Wagner’s Algorithm: How to Solve the Generalized Birthday Problem?

1. Introduction

The birthday problem is one of the most counterintuitive results in probability theory. In its classic form, it asks:

How many people must be in a room for there to be a better than 50% chance that at least two share the same birthday?

Surprisingly, only 23 people are needed for this probability to exceed 50%. While the classical birthday problem is a staple of introductory probability, its generalizations have far-reaching applications in cryptography, combinatorics, and computational complexity.

In this tutorial, we explore the generalized birthday problem and focus on Wagner’s algorithm. It solves the problem using techniques that balance exhaustive search and meet-in-the-middle strategies. We’ll also discuss how ideas from this algorithm have been implemented in modern proof-of-work systems, particularly in the Equihash algorithm used by cryptocurrencies like Zcash.

By the end of this article, we’ll understand the theoretical underpinnings and practical considerations of implementing Wagner’s algorithm.

2. Revisiting the Classical Birthday Problem

Before venturing into the generalized version, let’s briefly revisit the classical birthday problem.

To solve it, we first ask: in a group of n people, what is the probability that at least two individuals share the same birthday?

The opposite event is that all people have different birthday dates. The first person can have any of 365 days as their birthday. The second person must have a different birthday, reducing their choices to 364 out of 365, the third to 363, and so on. This results in the following probability:

(1) $\begin{equation*} $P(\text{all } n \text{ distinct}) = \prod_{i=0}^{n-1} \left(1 - \frac{i}{365}\right)$ \end{equation*}$

For n=23, this probability drops below 50%, meaning the chance of a shared birthday exceeds 50% for such a small group. This counterintuitive outcome highlights the nature of combinatorial explosion in probability theory.

3. The Generalized Birthday Problem

The generalized birthday problem extends the classic scenario. Instead of looking for matching birthdays, we can frame the problem as finding subsets of elements that satisfy a particular modular condition.

3.1. Mathematical Formulation

Given a set S={x1, x2, …, xn}, we want to find a non-empty subset S’ of S such that:

(2) $\begin{equation*} $\left( \sum_{x \in S'} x \right) \mod M = T$ \end{equation*}$

where:

M is a modulus
T is a target value

In cryptography, collision-finding problems like this form the basis of hash function attacks and proof-of-work mechanisms.

4. Wagner’s Algorithm: Overview

Wagner’s algorithm, introduced by David Wagner, is a meet-in-the-middle approach designed to solve the generalized birthday problem efficiently.

The key idea is to partition the set S into multiple subsets, solve smaller subproblems for each partition, and then combine the results to solve the overall problem. This approach reduces the exponential search space by finding a balanced trade-off between time and space complexity.

4.1. Meet-in-the-Middle Strategy

The meet-in-the-middle strategy is particularly effective when the problem can be divided into two or more parts we can solve independently. For the generalized birthday problem, the algorithm works in three phases:

We partition the original input set into k smaller subsets
For each subset, we compute all possible partial sums modulo M
Finally, the algorithm “meets in the middle” by searching for combinations of these partial sums that, when added together, yield the target T

The meet-in-the-middle strategy differs from divide-and-conquer in how it processes and combines subproblems. While divide-and-conquer recursively breaks a problem into smaller independent subproblems, solves them separately, and then merges the results (as seen in merge sort), meet-in-the-middle splits the problem into two halves, precomputes one half, and efficiently searches for matching solutions in the other.

This method significantly reduces the computational effort compared to a naive exhaustive search requiring examining $2^N$ subsets.

5. Step-by-Step Breakdown of Wagner’s Algorithm

5.1. Step 1: Partitioning the Set

Here, we split the set S into k groups, where, ideally, each group is of equal size.

The choice of k is critical, as it influences the algorithm’s time and space complexity. A common heuristic is to choose k so that the resulting search space in each partition is manageable.

5.2. Step 2: Computing Partial Sums

For each subset, we compute all possible partial sums modulo M.

However, since we are interested in sums modulo M, many of these sums will collide, allowing for potential reductions using hash tables or other lookup structures:

def compute_partial_sums(group, M):
    """
    Computes all possible subset sums modulo M for a given group.
    
    :param group: A list of numbers representing the group
    :param M: The modulus value
    :return: A dictionary mapping sum modulo M to the subset
    """
    
    partial_sums = {}  # Dictionary: key = sum mod M, value = subset representation

    # Iterate over all subsets of the group
    for subset in power_set(group):  
        sum_mod = sum(subset) % M  # Compute sum modulo M
        partial_sums[sum_mod] = subset  # Store subset in dictionary

    return partial_sums

5.3. Step 3: Merging the Partial Sums

The algorithm combines these results once it computes the partial sums for each group. The aim is to find one entry from each group such that their total sum modulo M equals T. We can conceptualize this as a multi-dimensional “matching” problem, where each dimension represents a group.

