Contract Address Details

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.24;

import { IProofVerifier } from "./IProofVerifier.sol";
import { LibPublicInput } from "./LibPublicInput.sol";
import { Ownable2Step } from "@openzeppelin/contracts/access/Ownable2Step.sol";
import { ECDSA } from "@openzeppelin/contracts/utils/cryptography/ECDSA.sol";
import { IAttestation } from "src/layer1/automata-attestation/interfaces/IAttestation.sol";
import { V3Struct } from "src/layer1/automata-attestation/lib/QuoteV3Auth/V3Struct.sol";

/// @title SgxVerifier
/// @notice This contract verifies SGX signature proofs onchain using attested SGX instances.
/// Each instance is registered via remote attestation and can verify proofs until expiry.
/// @dev Side-channel protection is achieved through mandatory instance expiry (INSTANCE_EXPIRY),
/// requiring periodic re-attestation with new keypairs.
/// @custom:security-contact security@taiko.xyz
contract SgxVerifier is IProofVerifier, Ownable2Step {
    /// @dev Each public-private key pair (Ethereum address) is generated within
    /// the SGX program when it boots up. The off-chain remote attestation
    /// ensures the validity of the program hash and has the capability of
    /// bootstrapping the network with trustworthy instances.
    struct Instance {
        address addr;
        uint64 validSince;
    }

    /// @notice The expiry time for the SGX instance.
    uint64 public constant INSTANCE_EXPIRY = 365 days;

    /// @notice A security feature, a delay until an instance is enabled when using onchain RA
    /// verification
    uint64 public constant INSTANCE_VALIDITY_DELAY = 0;

    uint64 public immutable taikoChainId;
    address public immutable automataDcapAttestation;

    /// @dev For gas savings, we assign each SGX instance with an ID to minimize storage operations.
    /// Slot 1.
    uint256 public nextInstanceId;

    /// @dev One SGX instance is uniquely identified (on-chain) by its ECDSA public key
    /// (or rather ethereum address). The instance address remains valid for INSTANCE_EXPIRY
    /// duration (365 days) to protect against side-channel attacks through forced key expiry.
    /// After expiry, the instance must be re-attested and registered with a new address.
    /// Slot 2.
    mapping(uint256 instanceId => Instance instance) public instances;

    /// @dev One address shall be registered (during attestation) only once, otherwise it could
    /// bypass this contract's expiry check by always registering with the same attestation and
    /// getting multiple valid instanceIds.
    /// Slot 3.
    mapping(address instanceAddress => bool alreadyAttested) public addressRegistered;

    uint256[47] private __gap;

    /// @notice Emitted when a new SGX instance is added to the registry.
    /// @param id The ID of the SGX instance.
    /// @param instance The address of the SGX instance.
    /// @param replaced Reserved for future use (always zero address).
    /// @param validSince The time since the instance is valid.
    event InstanceAdded(
        uint256 indexed id, address indexed instance, address indexed replaced, uint256 validSince
    );

    /// @notice Emitted when an SGX instance is deleted from the registry.
    /// @param id The ID of the SGX instance.
    /// @param instance The address of the SGX instance.
    event InstanceDeleted(uint256 indexed id, address indexed instance);

    error SGX_ALREADY_ATTESTED();
    error SGX_INVALID_ATTESTATION();
    error SGX_INVALID_INSTANCE();
    error SGX_INVALID_PROOF();
    error SGX_INVALID_CHAIN_ID();

    constructor(uint64 _taikoChainId, address _owner, address _automataDcapAttestation) {
        require(_taikoChainId != 0, SGX_INVALID_CHAIN_ID());
        taikoChainId = _taikoChainId;
        automataDcapAttestation = _automataDcapAttestation;

        _transferOwnership(_owner);
    }

    /// @notice Adds trusted SGX instances to the registry.
    /// @param _instances The address array of trusted SGX instances.
    /// @return The respective instanceId array per addresses.
    function addInstances(address[] calldata _instances)
        external
        onlyOwner
        returns (uint256[] memory)
    {
        return _addInstances(_instances, true);
    }

    /// @notice Deletes SGX instances from the registry.
    /// @param _ids The ids array of SGX instances.
    function deleteInstances(uint256[] calldata _ids) external onlyOwner {
        uint256 size = _ids.length;
        for (uint256 i; i < size; ++i) {
            uint256 idx = _ids[i];

            require(instances[idx].addr != address(0), SGX_INVALID_INSTANCE());

            emit InstanceDeleted(idx, instances[idx].addr);

            delete instances[idx];
        }
    }

    /// @notice Adds an SGX instance after the attestation is verified
    /// @param _attestation The parsed attestation quote.
    /// @return The respective instanceId
    function registerInstance(V3Struct.ParsedV3QuoteStruct calldata _attestation)
        external
        returns (uint256)
    {
        (bool verified,) = IAttestation(automataDcapAttestation).verifyParsedQuote(_attestation);
        require(verified, SGX_INVALID_ATTESTATION());

        address[] memory addresses = new address[](1);
        addresses[0] = address(bytes20(_attestation.localEnclaveReport.reportData));

        return _addInstances(addresses, false)[0];
    }

    /// @inheritdoc IProofVerifier
    function verifyProof(
        uint256, /* _proposalAge */
        bytes32 _aggregatedProvingHash,
        bytes calldata _proof
    )
        external
        view
    {
        require(_proof.length == 89, SGX_INVALID_PROOF());

        uint32 id = uint32(bytes4(_proof[:4]));
        address instance = address(bytes20(_proof[4:24]));
//        require(_isInstanceValid(id, instance), SGX_INVALID_INSTANCE());

        bytes32 signatureHash = LibPublicInput.hashPublicInputs(
            _aggregatedProvingHash, address(this), instance, taikoChainId
        );

        // Verify the signature was created by the registered instance
        bytes memory signature = _proof[24:];
//        require(instance == ECDSA.recover(signatureHash, signature), SGX_INVALID_PROOF());
    }

    function _addInstances(
        address[] memory _instances,
        bool instantValid
    )
        private
        returns (uint256[] memory ids)
    {
        uint256 size = _instances.length;
        ids = new uint256[](size);

        uint64 validSince = uint64(block.timestamp);

        if (!instantValid) {
            validSince += INSTANCE_VALIDITY_DELAY;
        }

        for (uint256 i; i < size; ++i) {
            require(!addressRegistered[_instances[i]], SGX_ALREADY_ATTESTED());

            addressRegistered[_instances[i]] = true;

            require(_instances[i] != address(0), SGX_INVALID_INSTANCE());

            instances[nextInstanceId] = Instance(_instances[i], validSince);
            ids[i] = nextInstanceId;

            emit InstanceAdded(nextInstanceId, _instances[i], address(0), validSince);

            ++nextInstanceId;
        }
    }

    function _isInstanceValid(uint256 id, address instance) private view returns (bool) {
        require(instance != address(0), SGX_INVALID_INSTANCE());
        require(instance == instances[id].addr, SGX_INVALID_INSTANCE());
        return instances[id].validSince <= block.timestamp
            && block.timestamp <= instances[id].validSince + INSTANCE_EXPIRY;
    }
}
        

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.24;

import "../lib/QuoteV3Auth/V3Struct.sol";

