Using Genetic Algorithm To Break A Mono- Alphabetic ... - IEEE Xplore

2010 IEEE Conference on Open Systems (ICOS 2010), December 5-7, 2010, Kuala Lumpur, Malaysia

Using Genetic Algorithm To Break A Mono - Alphabetic Substitution Cipher S. S. Omran

A. S. Al-Khalid

College of Elec. & Electronic Techniques Foundation of Technical Education [email protected]

College of Elec. & Electronic Techniques Foundation of Technical Education [email protected]

Abstract- Genetic algorithms (GAs) are a class of optimization algorithms. GAs attempt to solve problems through modeling a simplified version of genetic processes. There are many problems for which a Genetic Algorithm approach is useful. It is, however, undetermined if cryptanalysis is such a problem. Therefore, this work trying to explore the use of Genetic Algorithms in cryptography. The focus is to be on substitution cipher. The principles used in this cipher form the foundation for many of the modern cryptosystems. The frequency analysis is used as an essential factor in objective function.

D. M. Al-Saady Foundation of Technical Education [email protected]

In the substitution ciphers the value of character or character string is changed when transforming the plaintext into ciphertext, but the position of the original string and its value replacement correspond exactly in the plain and ciphertext [4,7,8,9,10,11]. For example, if we encrypt the plaintext genetic algorithm in cryptography using a single character substitution cipher with a certain key, the ciphertext will be as shown in Fig.2. Cryptographic ciphers

I. INTRODUCTION The field of cryptology today represents that branch of information theory which deals with the security of information confidentiality. Methods in cryptology may be subdivided into two classes, namely that of cryptography (methods applied by authorized information sharers to design and develop encryption schemes in order to ensure confidentiality of information) and that of crypt-analysis (mathematical and statistical attempts by unauthorized persons to break cipher in order to reveal the meaning of the underlying protected data). The ciphertext is created by choosing a permutation of the 26-character alphabet and using it to replace each letter in the plaintext message [1]. A Symmetric cryptography ciphers may in fact be sub classified into block ciphers (in which blocks of data, known as plaintext, are transformed into ciphertext which appears unintelligible to unauthorized persons) and stream ciphers (which involve streams of typically binary operations and are well suited for efficient computer implementation). In this paper we shall focus on block cipher just substitution cipher as shown in Fig.1 [2,3,4]. In 1993 Spillman [5] for the first time presented a genetic algorithm approach to break a substitution cipher using genetic algorithm. He has explored the possibility of random type search to discover the key (or key space) for a simple substitution cipher. In this paper different parameters of the genetic algorithm were tested such as the population size and the time required finishing the algorithm for different number of generations. II. SUBSTITUTION CIPHER

Block ciphers

Substitution ciphers

Stream ciphers

Transposition ciphers

Product ciphers

Fig.1 Schematic representation of cryptographic cipher classification Text Character: a b c d e f g h i j k l m n o p q r s t u v w x y z

genetic algorithm in cryptography ( key cipher ) : H W U G C T V A F K D Y Q P B R JLFI X M SOZ N (cipher text)

VCPCIEUHYVBLYIAQEPULZRIBVLHRAZ

Is a symmetric cryptography ciphers, the substitution cipher is classified into two parts (Mono alphabetic and Poly alphabetic) [2,3,6].

978-1-4244-9192-6/10/$26.00 ©2010 IEEE

Asymmetric Cryptographic ciphers

Symmetric Cryptographic ciphers

Fig.2 Example of a key of a single character substitution cipher

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i.

III. GENETIC ALGORITHMS

ii.

A Genetic Algorithm is a general method of solving problems to which no satisfactory, obvious, solution exists. It is based on the idea of emulating the evolution of a species in nature and so the various components of the algorithm are roughly analogous to aspects of natural evolution [2,3,6,12,13,14]. The process begins by creating a random initial generation of individuals, sometimes called chromosomes that in some way represent the problem being solved. Pairs of members of the current population are selected and “mated” with each other by means of a crossover operation to produce members for the succeeding generation. Randomly selected members of the current generation also undergo mutation, in which random portions of “genetic” material are exchanged. The fittest members of each generation, as determined by a fitness function are then selected for the succeeding generation as shown in Fig.3. The crossover and mutation operations are controlled by the crossover rate and mutation rate parameters, which determine the proportion of the population that undergoes these changes [1].

iii. 5.

Apply crossover to get children (10 pairs). Apply mutation of 0.02% to the new children. Apply replacement to get a new population (20 keys).

g. Go to 4. Output is the best solution. V. FITNESS MEASURE

The technique used to compare candidate keys is to compare statistics of the decrypted message with those of the language. The letter frequency is used for attacks against cryptographic cipher. The frequencies of letters are typically occur in natural languages are known and well documented [15, 16, 17]. Fig.4 shows the expected number of letters occurrences of letters in English language text of length 10000 characters. A natural choice as measure of fitness of a candidate key k for the cipher would be [3, 13, 18]. ,

,

,

Initialization

, , Evaluation

Selection

, ,

1

, ,

Mutation

Here, A denotes the language alphabet (i.e., for English, [A . . . .Z]), K and D denotes known language statistics and decrypted message statistics, respectively, and the indices u, b and t denote the unigram, bigram and trigram statistics, respectively. The values of α, β and γ allow assigning of different weights to each of the three n-gram types [13].