A practical approach is to combine the partial sum tables recursively. In each recursive step, we merge two tables to form a new table that contains the sums from combining both groups’ subsets.

The merge is performed while checking if any sums match the target when combined with other groups:

def wagner_algorithm(S, k, T, M):
    """
    Implements Wagner's algorithm to solve the generalized birthday problem.
    
    :param S: The set of elements to partition
    :param k: Number of partitions
    :param T: Target value
    :param M: Modulus value
    :return: A valid subset sum modulo M if found, else None
    """
    
    # Step 1: Partition the input set into k groups
    groups = partition_set(S, k)
    partial_sums_list = []

    # Step 2: Compute partial sums for each group
    for group in groups:
        partial_sums_list.append(compute_partial_sums(group, M))

    # Step 3: Merge tables iteratively
    while len(partial_sums_list) > 1:
        table1 = partial_sums_list.pop()
        table2 = partial_sums_list.pop()
        merged_table = merge_tables(table1, table2, M)
        partial_sums_list.append(merged_table)

    # Step 4: Retrieve the final table and return the valid subset if found
    final_table = partial_sums_list[0]
    return final_table.get(T)  # Returns None if no valid subset is found

5.4. Merging Tables

Here’s the merge_tables function that efficiently merges two tables while ensuring that the summed values are computed modulo M:

def merge_tables(table1, table2, M):
    """
    Merges two tables containing partial sums and returns a new table
    with all possible sums modulo M.
    
    :param table1: Dictionary {sum_mod_M: subset_representation} from first group
    :param table2: Dictionary {sum_mod_M: subset_representation} from second group
    :param M: Modulus value
    :return: Merged dictionary containing sums modulo M
    """
    merged_table = {}

    for sum1, subset1 in table1.items():
        for sum2, subset2 in table2.items():
            merged_sum = (sum1 + sum2) % M  # Combine sums and apply modulus
            merged_table[merged_sum] = subset1 + subset2  # Store combined subset

    return merged_table

For every pair of sums from the two tables, it computes their sum and takes the result modulo M, ensuring that all possible subset combinations are accounted for. The function stores the merged sums and the combined subset representation in a new table. This approach allows the algorithm to efficiently explore valid combinations without redundant computations, significantly reducing complexity compared to an exhaustive search.

The merging step applies the meet-in-the-middle technique and ensures that partial results are progressively combined, leading to a more manageable search space.

5.4. Example of Merging Partial Sum Tables

Consider a small set S={3, 7, 12, 18} and a modulus M=10. We partition S into two groups:

$\text{Group 1: } \{3, 7\} \text{Group 2: } \{12, 18\}$

We first compute the partial sums modulo M for each group. For the first group, we have:

$\text{Group 1: \{3, 7\}} 3 \mod 10 = 3 7 \mod 10 = 7 (3 + 7) \mod 10 = 0$

The corresponding table is:

$\text{Table 1:} \{3 : \{3\}, 7 : \{7\}, 0 : \{3, 7\} \}$

For the second group:

$\text{Group 2: \{12, 18\}} 12 \mod 10 = 2 18 \mod 10 = 8 (12 + 18) \mod 10 = 0 \text{Table 2: } \{2: \{12\}, 8: \{18\}, 0: \{12,18\}\}$

Now, we apply the merge_tables function, which combines every sum from Table 1 with every sum from Table 2, adding them together and taking the result modulo M:

$(3 + 2) \mod 10 = 5 \Rightarrow \{3,12\}$
$(3 + 8) \mod 10 = 1 \Rightarrow \{3,18\}$
$(3 + 0) \mod 10 = 3 \Rightarrow \{3,12,18\}$
$(7 + 2) \mod 10 = 9 \Rightarrow \{7,12\}$
$(7 + 8) \mod 10 = 5 \Rightarrow \{7,18\}$
$(7 + 0) \mod 10 = 7 \Rightarrow \{7,12,18\}$
$(0 + 2) \mod 10 = 2 \Rightarrow \{3,7,12\}$
$(0 + 8) \mod 10 = 8 \Rightarrow \{3,7,18\}$
$(0 + 0) \mod 10 = 0 \Rightarrow \{3,7,12,18\}$

The final merged table is:

$\text{Merged Table: }$
$\{5: \{3,12\}, 1: \{3,18\}, 3: \{3,12,18\}, 9: \{7,12\},$
$5: \{7,18\}, 7: \{7,12,18\}, 2: \{3,7,12\}, 8: \{3,7,18\}, 0: \{3,7,12,18\}\}$

6. Complexity Analysis

Wagner’s algorithm offers a significant improvement over brute-force methods. Let’s compare the complexities:

Method	Time Complexity	Space Complexity
Brute Force	O(2^N)	O(1)
Wagner’s Algorithm	O(k * 2^(N/k))	O(2^(N/k))

By setting $k = log_2N$ , Wagner’s algorithm achieves an optimal balance between space and time complexity.

