Blind Birthday Attack | Understand the problem

A Blind Birthday Attack is a cryptographic attack that exploits the birthday paradox to find collisions in hash functions without revealing the input values. It’s a more complex variant of the traditional birthday attack.

Birthday Paradox Recap:

The Birthday Paradox states that in a group of 23 people, there’s more than a 50% chance that at least two people share the same birthday. This counter-intuitive result arises because the number of possible pairs increases much faster than the number of individuals.

Birthday Attack in Cryptography:

A Birthday Attack exploits this paradox in the context of hash functions. Hash functions are designed to take an input and produce a fixed-size output (hash). A good hash function should minimize the chances of two different inputs producing the same hash (a collision).

However, due to the Birthday Paradox, if you hash enough random inputs, the probability of finding two inputs that produce the same hash grows surprisingly fast. For a hash function with an n-bit output, you can expect to find a collision after hashing around 2n^2 random inputs.

Key characteristics:

• Oracle Access: The attacker has access to an oracle that computes HMAC-SHA256 of two messages and returns the length of their common prefix.
• No Input Revelation: The attacker cannot see the actual HMAC values, only their common prefix length.
• Goal: Find two messages with the same HMAC value, even without knowing the secret key.

Attack Strategy

The attack typically involves building a “blind” binary search tree. Each node in the tree represents a range of possible HMAC values. By querying the oracle with carefully chosen message pairs, the attacker can gradually refine the search space and eventually find a collision.

Core idea:

• Start with a root node representing the entire HMAC space.
• Create child nodes by splitting the HMAC space in half.
• Query the oracle to determine which child node contains a collision.
• Recursively explore the promising child node.

Challenges and Countermeasures

• Complexity: The attack is computationally intensive and requires careful optimization.
• Oracle Limitations: The information provided by the oracle is limited, making the search process challenging.
• Countermeasures: Increasing the hash function output size, using stronger hash functions, and limiting the number of oracle queries can help mitigate the risk of blind birthday attacks.

Real-world Implications

While the blind birthday attack is a theoretical concept, it highlights the importance of carefully designing cryptographic systems to resist various attack vectors. Understanding these attacks helps in developing robust security measures.

Implementing the described “blind” binary search tree in C++ involves creating a structure to hold the messages and their HMACs, traversing and modifying the tree based on the HMAC prefixes, and finding collisions.

#include 
#include 
#include 
#include 
#include 
#include 

// Node structure for the binary search tree
struct Node {
    std::string message;
    std::string hmac;
    Node* left;
    Node* right;

    Node(std::string msg, std::string hmac_value) : message(msg), hmac(hmac_value), left(nullptr), right(nullptr) {}
};

// HMAC generation function using OpenSSL
std::string generate_hmac(const std::string& key, const std::string& message) {
    unsigned char* result;
    unsigned int len = 32;

    result = (unsigned char*)malloc(sizeof(char) * len);
    HMAC_CTX* ctx = HMAC_CTX_new();
    HMAC_Init_ex(ctx, key.c_str(), key.length(), EVP_sha256(), nullptr);
    HMAC_Update(ctx, (unsigned char*)message.c_str(), message.length());
    HMAC_Final(ctx, result, &len);
    HMAC_CTX_free(ctx);

    std::string hmac_str = std::string((char*)result, len);
    free(result);

    return hmac_str;
}

// Function to find the common prefix length of two strings
int common_prefix_length(const std::string& str1, const std::string& str2) {
    int length = 0;
    for (size_t i = 0; i < std::min(str1.size(), str2.size()); ++i) {
        if (str1[i] != str2[i]) {
            break;
        }
        length++;
    }
    return length;
}

// Function to add a message to the tree
void add_message_to_tree(Node*& root, const std::string& message, const std::string& hmac) {
    if (root == nullptr) {
        root = new Node(message, hmac);
        return;
    }

    Node* current = root;
    int level = 1;

    while (true) {
        int length = common_prefix_length(current->hmac, hmac);

        if (length == 256) {
            std::cout << "Collision found!" << std::endl;
            std::cout << "Message 1: " << current->message << std::endl;
            std::cout << "Message 2: " << message << std::endl;
            break;
        }

        if (length >= level) {
            if (current->right) {
                current = current->right;
            } else {
                current->right = new Node(message, hmac);
                break;
            }
        } else {
            if (current->left) {
                current = current->left;
            } else {
                current->left = new Node(message, hmac);
                break;
            }
        }

        level++;
    }
}

// Random string generator
std::string generate_random_message(int length) {
    static const char charset[] = "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789";
    std::string result;
    result.resize(length);

    std::random_device rd;
    std::mt19937 generator(rd());
    std::uniform_int_distribution<> dist(0, sizeof(charset) - 2);

    for (int i = 0; i < length; ++i) {
        result[i] = charset[dist(generator)];
    }

    return result;
}

int main() {
    const std::string key = "secretkey";
    Node* root = nullptr;

    // Keep generating random messages and adding them to the tree until a collision is found
    while (true) {
        std::string message = generate_random_message(32);
        std::string hmac = generate_hmac(key, message);

        add_message_to_tree(root, message, hmac);
    }

    return 0;
}

Explanation:

1. Node Structure:
   • Each Node represents a message and its corresponding HMAC.
   • Each node can have a left child or a right child.
2. HMAC Generation:
   • The generate_hmac function computes the HMAC of a given message using the secretkey. It uses the OpenSSL library to perform the HMAC calculation.
3. Tree Traversal and Message Addition:
   • The add_message_to_tree function handles the logic of adding a new message to the tree. It checks the common prefix length between the current node’s HMAC and the new HMAC to determine the correct position in the tree.
4. Random Message Generation:
   • The generate_random_message function generates random strings of a specified length to simulate different messages.
5. Collision Detection:
   • When two messages with identical HMACs are found, the program prints both messages, indicating a collision has occurred.

Dependencies:

To compile and run this program, you’ll need the OpenSSL library installed on your system. If you’re using a system with g++ and OpenSSL installed, you can compile this code with the following command:

g++ -o blind_birthday_attack blind_birthday_attack.cpp -lssl -lcrypto