openzeppelin-contracts/contracts/utils/cryptography/MerkleProof.sol

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.5.0-rc.0) (utils/cryptography/MerkleProof.sol)

pragma solidity ^0.8.0;

/**
 * @dev These functions deal with verification of Merkle Trees proofs.
 *
 * The proofs can be generated using the JavaScript library
 * https://github.com/miguelmota/merkletreejs[merkletreejs].
 * Note: the hashing algorithm should be keccak256 and pair sorting should be enabled.
 *
 * See `test/utils/cryptography/MerkleProof.test.js` for some examples.
 */
library MerkleProof {
    /**
     * @dev Returns true if a `leaf` can be proved to be a part of a Merkle tree
     * defined by `root`. For this, a `proof` must be provided, containing
     * sibling hashes on the branch from the leaf to the root of the tree. Each
     * pair of leaves and each pair of pre-images are assumed to be sorted.
     */
    function verify(
        bytes32[] memory proof,
        bytes32 root,
        bytes32 leaf
    ) internal pure returns (bool) {
        return processProof(proof, leaf) == root;
    }

    /**
     * @dev Returns the rebuilt hash obtained by traversing a Merklee tree up
     * from `leaf` using `proof`. A `proof` is valid if and only if the rebuilt
     * hash matches the root of the tree. When processing the proof, the pairs
     * of leafs & pre-images are assumed to be sorted.
     *
     * _Available since v4.4._
     */
    function processProof(bytes32[] memory proof, bytes32 leaf) internal pure returns (bytes32) {
        bytes32 computedHash = leaf;
        for (uint256 i = 0; i < proof.length; i++) {
            bytes32 proofElement = proof[i];
            if (computedHash <= proofElement) {
                // Hash(current computed hash + current element of the proof)
                computedHash = _efficientHash(computedHash, proofElement);
            } else {
                // Hash(current element of the proof + current computed hash)
                computedHash = _efficientHash(proofElement, computedHash);
            }
        }
        return computedHash;
    }

    function _efficientHash(bytes32 a, bytes32 b) private pure returns (bytes32 value) {
        assembly {
            mstore(0x00, a)
            mstore(0x20, b)
            value := keccak256(0x00, 0x40)
        }
    }
}