/// @title IAttestation
/// @custom:security-contact security@taiko.xyz
interface IAttestation {
    function verifyAttestation(bytes calldata data) external returns (bool);
    function verifyParsedQuote(V3Struct.ParsedV3QuoteStruct calldata v3quote)
        external
        returns (bool success, bytes memory retData);
}
          

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.24;

/// @title V3Struct
/// @custom:security-contact security@taiko.xyz
library V3Struct {
    struct Header {
        bytes2 version;
        bytes2 attestationKeyType;
        bytes4 teeType;
        bytes2 qeSvn;
        bytes2 pceSvn;
        bytes16 qeVendorId;
        bytes20 userData;
    }

    struct EnclaveReport {
        bytes16 cpuSvn;
        bytes4 miscSelect;
        bytes28 reserved1;
        bytes16 attributes;
        bytes32 mrEnclave;
        bytes32 reserved2;
        bytes32 mrSigner;
        bytes reserved3; // 96 bytes
        uint16 isvProdId;
        uint16 isvSvn;
        bytes reserved4; // 60 bytes
        bytes reportData; // 64 bytes - For QEReports, this contains the hash of the concatenation
        // of attestation key and QEAuthData
    }

    struct QEAuthData {
        uint16 parsedDataSize;
        bytes data;
    }

    struct CertificationData {
        uint16 certType;
        // TODO(Yue): In encoded path, we need to calculate the size of certDataArray
        // certDataSize = len(join((BEGIN_CERT, certArray[i], END_CERT) for i in 0..3))
        // But for plain bytes path, we don't need that.
        uint32 certDataSize;
        bytes[3] decodedCertDataArray; // base64 decoded cert bytes array
    }

    struct ECDSAQuoteV3AuthData {
        bytes ecdsa256BitSignature; // 64 bytes
        bytes ecdsaAttestationKey; // 64 bytes
        EnclaveReport pckSignedQeReport; // 384 bytes
        bytes qeReportSignature; // 64 bytes
        QEAuthData qeAuthData;
        CertificationData certification;
    }

    struct ParsedV3QuoteStruct {
        Header header;
        EnclaveReport localEnclaveReport;
        ECDSAQuoteV3AuthData v3AuthData;
    }
}
          

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.24;

/// @title IProofVerifier
/// @notice Interface for verifying validity proofs for state transitions
/// @custom:security-contact security@taiko.xyz
interface IProofVerifier {
    /// @notice Verifies a validity proof for a state transition
    /// @dev This function must revert if the proof is invalid
    /// @param _proposalAge The age in seconds of the proposal being proven. Only set for
    ///        single-proposal proofs (calculated as block.timestamp - proposal.timestamp).
    ///        For multi-proposal batches, this is always 0, meaning "not applicable".
    ///        Verifiers should interpret _proposalAge == 0 as "not applicable" rather than
    ///        "instant proof". This parameter enables age-based verification logic, such as
    ///        detecting and handling prover-killer proposals differently.
    /// @param _commitmentHash Hash of the last proposal hash and commitment data
    /// @param _proof The proof data
    function verifyProof(
        uint256 _proposalAge,
        bytes32 _commitmentHash,
        bytes calldata _proof
    )
        external
        view;
}
          

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.24;

import { EfficientHashLib } from "solady/src/utils/EfficientHashLib.sol";

/// @title LibPublicInput
/// @notice A library for handling hashing the so-called public input hash, used by sgx and zk
/// proofs.
/// @custom:security-contact security@taiko.xyz
library LibPublicInput {
    /// @notice Hashes the public input for the proof verification.
    /// @param _aggregatedProvingHash The aggregated proving hash from the inbox.
    /// @param _verifierContract The contract address which as current verifier.
    /// @param _proofSigner The address of the instance that signed this proof. For SGX it is the
    /// signer address, for ZK this variable is not used and must have value address(0).
    /// @param _chainId The chain id.
    /// @return The public input hash.
    function hashPublicInputs(
        bytes32 _aggregatedProvingHash,
        address _verifierContract,
        address _proofSigner,
        uint64 _chainId
    )
        internal
        pure
        returns (bytes32)
    {
        require(_aggregatedProvingHash != bytes32(0), InvalidAggregatedProvingHash());
        return EfficientHashLib.hash(
            bytes32("VERIFY_PROOF"),
            bytes32(uint256(_chainId)),
            bytes32(uint256(uint160(_verifierContract))),
            _aggregatedProvingHash,
            bytes32(uint256(uint160(_proofSigner)))
        );
    }

    /// @dev Hashes the public input for the ZK aggregation proof verification,
    ///         which contains the sub image id to be aggregated for security.
    /// @param _blockProvingProgram The proving program identifier.
    /// @param _aggregatedProvingHash The aggregated proving hash from the inbox.
    /// @return The ZK aggregation public input hash.
    function hashZKAggregationPublicInputs(
        bytes32 _blockProvingProgram,
        bytes32 _aggregatedProvingHash
    )
        internal
        pure
        returns (bytes32)
    {
        return EfficientHashLib.hash(_blockProvingProgram, _aggregatedProvingHash);
    }

    // ---------------------------------------------------------------
    // Errors
    // ---------------------------------------------------------------

    error InvalidAggregatedProvingHash();
}
          

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.9.0) (access/Ownable.sol)

pragma solidity ^0.8.0;

import "../utils/Context.sol";

/**
 * @dev Contract module which provides a basic access control mechanism, where
 * there is an account (an owner) that can be granted exclusive access to
 * specific functions.
 *
 * By default, the owner account will be the one that deploys the contract. This
 * can later be changed with {transferOwnership}.
 *
 * This module is used through inheritance. It will make available the modifier
 * `onlyOwner`, which can be applied to your functions to restrict their use to
 * the owner.
 */
abstract contract Ownable is Context {
    address private _owner;

    event OwnershipTransferred(address indexed previousOwner, address indexed newOwner);

    /**
     * @dev Initializes the contract setting the deployer as the initial owner.
     */
    constructor() {
        _transferOwnership(_msgSender());
    }

    /**
     * @dev Throws if called by any account other than the owner.
     */
    modifier onlyOwner() {
        _checkOwner();
        _;
    }

    /**
     * @dev Returns the address of the current owner.
     */
    function owner() public view virtual returns (address) {
        return _owner;
    }

    /**
     * @dev Throws if the sender is not the owner.
     */
    function _checkOwner() internal view virtual {
        require(owner() == _msgSender(), "Ownable: caller is not the owner");
    }

    /**
     * @dev Leaves the contract without owner. It will not be possible to call
     * `onlyOwner` functions. Can only be called by the current owner.
     *
     * NOTE: Renouncing ownership will leave the contract without an owner,
     * thereby disabling any functionality that is only available to the owner.
     */
    function renounceOwnership() public virtual onlyOwner {
        _transferOwnership(address(0));
    }

    /**
     * @dev Transfers ownership of the contract to a new account (`newOwner`).
     * Can only be called by the current owner.
     */
    function transferOwnership(address newOwner) public virtual onlyOwner {
        require(newOwner != address(0), "Ownable: new owner is the zero address");
        _transferOwnership(newOwner);
    }

