.
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								//
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								// Returns: 1: signature is valid
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								//          0: signature is invalid
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								// Args:    ctx:        pointer to a context object (cannot be NULL)
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								//  In:     sigdata:    pointer to a 64-byte signature (cannot be NULL)
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								//          msgdata:    pointer to a 32-byte message (cannot be NULL)
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								//          pubkeydata: pointer to public key data (cannot be NULL)
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								//          pubkeylen:  length of pubkeydata
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								static int secp256k1_ext_ecdsa_verify(
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									const secp256k1_context* ctx,
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									const unsigned char *sigdata,
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									const unsigned char *msgdata,
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									const unsigned char *pubkeydata,
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									size_t pubkeylen
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								) {
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									secp256k1_ecdsa_signature sig;
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									secp256k1_pubkey pubkey;
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									if (!secp256k1_ecdsa_signature_parse_compact(ctx, &sig, sigdata)) {
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										return 0;
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									}
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									if (!secp256k1_ec_pubkey_parse(ctx, &pubkey, pubkeydata, pubkeylen)) {
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										return 0;
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									}
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									return secp256k1_ecdsa_verify(ctx, &sig, msgdata, &pubkey);
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								}
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								// secp256k1_ext_reencode_pubkey decodes then encodes a public key. It can be used to
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								// convert between public key formats. The input/output formats are chosen depending on the
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								// length of the input/output buffers.
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								//
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								// Returns: 1: conversion successful
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								//          0: conversion unsuccessful
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								// Args:    ctx:        pointer to a context object (cannot be NULL)
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								//  Out:    out:        output buffer that will contain the reencoded key (cannot be NULL)
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								//  In:     outlen:     length of out (33 for compressed keys, 65 for uncompressed keys)
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								//          pubkeydata: the input public key (cannot be NULL)
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								//          pubkeylen:  length of pubkeydata
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								static int secp256k1_ext_reencode_pubkey(
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									const secp256k1_context* ctx,
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									unsigned char *out,
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									size_t outlen,
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									const unsigned char *pubkeydata,
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									size_t pubkeylen
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								) {
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									secp256k1_pubkey pubkey;
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									if (!secp256k1_ec_pubkey_parse(ctx, &pubkey, pubkeydata, pubkeylen)) {
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										return 0;
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									}
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									unsigned int flag = (outlen == 33) ? SECP256K1_EC_COMPRESSED : SECP256K1_EC_UNCOMPRESSED;
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									return secp256k1_ec_pubkey_serialize(ctx, out, &outlen, &pubkey, flag);
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								}
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								// secp256k1_ext_scalar_mul multiplies a point by a scalar in constant time.
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								//
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								// Returns: 1: multiplication was successful
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								//          0: scalar was invalid (zero or overflow)
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								// Args:    ctx:      pointer to a context object (cannot be NULL)
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								//  Out:    point:    the multiplied point (usually secret)
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								//  In:     point:    pointer to a 64-byte public point,
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								//                    encoded as two 256bit big-endian numbers.
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								//          scalar:   a 32-byte scalar with which to multiply the point
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								int secp256k1_ext_scalar_mul(const secp256k1_context* ctx, unsigned char *point, const unsigned char *scalar) {
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									int ret = 0;
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									int overflow = 0;
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									secp256k1_fe feX, feY;
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									secp256k1_gej res;
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									secp256k1_ge ge;
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									secp256k1_scalar s;
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									ARG_CHECK(point != NULL);
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									ARG_CHECK(scalar != NULL);
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									(void)ctx;
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									secp256k1_fe_set_b32(&feX, point);
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									secp256k1_fe_set_b32(&feY, point+32);
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									secp256k1_ge_set_xy(&ge, &feX, &feY);
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									secp256k1_scalar_set_b32(&s, scalar, &overflow);
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									if (overflow || secp256k1_scalar_is_zero(&s)) {
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										ret = 0;
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									} else {
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										secp256k1_ecmult_const(&res, &ge, &s);
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										secp256k1_ge_set_gej(&ge, &res);
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										/* Note: can't use secp256k1_pubkey_save here because it is not constant time. */
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										secp256k1_fe_normalize(&ge.x);
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										secp256k1_fe_normalize(&ge.y);
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										secp256k1_fe_get_b32(point, &ge.x);
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										secp256k1_fe_get_b32(point+32, &ge.y);
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										ret = 1;
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									}
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									secp256k1_scalar_clear(&s);
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									return ret;
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								}
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