Genetic-Algorithm-Based Collision Search in DES Internal Hash Functions and S-Boxes

1. Introduction

Digital signatures authenticate electronic documents and verify their origin. Their security depends not only on the encryption method but also on the hash function used to represent the message. A cryptographic hash function maps an arbitrary-length input to a fixed-length output, and a collision occurs when two distinct inputs produce the same result [1,2,3].

Collision analysis is useful both for evaluating cryptographic robustness and for revealing structural weaknesses. Traditional approaches include birthday attacks, differential cryptanalysis, and linear cryptanalysis. Linear cryptanalysis has been studied extensively for DES-like block ciphers and related constructions [4,5]. These techniques can become computationally expensive as the output size grows.

Evolutionary algorithms provide an alternative because they can explore large search spaces without requiring a complete analytical model of the target function. Genetic algorithms have been applied to cryptographic key generation and cryptanalysis, including DES-like systems [6,7]. Related artificial-life methods have also been investigated for automated test-data generation [8].

More broadly, evolutionary computation and machine learning have demonstrated considerable success in solving complex optimization and search problems across diverse domains. Fundamental developments in genetic algorithms and evolutionary computation have established powerful frameworks for exploring large, nonlinear, and highly constrained search spaces [9,10,11,12]. These methods have subsequently been applied successfully to face detection and recognition [13,14,15,16], biometric identification [17,18], computer vision and deformable texture analysis [19,20], healthcare prediction and intelligent medical image analysis [21,22,23,24,25,26,27], autonomous decision making using deep reinforcement learning [28], and machine-learning software development [29,30]. Similar optimization principles have also played an important role in modern cryptography and security analysis [31]. Collectively, these studies demonstrate that evolutionary search provides an effective framework for solving difficult optimization problems, motivating its application to collision discovery in nonlinear cryptographic transformations such as the DES round function and its S-boxes.

This study applies a genetic algorithm to collision search in internal DES transformations. The first approach searches simultaneously for two colliding message–key pairs at the DES round-function level. The second searches for collisions at the S-box level and then derives a compatible key. The second approach is considerably more practical.

2. DES Encryption

DES is a symmetric block cipher operating on 64-bit blocks with a 56-bit effective key. It was standardized in the United States as Federal Information Processing Standard Publication 46 [32]. Although DES is no longer considered secure for modern applications, its Feistel structure and substitution boxes remain important in the study of block-cipher design [2,3].

A plaintext block is first processed by an initial permutation, divided into left and right halves, and passed through 16 Feistel rounds. In round i,

$$L_i=R_{i-1},$$

(1)

$$R_i=L_{i-1}\oplus f(R_{i-1},K_i),$$

(2)

where $K_i$ is the 48-bit round key.

Figure 1. Overall DES structure with initial permutation, 16 Feistel rounds, and inverse initial permutation.

Figure 1. Overall DES structure with initial permutation, 16 Feistel rounds, and inverse initial permutation.

2.1. DES Round Function

The DES round function has five principal stages:

1.

expansion of the 32-bit right half to 48 bits;

2.

bitwise exclusive-OR with the 48-bit round key;

3.

division into eight 6-bit blocks;

4.

substitution through eight S-boxes, each mapping 6 bits to 4 bits; and

5.

permutation of the resulting 32-bit output.

The round function is

$$f(R,K)=P\left(S\left(E\left(R\right)\oplus K\right)\right),$$

(3)

where E is the expansion permutation, S denotes the eight S-box transformations, and P is the final permutation.

Figure 2. Internal structure of the DES round function.

Figure 2. Internal structure of the DES round function.

2.2. DES S-Boxes

Substitution boxes are central to the nonlinear security properties of block ciphers. S-box design criteria and construction methods have therefore been studied extensively [1,33].

Each DES S-box maps a 6-bit input to a 4-bit output. Since 64 possible inputs map to only 16 outputs, collisions necessarily exist at the S-box level. For an input $b_1{b}_2{b}_3{b}_4{b}_5{b}_6$, the row is determined by $b_1{b}_6$ and the column by $b_2{b}_3{b}_4{b}_5$.

For example, if the S-box input is 011011, then the row is ${01}_2=1$ and the column is ${1101}_2=13$. The S-box output is obtained from the corresponding lookup-table entry.

This many-to-one mapping makes the S-box layer a natural location for collision search.

3. Genetic-Algorithm-Based Collision Search

3.1. First Approach: Message–Key Pair Search

The first approach searches for

$$(M_1,K_1)\ne (M_2,K_2)$$

such that

$$f(M_1,K_1)=f(M_2,K_2).$$

(4)

Each chromosome contains two messages and two keys. Fitness is based on the Hamming distance between the corresponding outputs:

$$d_H(H_1,H_2)={\displaystyle \sum _{i=1}^n}\mathbf{1}[H_{1,i}\ne H_{2,i}],$$

(5)

where $\mathbf{1}[·]$ is the indicator function. The objective is to minimize $d_H$ to zero.

3.1.1. Fitness Function

A simplified version of the implemented fitness function is shown below.

3.1.2. Mutation

Mutation flips message and key bits with a specified probability. A high mutation rate is used initially to encourage broad exploration. As the Hamming distance decreases, the mutation rate is reduced to support local refinement.

3.1.3. Crossover

A crossover operator exchanges chromosome segments between parent candidates. Similar genetic operators have been used in cryptographic search and DES-like cryptanalysis [6,7]. Although crossover increases diversity, it can also cause rapid population uniformity if combined with overly aggressive selection.