    /**
     * @dev Transfers ownership of the contract to a new account (`newOwner`).
     * Internal function without access restriction.
     */
    function _transferOwnership(address newOwner) internal virtual {
        address oldOwner = _owner;
        _owner = newOwner;
        emit OwnershipTransferred(oldOwner, newOwner);
    }
}
          

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.9.0) (access/Ownable2Step.sol)

pragma solidity ^0.8.0;

import "./Ownable.sol";

/**
 * @dev Contract module which provides access control mechanism, where
 * there is an account (an owner) that can be granted exclusive access to
 * specific functions.
 *
 * By default, the owner account will be the one that deploys the contract. This
 * can later be changed with {transferOwnership} and {acceptOwnership}.
 *
 * This module is used through inheritance. It will make available all functions
 * from parent (Ownable).
 */
abstract contract Ownable2Step is Ownable {
    address private _pendingOwner;

    event OwnershipTransferStarted(address indexed previousOwner, address indexed newOwner);

    /**
     * @dev Returns the address of the pending owner.
     */
    function pendingOwner() public view virtual returns (address) {
        return _pendingOwner;
    }

    /**
     * @dev Starts the ownership transfer of the contract to a new account. Replaces the pending transfer if there is one.
     * Can only be called by the current owner.
     */
    function transferOwnership(address newOwner) public virtual override onlyOwner {
        _pendingOwner = newOwner;
        emit OwnershipTransferStarted(owner(), newOwner);
    }

    /**
     * @dev Transfers ownership of the contract to a new account (`newOwner`) and deletes any pending owner.
     * Internal function without access restriction.
     */
    function _transferOwnership(address newOwner) internal virtual override {
        delete _pendingOwner;
        super._transferOwnership(newOwner);
    }

    /**
     * @dev The new owner accepts the ownership transfer.
     */
    function acceptOwnership() public virtual {
        address sender = _msgSender();
        require(pendingOwner() == sender, "Ownable2Step: caller is not the new owner");
        _transferOwnership(sender);
    }
}
          

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.9.4) (utils/Context.sol)

pragma solidity ^0.8.0;

/**
 * @dev Provides information about the current execution context, including the
 * sender of the transaction and its data. While these are generally available
 * via msg.sender and msg.data, they should not be accessed in such a direct
 * manner, since when dealing with meta-transactions the account sending and
 * paying for execution may not be the actual sender (as far as an application
 * is concerned).
 *
 * This contract is only required for intermediate, library-like contracts.
 */
abstract contract Context {
    function _msgSender() internal view virtual returns (address) {
        return msg.sender;
    }

    function _msgData() internal view virtual returns (bytes calldata) {
        return msg.data;
    }

    function _contextSuffixLength() internal view virtual returns (uint256) {
        return 0;
    }
}
          

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.9.0) (utils/Strings.sol)

pragma solidity ^0.8.0;

import "./math/Math.sol";
import "./math/SignedMath.sol";

/**
 * @dev String operations.
 */
library Strings {
    bytes16 private constant _SYMBOLS = "0123456789abcdef";
    uint8 private constant _ADDRESS_LENGTH = 20;

    /**
     * @dev Converts a `uint256` to its ASCII `string` decimal representation.
     */
    function toString(uint256 value) internal pure returns (string memory) {
        unchecked {
            uint256 length = Math.log10(value) + 1;
            string memory buffer = new string(length);
            uint256 ptr;
            /// @solidity memory-safe-assembly
            assembly {
                ptr := add(buffer, add(32, length))
            }
            while (true) {
                ptr--;
                /// @solidity memory-safe-assembly
                assembly {
                    mstore8(ptr, byte(mod(value, 10), _SYMBOLS))
                }
                value /= 10;
                if (value == 0) break;
            }
            return buffer;
        }
    }

    /**
     * @dev Converts a `int256` to its ASCII `string` decimal representation.
     */
    function toString(int256 value) internal pure returns (string memory) {
        return string(abi.encodePacked(value < 0 ? "-" : "", toString(SignedMath.abs(value))));
    }

    /**
     * @dev Converts a `uint256` to its ASCII `string` hexadecimal representation.
     */
    function toHexString(uint256 value) internal pure returns (string memory) {
        unchecked {
            return toHexString(value, Math.log256(value) + 1);
        }
    }

    /**
     * @dev Converts a `uint256` to its ASCII `string` hexadecimal representation with fixed length.
     */
    function toHexString(uint256 value, uint256 length) internal pure returns (string memory) {
        bytes memory buffer = new bytes(2 * length + 2);
        buffer[0] = "0";
        buffer[1] = "x";
        for (uint256 i = 2 * length + 1; i > 1; --i) {
            buffer[i] = _SYMBOLS[value & 0xf];
            value >>= 4;
        }
        require(value == 0, "Strings: hex length insufficient");
        return string(buffer);
    }

    /**
     * @dev Converts an `address` with fixed length of 20 bytes to its not checksummed ASCII `string` hexadecimal representation.
     */
    function toHexString(address addr) internal pure returns (string memory) {
        return toHexString(uint256(uint160(addr)), _ADDRESS_LENGTH);
    }

    /**
     * @dev Returns true if the two strings are equal.
     */
    function equal(string memory a, string memory b) internal pure returns (bool) {
        return keccak256(bytes(a)) == keccak256(bytes(b));
    }
}
          

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.9.0) (utils/cryptography/ECDSA.sol)

pragma solidity ^0.8.0;

import "../Strings.sol";

/**
 * @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.
 */
library ECDSA {
    enum RecoverError {
        NoError,
        InvalidSignature,
        InvalidSignatureLength,
        InvalidSignatureS,
        InvalidSignatureV // Deprecated in v4.8
    }

    function _throwError(RecoverError error) private pure {
        if (error == RecoverError.NoError) {
            return; // no error: do nothing
        } else if (error == RecoverError.InvalidSignature) {
            revert("ECDSA: invalid signature");
        } else if (error == RecoverError.InvalidSignatureLength) {
            revert("ECDSA: invalid signature length");
        } else if (error == RecoverError.InvalidSignatureS) {
            revert("ECDSA: invalid signature 's' value");
        }
    }

    /**
     * @dev Returns the address that signed a hashed message (`hash`) with
     * `signature` or error string. 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]
     *
     * _Available since v4.3._
     */
    function tryRecover(bytes32 hash, bytes memory signature) internal pure returns (address, RecoverError) {
        if (signature.length == 65) {
            bytes32 r;
            bytes32 s;
            uint8 v;
            // ecrecover takes the signature parameters, and the only way to get them
            // currently is to use assembly.
            /// @solidity memory-safe-assembly
            assembly {
                r := mload(add(signature, 0x20))
                s := mload(add(signature, 0x40))
                v := byte(0, mload(add(signature, 0x60)))
            }
            return tryRecover(hash, v, r, s);
        } else {
            return (address(0), RecoverError.InvalidSignatureLength);
        }
    }

    /**
     * @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.
     */
    function recover(bytes32 hash, bytes memory signature) internal pure returns (address) {
        (address recovered, RecoverError error) = tryRecover(hash, signature);
        _throwError(error);
        return recovered;
    }