7. Implementation Considerations

When implementing Wagner’s algorithm, we must address several practical issues.

7.1. Memory Management

Storing partial sum tables for each group can become memory-intensive.

Efficient data structures such as hash maps or balanced trees can help manage memory usage while allowing quick lookups. In languages like C++, careful memory allocation and deallocation management are crucial.

7.2. Parallelization

The independent computation of partial sums for each group lends itself well to parallelization.

Leveraging multi-threading or distributed computing can further reduce the overall runtime. Modern programming environments and libraries provide constructs that make such parallelization easier and more efficient.

7.3. Optimization Techniques

Various optimization techniques can improve the algorithm’s performance. One method is to prune during the merge process by discarding partial sums that can’t lead to a solution. This can be done by checking whether a partial sum, combined with the remaining possible sums, can still reach the target modulo M. If not, it is eliminated early to reduce unnecessary computations.

Another approach is to sort the partial sums in advance, which can speed up the merge phase when a binary search is used for matching sums.

In addition, when M is large, efficient modular arithmetic becomes important; precomputing modular inverses or using specific libraries can help speed up the process.

8. Application in Cryptography: Equihash

One of the most notable applications of Wagner’s algorithm is in the Equihash proof-of-work algorithm, which underpins cryptocurrencies like Zcash.

Equihash leverages the generalized birthday problem to ensure that mining requires significant memory and computational resources, thereby reducing the feasibility of ASIC-based mining.

8.1. How Equihash Uses the Generalized Birthday Problem

Equihash is designed to be memory-hard, meaning it deliberately requires significant memory to solve. This characteristic ensures that the algorithm is more resistant to specialized hardware attacks.

The mining process in Equihash involves finding solutions to a problem structured similarly to the generalized birthday problem, though with a different mathematical formulation. Instead of finding a subset whose sum modulo M equals T as in Wagner’s algorithm, Equihash requires miners to find a subset of hashed values that satisfies a specific collision condition defined by the proof-of-work scheme.

So, given a large set of hashed inputs, the task is to find a subset whose combined value (under a specific Equihash-defined operation) meets the difficulty criterion required for mining. While Equihash leverages the meet-in-the-middle strategy from Wagner’s algorithm, it replaces the sum modulo M constraint with a cryptographic condition that determines valid solutions.

By partitioning the input space into smaller parts and merging results efficiently, Equihash reduces computational overhead while ensuring that a large amount of memory is required. This balance of computational effort and memory usage makes ASIC-based optimizations more difficult, preserving the decentralized nature of mining.

8.2. Advantages for Cryptocurrency Mining

The incorporation of Wagner’s algorithm in Equihash provides several advantages:

ASIC resistance: Equihash requires substantial memory, which levels the playing field between general-purpose hardware and specialized ASICs. This helps prevent the centralization of mining power.
Scalability: we can adjust the algorithm by changing the parameters (such as k and M) to fine-tune the balance between memory usage and computational workload
Security: the inherent difficulty of the generalized birthday problem provides an additional layer of security. The probability of finding a valid subset without performing significant work is extremely low, thus ensuring the integrity of the proof-of-work system.

9. Beyond Equihash: Other Applications of Wagner’s Algorithm

While Equihash is the most prominent application, Wagner’s algorithm finds relevance in other domains as well:

9.1. Cryptanalysis

The algorithm attacks cryptographic schemes that rely on the hardness of the generalized birthday problem.

In cryptanalysis, researchers study these algorithms to understand their limits and to design systems that remain secure against such attacks.

9.2. Combinatorial Optimization

We can adapt the meet-in-the-middle technique to solve combinatorial problems where the solution space is too large for an exhaustive search.

By breaking the problem into smaller parts, solutions can be found more efficiently.

9.3. Data Mining and Big Data

In scenarios where large datasets are involved, efficient search algorithms are essential.

Techniques inspired by Wagner’s algorithm can search for specific patterns or correlations in big data, where brute-force methods are computationally prohibitive.

10. Conclusion

In this article, we discussed Wagner’s algorithm, a powerful approach to solving the generalized birthday problem that strikes the optimal balance between space and time complexity.

Its key features are:

it reduces search space significantly compared to brute-force methods
it plays a key role in Equihash, ensuring ASIC resistance in Zcash mining
it has far-reaching applications in cryptanalysis, combinatorial problems, and big data

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