3.1.4. Immigration

An immigration operator replaces a small percentage of chromosomes with newly generated random candidates. In the experiments, the immigration probability was approximately 5%. This mechanism helps prevent premature convergence and mating-pool stagnation.

3.1.5. Selection

Chromosomes are ranked by fitness, and the stronger candidates are retained. To preserve diversity, a small probability is assigned to retaining lower-ranked candidates, because weak parents may still generate useful offspring.

3.1.6. Outcome of the First Approach

The first approach reduced Hamming distance quickly, often from approximately 20 bits to 4 or 5 bits. Extended runs reached distances of 1 or 2 bits, but exact collisions were not obtained reliably within practical computation times. This indicates that simultaneous optimization over two messages and two keys produces a very large search space.

3.2. Second Approach: S-Box Collision Search

The second approach selects a random message $R_1$ and key $K_1$ and computes

$$X_1=E\left(R_1\right)\oplus K_1,$$

(6)

$$Y=S\left(X_1\right).$$

(7)

The genetic algorithm searches for $X_2\ne X_1$ such that

$$S\left(X_2\right)=Y.$$

(8)

After identifying a colliding S-box input, a compatible key is derived by

$$K_2=E\left(R_2\right)\oplus X_2.$$

(9)

Thus,

$$S\left(E\left(R_1\right)\oplus K_1\right)=S\left(E\left(R_2\right)\oplus K_2\right).$$

(10)

Figure 3. Second collision-search approach. A colliding S-box input is identified and a compatible key is then derived.

Figure 3. Second collision-search approach. A colliding S-box input is identified and a compatible key is then derived.

3.3. Genetic Operators in the Second Approach

The chromosomes in the second approach contain candidate messages rather than complete message–key pairs. Mutation, crossover, immigration, and fitness-based selection are applied in each generation.

4. Experimental Results

4.1. Effect of Mutation Rate

Mutation rate had a strong effect on convergence. Very high mutation rates improved global exploration but disrupted promising partial solutions. Lower and adaptive mutation rates generally produced faster convergence.

The most effective observed schedule was approximately:

• 40% at the beginning;
• 30% when Hamming distance was below 10;
• 20% when Hamming distance was below 5; and
• 10% when Hamming distance was below 4.

This schedule balanced global exploration and local refinement.

4.2. Effect of Population Size

Increasing the mating-pool population substantially improved the probability and speed of convergence. Populations near 800–900 chromosomes often produced collisions within several minutes.

4.3. Collision Examples

One observed collision pair was

$$\begin{array}{cc}\hfill R_1& =101011010111100101110101000100101101101011000011,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill R_2& =101011011110010010010110000100101101101011111000,\hfill \end{array}$$

(12)

with the common S-box output

$$10011010110101010100111101001111.$$

Another observed pair was

$$\begin{array}{cc}\hfill R_1& =100001111011010001110011110100100000110011001101,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill R_2& =001010011100011100110011000111001000011000100000,\hfill \end{array}$$

(14)

with the common output

$$11110101001001001100100101010111.$$

4.4. Software Output

Figure 4 shows a successful collision with zero Hamming distance.

Figure 5 shows the derived key and colliding message pair.

5. Implementation

The algorithm was implemented in C# using the Microsoft .NET environment. The implementation follows an object-oriented design.

5.1. Main Classes

The software contains five principal classes:

• Chromosome, which stores candidate messages, keys, outputs, fitness values, and mutation logic;
• Pool, which manages sorting, crossover, immigration, mutation, and selection;
• Form1, which provides the graphical user interface and controls execution;
• RandNum, which generates pseudo-random numbers; and
• HashDESOne, which implements the internal DES transformation used in collision search.

The modular structure allows the DES-specific transformation to be replaced by another hash or substitution function.

5.2. Execution Procedure

The user first selects the population size and monitoring interval. A random mating pool is then generated, and the genetic algorithm runs until either a collision is found or the process is manually stopped.

The Hamming distance is displayed continuously, and the target value is zero. Because the method is stochastic, convergence time varies. Typical successful runs required approximately 10–20 minutes, although some runs did not converge within several hours.

6. Discussion

The first approach searches a very large space because it evolves two messages and two keys simultaneously. Although it reduces Hamming distance effectively, it does not consistently find exact collisions.

The second approach is more successful because it exploits the many-to-one structure of DES S-boxes. By finding a colliding S-box input first and deriving a compatible key afterward, the dimensionality of the search is reduced substantially.

The results also show that adaptive mutation is more effective than a uniformly high mutation rate and that larger populations improve diversity and convergence. These observations are consistent with previous uses of genetic search in cryptography [6,7].

This work does not imply that modern cryptographic systems can be broken using the presented implementation. DES is obsolete, and S-box-level collisions are structurally inevitable because each S-box maps 64 possible inputs to 16 outputs. The contribution is methodological: evolutionary search can be applied to constrained nonlinear cryptographic transformations.

7. Conclusions

This paper investigated genetic-algorithm-based collision search in DES internal transformations. Two approaches were evaluated. The first searched for colliding message–key pairs at the round-function level but did not reliably produce exact collisions. The second searched for collisions at the S-box level and then derived compatible keys. This strategy was successful within practical time for sufficiently large populations. Future work should investigate parallel implementations, more advanced evolutionary operators, multi-objective fitness functions, and reduced-round modern cryptographic structures.