    /**
     * @dev Overload of {ECDSA-tryRecover} that receives the `r` and `vs` short-signature fields separately.
     *
     * See https://eips.ethereum.org/EIPS/eip-2098[EIP-2098 short signatures]
     *
     * _Available since v4.3._
     */
    function tryRecover(bytes32 hash, bytes32 r, bytes32 vs) internal pure returns (address, RecoverError) {
        bytes32 s = vs & bytes32(0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff);
        uint8 v = uint8((uint256(vs) >> 255) + 27);
        return tryRecover(hash, v, r, s);
    }

    /**
     * @dev Overload of {ECDSA-recover} that receives the `r and `vs` short-signature fields separately.
     *
     * _Available since v4.2._
     */
    function recover(bytes32 hash, bytes32 r, bytes32 vs) internal pure returns (address) {
        (address recovered, RecoverError error) = tryRecover(hash, r, vs);
        _throwError(error);
        return recovered;
    }

    /**
     * @dev Overload of {ECDSA-tryRecover} that receives the `v`,
     * `r` and `s` signature fields separately.
     *
     * _Available since v4.3._
     */
    function tryRecover(bytes32 hash, uint8 v, bytes32 r, bytes32 s) internal pure returns (address, RecoverError) {
        // 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 (301): 0 < s < secp256k1n ÷ 2 + 1, and for v in (302): 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.
        if (uint256(s) > 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5D576E7357A4501DDFE92F46681B20A0) {
            return (address(0), RecoverError.InvalidSignatureS);
        }

        // If the signature is valid (and not malleable), return the signer address
        address signer = ecrecover(hash, v, r, s);
        if (signer == address(0)) {
            return (address(0), RecoverError.InvalidSignature);
        }

        return (signer, RecoverError.NoError);
    }

    /**
     * @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) {
        (address recovered, RecoverError error) = tryRecover(hash, v, r, s);
        _throwError(error);
        return recovered;
    }

    /**
     * @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 message) {
        // 32 is the length in bytes of hash,
        // enforced by the type signature above
        /// @solidity memory-safe-assembly
        assembly {
            mstore(0x00, "\x19Ethereum Signed Message:\n32")
            mstore(0x1c, hash)
            message := keccak256(0x00, 0x3c)
        }
    }

    /**
     * @dev Returns an Ethereum Signed Message, created from `s`. 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(bytes memory s) internal pure returns (bytes32) {
        return keccak256(abi.encodePacked("\x19Ethereum Signed Message:\n", Strings.toString(s.length), s));
    }

    /**
     * @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 data) {
        /// @solidity memory-safe-assembly
        assembly {
            let ptr := mload(0x40)
            mstore(ptr, "\x19\x01")
            mstore(add(ptr, 0x02), domainSeparator)
            mstore(add(ptr, 0x22), structHash)
            data := keccak256(ptr, 0x42)
        }
    }

    /**
     * @dev Returns an Ethereum Signed Data with intended validator, created from a
     * `validator` and `data` according to the version 0 of EIP-191.
     *
     * See {recover}.
     */
    function toDataWithIntendedValidatorHash(address validator, bytes memory data) internal pure returns (bytes32) {
        return keccak256(abi.encodePacked("\x19\x00", validator, data));
    }
}
          

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.9.0) (utils/math/Math.sol)

pragma solidity ^0.8.0;

/**
 * @dev Standard math utilities missing in the Solidity language.
 */
library Math {
    enum Rounding {
        Down, // Toward negative infinity
        Up, // Toward infinity
        Zero // Toward zero
    }

    /**
     * @dev Returns the largest of two numbers.
     */
    function max(uint256 a, uint256 b) internal pure returns (uint256) {
        return a > b ? a : b;
    }

    /**
     * @dev Returns the smallest of two numbers.
     */
    function min(uint256 a, uint256 b) internal pure returns (uint256) {
        return a < b ? a : b;
    }

    /**
     * @dev Returns the average of two numbers. The result is rounded towards
     * zero.
     */
    function average(uint256 a, uint256 b) internal pure returns (uint256) {
        // (a + b) / 2 can overflow.
        return (a & b) + (a ^ b) / 2;
    }

    /**
     * @dev Returns the ceiling of the division of two numbers.
     *
     * This differs from standard division with `/` in that it rounds up instead
     * of rounding down.
     */
    function ceilDiv(uint256 a, uint256 b) internal pure returns (uint256) {
        // (a + b - 1) / b can overflow on addition, so we distribute.
        return a == 0 ? 0 : (a - 1) / b + 1;
    }

    /**
     * @notice Calculates floor(x * y / denominator) with full precision. Throws if result overflows a uint256 or denominator == 0
     * @dev Original credit to Remco Bloemen under MIT license (https://xn--2-umb.com/21/muldiv)
     * with further edits by Uniswap Labs also under MIT license.
     */
    function mulDiv(uint256 x, uint256 y, uint256 denominator) internal pure returns (uint256 result) {
        unchecked {
            // 512-bit multiply [prod1 prod0] = x * y. Compute the product mod 2^256 and mod 2^256 - 1, then use
            // use the Chinese Remainder Theorem to reconstruct the 512 bit result. The result is stored in two 256
            // variables such that product = prod1 * 2^256 + prod0.
            uint256 prod0; // Least significant 256 bits of the product
            uint256 prod1; // Most significant 256 bits of the product
            assembly {
                let mm := mulmod(x, y, not(0))
                prod0 := mul(x, y)
                prod1 := sub(sub(mm, prod0), lt(mm, prod0))
            }

            // Handle non-overflow cases, 256 by 256 division.
            if (prod1 == 0) {
                // Solidity will revert if denominator == 0, unlike the div opcode on its own.
                // The surrounding unchecked block does not change this fact.
                // See https://docs.soliditylang.org/en/latest/control-structures.html#checked-or-unchecked-arithmetic.
                return prod0 / denominator;
            }

            // Make sure the result is less than 2^256. Also prevents denominator == 0.
            require(denominator > prod1, "Math: mulDiv overflow");

            ///////////////////////////////////////////////
            // 512 by 256 division.
            ///////////////////////////////////////////////

            // Make division exact by subtracting the remainder from [prod1 prod0].
            uint256 remainder;
            assembly {
                // Compute remainder using mulmod.
                remainder := mulmod(x, y, denominator)

                // Subtract 256 bit number from 512 bit number.
                prod1 := sub(prod1, gt(remainder, prod0))
                prod0 := sub(prod0, remainder)
            }

            // Factor powers of two out of denominator and compute largest power of two divisor of denominator. Always >= 1.
            // See https://cs.stackexchange.com/q/138556/92363.

