library ECDSA {
	/**
	 * @dev Returns the address that signed a hashed message (`hash`) with
	 * `signature`. This address can then be used for verification purposes.
	 *
	 * The `ecrecover` EVM opcode allows for malleable (non-unique) signatures:
	 * this function rejects them by requiring the `s` value to be in the lower
	 * half order, and the `v` value to be either 27 or 28.
	 *
	 * IMPORTANT: `hash` _must_ be the result of a hash operation for the
	 * verification to be secure: it is possible to craft signatures that
	 * recover to arbitrary addresses for non-hashed data. A safe way to ensure
	 * this is by receiving a hash of the original message (which may otherwise
	 * be too long), and then calling {toEthSignedMessageHash} on it.
	 *
	 * Documentation for signature generation:
	 * - with https://web3js.readthedocs.io/en/v1.3.4/web3-eth-accounts.html#sign[Web3.js]
	 * - with https://docs.ethers.io/v5/api/signer/#Signer-signMessage[ethers]
	 */
	function recover(bytes32 hash, bytes memory signature) internal pure returns (address) {
		// Divide the signature in r, s and v variables
		bytes32 r;
		bytes32 s;
		uint8 v;

		// Check the signature length
		// - case 65: r,s,v signature (standard)
		// - case 64: r,vs signature (cf https://eips.ethereum.org/EIPS/eip-2098) _Available since v4.1._
		if (signature.length == 65) {
			// ecrecover takes the signature parameters, and the only way to get them
			// currently is to use assembly.
			assembly {
				r := mload(add(signature, 0x20))
				s := mload(add(signature, 0x40))
				v := byte(0, mload(add(signature, 0x60)))
			}
		}
		else if (signature.length == 64) {
			// ecrecover takes the signature parameters, and the only way to get them
			// currently is to use assembly.
			assembly {
				let vs := mload(add(signature, 0x40))
				r := mload(add(signature, 0x20))
				s := and(vs, 0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff)
				v := add(shr(255, vs), 27)
			}
		}
		else {
			revert("invalid signature length");
		}

		return recover(hash, v, r, s);
	}

	/**
	 * @dev Overload of {ECDSA-recover} that receives the `v`,
	 * `r` and `s` signature fields separately.
	 */
	function recover(
		bytes32 hash,
		uint8 v,
		bytes32 r,
		bytes32 s
	) internal pure returns (address) {
		// EIP-2 still allows signature malleability for ecrecover(). Remove this possibility and make the signature
		// unique. Appendix F in the Ethereum Yellow paper (https://ethereum.github.io/yellowpaper/paper.pdf), defines
		// the valid range for s in (281): 0 < s < secp256k1n ÷ 2 + 1, and for v in (282): v ∈ {27, 28}. Most
		// signatures from current libraries generate a unique signature with an s-value in the lower half order.
		//
		// If your library generates malleable signatures, such as s-values in the upper range, calculate a new s-value
		// with 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141 - s1 and flip v from 27 to 28 or
		// vice versa. If your library also generates signatures with 0/1 for v instead 27/28, add 27 to v to accept
		// these malleable signatures as well.
		require(
			uint256(s) <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5D576E7357A4501DDFE92F46681B20A0,
			"invalid signature 's' value"
		);
		require(v == 27 || v == 28, "invalid signature 'v' value");

		// If the signature is valid (and not malleable), return the signer address
		address signer = ecrecover(hash, v, r, s);
		require(signer != address(0), "invalid signature");

		return signer;
	}

	/**
	 * @dev Returns an Ethereum Signed Message, created from a `hash`. This
	 * produces hash corresponding to the one signed with the
	 * https://eth.wiki/json-rpc/API#eth_sign[`eth_sign`]
	 * JSON-RPC method as part of EIP-191.
	 *
	 * See {recover}.
	 */
	function toEthSignedMessageHash(bytes32 hash) internal pure returns (bytes32) {
		// 32 is the length in bytes of hash,
		// enforced by the type signature above
		return keccak256(abi.encodePacked("\x19Ethereum Signed Message:\n32", hash));
	}

	/**
	 * @dev Returns an Ethereum Signed Typed Data, created from a
	 * `domainSeparator` and a `structHash`. This produces hash corresponding
	 * to the one signed with the
	 * https://eips.ethereum.org/EIPS/eip-712[`eth_signTypedData`]
	 * JSON-RPC method as part of EIP-712.
	 *
	 * See {recover}.
	 */
	function toTypedDataHash(bytes32 domainSeparator, bytes32 structHash) internal pure returns (bytes32) {
		return keccak256(abi.encodePacked("\x19\x01", domainSeparator, structHash));
	}
}
/**
 * @dev Elliptic Curve Digital Signature Algorithm (ECDSA) operations.
 *
 * These functions can be used to verify that a message was signed by the holder
 * of the private keys of a given address.
 *
 * @dev Copy of the Zeppelin's library:
 *      https://github.com/OpenZeppelin/openzeppelin-contracts/blob/b0cf6fbb7a70f31527f36579ad644e1cf12fdf4e/contracts/utils/cryptography/ECDSA.sol
 */
function toTypedDataHash(bytes32 domainSeparator, bytes32 structHash) internal pure returns (bytes32) {
	return keccak256(abi.encodePacked("\x19\x01", domainSeparator, structHash));
}

/**
 	 * @dev Returns an Ethereum Signed Typed Data, created from a
 	 * `domainSeparator` and a `structHash`. This produces hash corresponding
 	 * to the one signed with the
 	 * https://eips.ethereum.org/EIPS/eip-712[`eth_signTypedData`]
 	 * JSON-RPC method as part of EIP-712.
 	 *
 	 * See {recover}.
 	 */