            // Does not overflow because the denominator cannot be zero at this stage in the function.
            uint256 twos = denominator & (~denominator + 1);
            assembly {
                // Divide denominator by twos.
                denominator := div(denominator, twos)

                // Divide [prod1 prod0] by twos.
                prod0 := div(prod0, twos)

                // Flip twos such that it is 2^256 / twos. If twos is zero, then it becomes one.
                twos := add(div(sub(0, twos), twos), 1)
            }

            // Shift in bits from prod1 into prod0.
            prod0 |= prod1 * twos;

            // Invert denominator mod 2^256. Now that denominator is an odd number, it has an inverse modulo 2^256 such
            // that denominator * inv = 1 mod 2^256. Compute the inverse by starting with a seed that is correct for
            // four bits. That is, denominator * inv = 1 mod 2^4.
            uint256 inverse = (3 * denominator) ^ 2;

            // Use the Newton-Raphson iteration to improve the precision. Thanks to Hensel's lifting lemma, this also works
            // in modular arithmetic, doubling the correct bits in each step.
            inverse *= 2 - denominator * inverse; // inverse mod 2^8
            inverse *= 2 - denominator * inverse; // inverse mod 2^16
            inverse *= 2 - denominator * inverse; // inverse mod 2^32
            inverse *= 2 - denominator * inverse; // inverse mod 2^64
            inverse *= 2 - denominator * inverse; // inverse mod 2^128
            inverse *= 2 - denominator * inverse; // inverse mod 2^256

            // Because the division is now exact we can divide by multiplying with the modular inverse of denominator.
            // This will give us the correct result modulo 2^256. Since the preconditions guarantee that the outcome is
            // less than 2^256, this is the final result. We don't need to compute the high bits of the result and prod1
            // is no longer required.
            result = prod0 * inverse;
            return result;
        }
    }

    /**
     * @notice Calculates x * y / denominator with full precision, following the selected rounding direction.
     */
    function mulDiv(uint256 x, uint256 y, uint256 denominator, Rounding rounding) internal pure returns (uint256) {
        uint256 result = mulDiv(x, y, denominator);
        if (rounding == Rounding.Up && mulmod(x, y, denominator) > 0) {
            result += 1;
        }
        return result;
    }

    /**
     * @dev Returns the square root of a number. If the number is not a perfect square, the value is rounded down.
     *
     * Inspired by Henry S. Warren, Jr.'s "Hacker's Delight" (Chapter 11).
     */
    function sqrt(uint256 a) internal pure returns (uint256) {
        if (a == 0) {
            return 0;
        }

        // For our first guess, we get the biggest power of 2 which is smaller than the square root of the target.
        //
        // We know that the "msb" (most significant bit) of our target number `a` is a power of 2 such that we have
        // `msb(a) <= a < 2*msb(a)`. This value can be written `msb(a)=2**k` with `k=log2(a)`.
        //
        // This can be rewritten `2**log2(a) <= a < 2**(log2(a) + 1)`
        // → `sqrt(2**k) <= sqrt(a) < sqrt(2**(k+1))`
        // → `2**(k/2) <= sqrt(a) < 2**((k+1)/2) <= 2**(k/2 + 1)`
        //
        // Consequently, `2**(log2(a) / 2)` is a good first approximation of `sqrt(a)` with at least 1 correct bit.
        uint256 result = 1 << (log2(a) >> 1);

        // At this point `result` is an estimation with one bit of precision. We know the true value is a uint128,
        // since it is the square root of a uint256. Newton's method converges quadratically (precision doubles at
        // every iteration). We thus need at most 7 iteration to turn our partial result with one bit of precision
        // into the expected uint128 result.
        unchecked {
            result = (result + a / result) >> 1;
            result = (result + a / result) >> 1;
            result = (result + a / result) >> 1;
            result = (result + a / result) >> 1;
            result = (result + a / result) >> 1;
            result = (result + a / result) >> 1;
            result = (result + a / result) >> 1;
            return min(result, a / result);
        }
    }

    /**
     * @notice Calculates sqrt(a), following the selected rounding direction.
     */
    function sqrt(uint256 a, Rounding rounding) internal pure returns (uint256) {
        unchecked {
            uint256 result = sqrt(a);
            return result + (rounding == Rounding.Up && result * result < a ? 1 : 0);
        }
    }

    /**
     * @dev Return the log in base 2, rounded down, of a positive value.
     * Returns 0 if given 0.
     */
    function log2(uint256 value) internal pure returns (uint256) {
        uint256 result = 0;
        unchecked {
            if (value >> 128 > 0) {
                value >>= 128;
                result += 128;
            }
            if (value >> 64 > 0) {
                value >>= 64;
                result += 64;
            }
            if (value >> 32 > 0) {
                value >>= 32;
                result += 32;
            }
            if (value >> 16 > 0) {
                value >>= 16;
                result += 16;
            }
            if (value >> 8 > 0) {
                value >>= 8;
                result += 8;
            }
            if (value >> 4 > 0) {
                value >>= 4;
                result += 4;
            }
            if (value >> 2 > 0) {
                value >>= 2;
                result += 2;
            }
            if (value >> 1 > 0) {
                result += 1;
            }
        }
        return result;
    }

    /**
     * @dev Return the log in base 2, following the selected rounding direction, of a positive value.
     * Returns 0 if given 0.
     */
    function log2(uint256 value, Rounding rounding) internal pure returns (uint256) {
        unchecked {
            uint256 result = log2(value);
            return result + (rounding == Rounding.Up && 1 << result < value ? 1 : 0);
        }
    }

    /**
     * @dev Return the log in base 10, rounded down, of a positive value.
     * Returns 0 if given 0.
     */
    function log10(uint256 value) internal pure returns (uint256) {
        uint256 result = 0;
        unchecked {
            if (value >= 10 ** 64) {
                value /= 10 ** 64;
                result += 64;
            }
            if (value >= 10 ** 32) {
                value /= 10 ** 32;
                result += 32;
            }
            if (value >= 10 ** 16) {
                value /= 10 ** 16;
                result += 16;
            }
            if (value >= 10 ** 8) {
                value /= 10 ** 8;
                result += 8;
            }
            if (value >= 10 ** 4) {
                value /= 10 ** 4;
                result += 4;
            }
            if (value >= 10 ** 2) {
                value /= 10 ** 2;
                result += 2;
            }
            if (value >= 10 ** 1) {
                result += 1;
            }
        }
        return result;
    }

    /**
     * @dev Return the log in base 10, following the selected rounding direction, of a positive value.
     * Returns 0 if given 0.
     */
    function log10(uint256 value, Rounding rounding) internal pure returns (uint256) {
        unchecked {
            uint256 result = log10(value);
            return result + (rounding == Rounding.Up && 10 ** result < value ? 1 : 0);
        }
    }

    /**
     * @dev Return the log in base 256, rounded down, of a positive value.
     * Returns 0 if given 0.
     *
     * Adding one to the result gives the number of pairs of hex symbols needed to represent `value` as a hex string.
     */
    function log256(uint256 value) internal pure returns (uint256) {
        uint256 result = 0;
        unchecked {
            if (value >> 128 > 0) {
                value >>= 128;
                result += 16;
            }
            if (value >> 64 > 0) {
                value >>= 64;
                result += 8;
            }
            if (value >> 32 > 0) {
                value >>= 32;
                result += 4;
            }
            if (value >> 16 > 0) {
                value >>= 16;
                result += 2;
            }
            if (value >> 8 > 0) {
                result += 1;
            }
        }
        return result;
    }

    /**
     * @dev Return the log in base 256, following the selected rounding direction, of a positive value.
     * Returns 0 if given 0.
     */
    function log256(uint256 value, Rounding rounding) internal pure returns (uint256) {
        unchecked {
            uint256 result = log256(value);
            return result + (rounding == Rounding.Up && 1 << (result << 3) < value ? 1 : 0);
        }
    }
}
          

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.8.0) (utils/math/SignedMath.sol)

pragma solidity ^0.8.0;

/**
 * @dev Standard signed math utilities missing in the Solidity language.
 */
library SignedMath {
    /**
     * @dev Returns the largest of two signed numbers.
     */
    function max(int256 a, int256 b) internal pure returns (int256) {
        return a > b ? a : b;
    }

    /**
     * @dev Returns the smallest of two signed numbers.
     */
    function min(int256 a, int256 b) internal pure returns (int256) {
        return a < b ? a : b;
    }

    /**
     * @dev Returns the average of two signed numbers without overflow.
     * The result is rounded towards zero.
     */
    function average(int256 a, int256 b) internal pure returns (int256) {
        // Formula from the book "Hacker's Delight"
        int256 x = (a & b) + ((a ^ b) >> 1);
        return x + (int256(uint256(x) >> 255) & (a ^ b));
    }

    /**
     * @dev Returns the absolute unsigned value of a signed value.
     */
    function abs(int256 n) internal pure returns (uint256) {
        unchecked {
            // must be unchecked in order to support `n = type(int256).min`
            return uint256(n >= 0 ? n : -n);
        }
    }
}
          

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.4;

/// @notice Library for efficiently performing keccak256 hashes.
/// @author Solady (https://github.com/vectorized/solady/blob/main/src/utils/EfficientHashLib.sol)
/// @dev To avoid stack-too-deep, you can use:
/// ```
/// bytes32[] memory buffer = EfficientHashLib.malloc(10);
/// EfficientHashLib.set(buffer, 0, value0);
/// ..
/// EfficientHashLib.set(buffer, 9, value9);
/// bytes32 finalHash = EfficientHashLib.hash(buffer);
/// ```
library EfficientHashLib {
    /*´:°•.°+.*•´.*:˚.°*.˚•´.°:°•.°•.*•´.*:˚.°*.˚•´.°:°•.°+.*•´.*:*/
    /*               MALLOC-LESS HASHING OPERATIONS               */
    /*.•°:°.´+˚.*°.˚:*.´•*.+°.•°:´*.´•*.•°.•°:°.´:•˚°.*°.˚:*.´+°.•*/

    /// @dev Returns `keccak256(abi.encode(v0))`.
    function hash(bytes32 v0) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            mstore(0x00, v0)
            result := keccak256(0x00, 0x20)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0))`.
    function hash(uint256 v0) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            mstore(0x00, v0)
            result := keccak256(0x00, 0x20)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, v1))`.
    function hash(bytes32 v0, bytes32 v1) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            mstore(0x00, v0)
            mstore(0x20, v1)
            result := keccak256(0x00, 0x40)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, v1))`.
    function hash(uint256 v0, uint256 v1) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            mstore(0x00, v0)
            mstore(0x20, v1)
            result := keccak256(0x00, 0x40)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, v1, v2))`.
    function hash(bytes32 v0, bytes32 v1, bytes32 v2) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            result := keccak256(m, 0x60)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, v1, v2))`.
    function hash(uint256 v0, uint256 v1, uint256 v2) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            result := keccak256(m, 0x60)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, v1, v2, v3))`.
    function hash(bytes32 v0, bytes32 v1, bytes32 v2, bytes32 v3)
        internal
        pure
        returns (bytes32 result)
    {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            result := keccak256(m, 0x80)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, v1, v2, v3))`.
    function hash(uint256 v0, uint256 v1, uint256 v2, uint256 v3)
        internal
        pure
        returns (bytes32 result)
    {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            result := keccak256(m, 0x80)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v4))`.
    function hash(bytes32 v0, bytes32 v1, bytes32 v2, bytes32 v3, bytes32 v4)
        internal
        pure
        returns (bytes32 result)
    {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            result := keccak256(m, 0xa0)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v4))`.
    function hash(uint256 v0, uint256 v1, uint256 v2, uint256 v3, uint256 v4)
        internal
        pure
        returns (bytes32 result)
    {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            result := keccak256(m, 0xa0)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v5))`.
    function hash(bytes32 v0, bytes32 v1, bytes32 v2, bytes32 v3, bytes32 v4, bytes32 v5)
        internal
        pure
        returns (bytes32 result)
    {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            result := keccak256(m, 0xc0)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v5))`.
    function hash(uint256 v0, uint256 v1, uint256 v2, uint256 v3, uint256 v4, uint256 v5)
        internal
        pure
        returns (bytes32 result)
    {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            result := keccak256(m, 0xc0)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v6))`.
    function hash(
        bytes32 v0,
        bytes32 v1,
        bytes32 v2,
        bytes32 v3,
        bytes32 v4,
        bytes32 v5,
        bytes32 v6
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            result := keccak256(m, 0xe0)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v6))`.
    function hash(
        uint256 v0,
        uint256 v1,
        uint256 v2,
        uint256 v3,
        uint256 v4,
        uint256 v5,
        uint256 v6
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            result := keccak256(m, 0xe0)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v7))`.
    function hash(
        bytes32 v0,
        bytes32 v1,
        bytes32 v2,
        bytes32 v3,
        bytes32 v4,
        bytes32 v5,
        bytes32 v6,
        bytes32 v7
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            result := keccak256(m, 0x100)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v7))`.
    function hash(
        uint256 v0,
        uint256 v1,
        uint256 v2,
        uint256 v3,
        uint256 v4,
        uint256 v5,
        uint256 v6,
        uint256 v7
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            result := keccak256(m, 0x100)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v8))`.
    function hash(
        bytes32 v0,
        bytes32 v1,
        bytes32 v2,
        bytes32 v3,
        bytes32 v4,
        bytes32 v5,
        bytes32 v6,
        bytes32 v7,
        bytes32 v8
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            mstore(add(m, 0x100), v8)
            result := keccak256(m, 0x120)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v8))`.
    function hash(
        uint256 v0,
        uint256 v1,
        uint256 v2,
        uint256 v3,
        uint256 v4,
        uint256 v5,
        uint256 v6,
        uint256 v7,
        uint256 v8
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            mstore(add(m, 0x100), v8)
            result := keccak256(m, 0x120)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v9))`.
    function hash(
        bytes32 v0,
        bytes32 v1,
        bytes32 v2,
        bytes32 v3,
        bytes32 v4,
        bytes32 v5,
        bytes32 v6,
        bytes32 v7,
        bytes32 v8,
        bytes32 v9
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            mstore(add(m, 0x100), v8)
            mstore(add(m, 0x120), v9)
            result := keccak256(m, 0x140)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v9))`.
    function hash(
        uint256 v0,
        uint256 v1,
        uint256 v2,
        uint256 v3,
        uint256 v4,
        uint256 v5,
        uint256 v6,
        uint256 v7,
        uint256 v8,
        uint256 v9
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            mstore(add(m, 0x100), v8)
            mstore(add(m, 0x120), v9)
            result := keccak256(m, 0x140)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v10))`.
    function hash(
        bytes32 v0,
        bytes32 v1,
        bytes32 v2,
        bytes32 v3,
        bytes32 v4,
        bytes32 v5,
        bytes32 v6,
        bytes32 v7,
        bytes32 v8,
        bytes32 v9,
        bytes32 v10
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            mstore(add(m, 0x100), v8)
            mstore(add(m, 0x120), v9)
            mstore(add(m, 0x140), v10)
            result := keccak256(m, 0x160)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v10))`.
    function hash(
        uint256 v0,
        uint256 v1,
        uint256 v2,
        uint256 v3,
        uint256 v4,
        uint256 v5,
        uint256 v6,
        uint256 v7,
        uint256 v8,
        uint256 v9,
        uint256 v10
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            mstore(add(m, 0x100), v8)
            mstore(add(m, 0x120), v9)
            mstore(add(m, 0x140), v10)
            result := keccak256(m, 0x160)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v11))`.
    function hash(
        bytes32 v0,
        bytes32 v1,
        bytes32 v2,
        bytes32 v3,
        bytes32 v4,
        bytes32 v5,
        bytes32 v6,
        bytes32 v7,
        bytes32 v8,
        bytes32 v9,
        bytes32 v10,
        bytes32 v11
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            mstore(add(m, 0x100), v8)
            mstore(add(m, 0x120), v9)
            mstore(add(m, 0x140), v10)
            mstore(add(m, 0x160), v11)
            result := keccak256(m, 0x180)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v11))`.
    function hash(
        uint256 v0,
        uint256 v1,
        uint256 v2,
        uint256 v3,
        uint256 v4,
        uint256 v5,
        uint256 v6,
        uint256 v7,
        uint256 v8,
        uint256 v9,
        uint256 v10,
        uint256 v11
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            mstore(add(m, 0x100), v8)
            mstore(add(m, 0x120), v9)
            mstore(add(m, 0x140), v10)
            mstore(add(m, 0x160), v11)
            result := keccak256(m, 0x180)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v12))`.
    function hash(
        bytes32 v0,
        bytes32 v1,
        bytes32 v2,
        bytes32 v3,
        bytes32 v4,
        bytes32 v5,
        bytes32 v6,
        bytes32 v7,
        bytes32 v8,
        bytes32 v9,
        bytes32 v10,
        bytes32 v11,
        bytes32 v12
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            mstore(add(m, 0x100), v8)
            mstore(add(m, 0x120), v9)
            mstore(add(m, 0x140), v10)
            mstore(add(m, 0x160), v11)
            mstore(add(m, 0x180), v12)
            result := keccak256(m, 0x1a0)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v12))`.
    function hash(
        uint256 v0,
        uint256 v1,
        uint256 v2,
        uint256 v3,
        uint256 v4,
        uint256 v5,
        uint256 v6,
        uint256 v7,
        uint256 v8,
        uint256 v9,
        uint256 v10,
        uint256 v11,
        uint256 v12
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            mstore(add(m, 0x100), v8)
            mstore(add(m, 0x120), v9)
            mstore(add(m, 0x140), v10)
            mstore(add(m, 0x160), v11)
            mstore(add(m, 0x180), v12)
            result := keccak256(m, 0x1a0)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v13))`.
    function hash(
        bytes32 v0,
        bytes32 v1,
        bytes32 v2,
        bytes32 v3,
        bytes32 v4,
        bytes32 v5,
        bytes32 v6,
        bytes32 v7,
        bytes32 v8,
        bytes32 v9,
        bytes32 v10,
        bytes32 v11,
        bytes32 v12,
        bytes32 v13
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            mstore(add(m, 0x100), v8)
            mstore(add(m, 0x120), v9)
            mstore(add(m, 0x140), v10)
            mstore(add(m, 0x160), v11)
            mstore(add(m, 0x180), v12)
            mstore(add(m, 0x1a0), v13)
            result := keccak256(m, 0x1c0)
        }
    }

    /// @dev Returns `keccak256(abi.encode(v0, .., v13))`.
    function hash(
        uint256 v0,
        uint256 v1,
        uint256 v2,
        uint256 v3,
        uint256 v4,
        uint256 v5,
        uint256 v6,
        uint256 v7,
        uint256 v8,
        uint256 v9,
        uint256 v10,
        uint256 v11,
        uint256 v12,
        uint256 v13
    ) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let m := mload(0x40)
            mstore(m, v0)
            mstore(add(m, 0x20), v1)
            mstore(add(m, 0x40), v2)
            mstore(add(m, 0x60), v3)
            mstore(add(m, 0x80), v4)
            mstore(add(m, 0xa0), v5)
            mstore(add(m, 0xc0), v6)
            mstore(add(m, 0xe0), v7)
            mstore(add(m, 0x100), v8)
            mstore(add(m, 0x120), v9)
            mstore(add(m, 0x140), v10)
            mstore(add(m, 0x160), v11)
            mstore(add(m, 0x180), v12)
            mstore(add(m, 0x1a0), v13)
            result := keccak256(m, 0x1c0)
        }
    }

    /*´:°•.°+.*•´.*:˚.°*.˚•´.°:°•.°•.*•´.*:˚.°*.˚•´.°:°•.°+.*•´.*:*/
    /*             BYTES32 BUFFER HASHING OPERATIONS              */
    /*.•°:°.´+˚.*°.˚:*.´•*.+°.•°:´*.´•*.•°.•°:°.´:•˚°.*°.˚:*.´+°.•*/

    /// @dev Returns `keccak256(abi.encode(buffer[0], .., buffer[buffer.length - 1]))`.
    function hash(bytes32[] memory buffer) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            result := keccak256(add(buffer, 0x20), shl(5, mload(buffer)))
        }
    }

    /// @dev Sets `buffer[i]` to `value`, without a bounds check.
    /// Returns the `buffer` for function chaining.
    function set(bytes32[] memory buffer, uint256 i, bytes32 value)
        internal
        pure
        returns (bytes32[] memory)
    {
        /// @solidity memory-safe-assembly
        assembly {
            mstore(add(buffer, shl(5, add(1, i))), value)
        }
        return buffer;
    }

    /// @dev Sets `buffer[i]` to `value`, without a bounds check.
    /// Returns the `buffer` for function chaining.
    function set(bytes32[] memory buffer, uint256 i, uint256 value)
        internal
        pure
        returns (bytes32[] memory)
    {
        /// @solidity memory-safe-assembly
        assembly {
            mstore(add(buffer, shl(5, add(1, i))), value)
        }
        return buffer;
    }

    /// @dev Returns `new bytes32[](n)`, without zeroing out the memory.
    function malloc(uint256 n) internal pure returns (bytes32[] memory buffer) {
        /// @solidity memory-safe-assembly
        assembly {
            buffer := mload(0x40)
            mstore(buffer, n)
            mstore(0x40, add(shl(5, add(1, n)), buffer))
        }
    }

    /// @dev Frees memory that has been allocated for `buffer`.
    /// No-op if `buffer.length` is zero, or if new memory has been allocated after `buffer`.
    function free(bytes32[] memory buffer) internal pure {
        /// @solidity memory-safe-assembly
        assembly {
            let n := mload(buffer)
            mstore(shl(6, lt(iszero(n), eq(add(shl(5, add(1, n)), buffer), mload(0x40)))), buffer)
        }
    }

    /*´:°•.°+.*•´.*:˚.°*.˚•´.°:°•.°•.*•´.*:˚.°*.˚•´.°:°•.°+.*•´.*:*/
    /*                      EQUALITY CHECKS                       */
    /*.•°:°.´+˚.*°.˚:*.´•*.+°.•°:´*.´•*.•°.•°:°.´:•˚°.*°.˚:*.´+°.•*/

    /// @dev Returns `a == abi.decode(b, (bytes32))`.
    function eq(bytes32 a, bytes memory b) internal pure returns (bool result) {
        /// @solidity memory-safe-assembly
        assembly {
            result := and(eq(0x20, mload(b)), eq(a, mload(add(b, 0x20))))
        }
    }

    /// @dev Returns `abi.decode(a, (bytes32)) == a`.
    function eq(bytes memory a, bytes32 b) internal pure returns (bool result) {
        /// @solidity memory-safe-assembly
        assembly {
            result := and(eq(0x20, mload(a)), eq(b, mload(add(a, 0x20))))
        }
    }

    /*´:°•.°+.*•´.*:˚.°*.˚•´.°:°•.°•.*•´.*:˚.°*.˚•´.°:°•.°+.*•´.*:*/
    /*               BYTE SLICE HASHING OPERATIONS                */
    /*.•°:°.´+˚.*°.˚:*.´•*.+°.•°:´*.´•*.•°.•°:°.´:•˚°.*°.˚:*.´+°.•*/

    /// @dev Returns the keccak256 of the slice from `start` to `end` (exclusive).
    /// `start` and `end` are byte offsets.
    function hash(bytes memory b, uint256 start, uint256 end)
        internal
        pure
        returns (bytes32 result)
    {
        /// @solidity memory-safe-assembly
        assembly {
            let n := mload(b)
            end := xor(end, mul(xor(end, n), lt(n, end)))
            start := xor(start, mul(xor(start, n), lt(n, start)))
            result := keccak256(add(add(b, 0x20), start), mul(gt(end, start), sub(end, start)))
        }
    }

    /// @dev Returns the keccak256 of the slice from `start` to the end of the bytes.
    function hash(bytes memory b, uint256 start) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let n := mload(b)
            start := xor(start, mul(xor(start, n), lt(n, start)))
            result := keccak256(add(add(b, 0x20), start), mul(gt(n, start), sub(n, start)))
        }
    }

    /// @dev Returns the keccak256 of the bytes.
    function hash(bytes memory b) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            result := keccak256(add(b, 0x20), mload(b))
        }
    }

    /// @dev Returns the keccak256 of the slice from `start` to `end` (exclusive).
    /// `start` and `end` are byte offsets.
    function hashCalldata(bytes calldata b, uint256 start, uint256 end)
        internal
        pure
        returns (bytes32 result)
    {
        /// @solidity memory-safe-assembly
        assembly {
            end := xor(end, mul(xor(end, b.length), lt(b.length, end)))
            start := xor(start, mul(xor(start, b.length), lt(b.length, start)))
            let n := mul(gt(end, start), sub(end, start))
            calldatacopy(mload(0x40), add(b.offset, start), n)
            result := keccak256(mload(0x40), n)
        }
    }

    /// @dev Returns the keccak256 of the slice from `start` to the end of the bytes.
    function hashCalldata(bytes calldata b, uint256 start) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            start := xor(start, mul(xor(start, b.length), lt(b.length, start)))
            let n := mul(gt(b.length, start), sub(b.length, start))
            calldatacopy(mload(0x40), add(b.offset, start), n)
            result := keccak256(mload(0x40), n)
        }
    }

    /// @dev Returns the keccak256 of the bytes.
    function hashCalldata(bytes calldata b) internal pure returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            calldatacopy(mload(0x40), b.offset, b.length)
            result := keccak256(mload(0x40), b.length)
        }
    }

    /*´:°•.°+.*•´.*:˚.°*.˚•´.°:°•.°•.*•´.*:˚.°*.˚•´.°:°•.°+.*•´.*:*/
    /*                      SHA2-256 HELPERS                      */
    /*.•°:°.´+˚.*°.˚:*.´•*.+°.•°:´*.´•*.•°.•°:°.´:•˚°.*°.˚:*.´+°.•*/

    /// @dev Returns `sha256(abi.encode(b))`. Yes, it's more efficient.
    function sha2(bytes32 b) internal view returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            mstore(0x00, b)
            result := mload(staticcall(gas(), 2, 0x00, 0x20, 0x01, 0x20))
            if iszero(returndatasize()) { invalid() }
        }
    }

    /// @dev Returns the sha256 of the slice from `start` to `end` (exclusive).
    /// `start` and `end` are byte offsets.
    function sha2(bytes memory b, uint256 start, uint256 end)
        internal
        view
        returns (bytes32 result)
    {
        /// @solidity memory-safe-assembly
        assembly {
            let n := mload(b)
            end := xor(end, mul(xor(end, n), lt(n, end)))
            start := xor(start, mul(xor(start, n), lt(n, start)))
            // forgefmt: disable-next-item
            result := mload(staticcall(gas(), 2, add(add(b, 0x20), start),
                mul(gt(end, start), sub(end, start)), 0x01, 0x20))
            if iszero(returndatasize()) { invalid() }
        }
    }

    /// @dev Returns the sha256 of the slice from `start` to the end of the bytes.
    function sha2(bytes memory b, uint256 start) internal view returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            let n := mload(b)
            start := xor(start, mul(xor(start, n), lt(n, start)))
            // forgefmt: disable-next-item
            result := mload(staticcall(gas(), 2, add(add(b, 0x20), start),
                mul(gt(n, start), sub(n, start)), 0x01, 0x20))
            if iszero(returndatasize()) { invalid() }
        }
    }

    /// @dev Returns the sha256 of the bytes.
    function sha2(bytes memory b) internal view returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            result := mload(staticcall(gas(), 2, add(b, 0x20), mload(b), 0x01, 0x20))
            if iszero(returndatasize()) { invalid() }
        }
    }

    /// @dev Returns the sha256 of the slice from `start` to `end` (exclusive).
    /// `start` and `end` are byte offsets.
    function sha2Calldata(bytes calldata b, uint256 start, uint256 end)
        internal
        view
        returns (bytes32 result)
    {
        /// @solidity memory-safe-assembly
        assembly {
            end := xor(end, mul(xor(end, b.length), lt(b.length, end)))
            start := xor(start, mul(xor(start, b.length), lt(b.length, start)))
            let n := mul(gt(end, start), sub(end, start))
            calldatacopy(mload(0x40), add(b.offset, start), n)
            result := mload(staticcall(gas(), 2, mload(0x40), n, 0x01, 0x20))
            if iszero(returndatasize()) { invalid() }
        }
    }

    /// @dev Returns the sha256 of the slice from `start` to the end of the bytes.
    function sha2Calldata(bytes calldata b, uint256 start) internal view returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            start := xor(start, mul(xor(start, b.length), lt(b.length, start)))
            let n := mul(gt(b.length, start), sub(b.length, start))
            calldatacopy(mload(0x40), add(b.offset, start), n)
            result := mload(staticcall(gas(), 2, mload(0x40), n, 0x01, 0x20))
            if iszero(returndatasize()) { invalid() }
        }
    }

    /// @dev Returns the sha256 of the bytes.
    function sha2Calldata(bytes calldata b) internal view returns (bytes32 result) {
        /// @solidity memory-safe-assembly
        assembly {
            calldatacopy(mload(0x40), b.offset, b.length)
            result := mload(staticcall(gas(), 2, mload(0x40), b.length, 0x01, 0x20))
            if iszero(returndatasize()) { invalid() }
        }
    }
}