Mating

Termination

Fig.3 The basic genetic algorithm IV. PROPOSED ALGORITHM The following is an outline of proposed algorithm: 1. Input the cipher text to the algorithm and relative character frequencies. 2. Initialize the algorithm parameters: maximum number of generations (M). 3. Generate the population p(0) keys randomly (for example 20 keys) each one with length of 26 letters. 4. For 1 to (M) do: a. Decrypt the cipher text by the 20 generated keys. b. Calculate the suitability of each key from every decrypted text using the formula of fitness. c. Sort the keys based on the increased fitness values. d. Keep 20% (2 pairs) of best fittest of p(0) for next generation. e. Use stochastically selection to choose 8 pairs from the 20 keys (parents). f. For 1 to 10 pairs do:

Fig.4 Relative frequency of letters in English text VI. IMPLEMENTING THE ATTACK FOR MONO - ALPHABETIC SUBSTITUTION CIPHER

The attack is implemented by generating an initial candidate key pool p(0) of even cardinality, consisting of permutations of the set { a, b, c, . . . z }. The first generation is generated

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in the random binary vector. The outstanding permutation entries are filled into child 1 in the order in which they occur in parent 2. Similarly, child 2 was formed by copying the entries (in the same positions) from parent 2 corresponding to zero entries in the random binary vector. The outstanding permutation entries are filled into child 2 in the order in which they occur in parent 1. This process is illustrated in Fig.7 [12, 19, 20]. After mutation has taken place the resulting set of candidate keys form the new pool population P(1). The crossover and mutation procedures are applied to this new key pool in order to produce the population P(2), and so forth, until some final population P(T) is reached (either after a prespecified number of generations, or when the minimum candidate key fitness exceeds some acceptable threshold).

randomly using a simple uniform random generator. Thereafter, the cipher text is decrypted using each permutation as a key, enabling us to assign a measure of fitness by using equation (1) to each candidate key. Pairs of candidate key are then stochastically selected for producing offspring after applying a method of crossover to each pair. The stochastic selection method is applied by choosing some pairs from the candidate which they have best fitness. Then Stochastic Universal Sampling works by making a single spin of the roulette wheel. This provides a starting position and the first selected individual. The selection process then proceeds by advancing all the way around the wheel in equal sized steps, where the step size is determined by the number of individuals to be selected. So if we are selecting n individuals we will advance by 1/n x 360 degrees for each selection. Note that this does not mean that every candidate on the wheel will be selected. Some weak individuals will have very thin slices of the wheel and these might be stepped over completely depending on the random starting position, as shown in Fig.5 [12].

P1

PKXAMLTIUBESJFG

HCORQYDVWNZ

P2

XRPUWZANMGOVSCQ TDFBJEHLKYI Crossover point

Intermediate child 1 PKXAMLTIUBESJFG

TDFBJEHLKYI

Intermediate child 2 XRPUWZANMGOVSCQ HCORQYDVWNZ Fig.5 Stochastic selection Crossover is the process of taking two parent solutions and producing from them a child. After the selection process, the population is enriched with better individuals. Crossover is a recombination operator that proceeds in three steps: i. The reproduction operator selects at random a pair of two individual keys for the mating. ii. A cross site is selected at random along the key length. iii. Finally, the position values are swapped between the two keys following the cross site. The traditional genetic algorithm uses single point crossover (which is used in this paper), when the two mating chromosomes are cut once at corresponding points and the sections after the cuts exchanged. Here, a cross site or crossover point is selected randomly along the length of the mated key and letters next to the cross sites are exchanged. If any of the exchanged chromosomes are already appears in the child, then these positions of chromosomes are left blank, then the letters that do not appear in a child are inserted in it. The outcome of this procedure is shown in Fig.6 [12, 18,19].

Intermediate child 11 PKXAMLTIUBESJFG

□D□□□□H□□Y□

Intermediate child 22 XRPUWZANMGOVSCQ

H□□□□YD□□□□

Child 1 PKXAMLTIUBESJFG

CDNOQRHVWYZ

Child 2 XRPUWZANMGOVSCQ HBEFIYDJKLT Fig.6 Applying crossover between two parents

Generating random binary number (26 binary numbers (0, 1))

110 0 0 1 0 10 1 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 First Parent (chosen randomly)

PRXAWLTIUGEVJFQHDOBCYMSKNZ Second Parent (chosen randomly)

XKPHMZANUBQVSFGTCORJEDLWYI 1st Child

After crossover, some keys are subjected to mutation. Mutation prevents the algorithm to be trapped in a local minimum. A random binary vector of the same length of the cipher key is generated and then using this vector to produce two offspring key from two parent candidate keys in the following way. Child 1 was formed by copying the entries (in the same position) from parent 1 corresponding to unit entries

PRXKMLZIAGUVSFQHDOBTYCJENW 2nd Child

RXPHMTAGUEQVSFOBCYKJNDLWZI

Fig.7 Applying mutation for two parents

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solution is for key 4 where the correct number of correct letters is 17. Bold lettering is used in table (1) for those letters in the population that appeared to be correct. The true key which is RNKIYUJEFCSZGOATMLPDWHVQBX is not in the final pool, this is due to the limited length of the ciphertext. Fig.9 shows the time required (elapsed) to finish the algorithm for different number of populations. The time required is increased as the number of populations is increased.

VII. RESULTS The attack to a mono-alphabetic substitution cipher was implemented for different number of populations. Fig.8 shows the relation between the fitness and number of generations, for population size of 20, 40, 60, and 80. It is clear from Fig.8 that the best fitness is reached after 300-400 generations. Table 1 shows the values of fitness for a 20 population (keys) and the number of correct letters obtained. It is clear that the best

No

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

100 200 300 400 500 600 700 800 900 1000 number of generations

fitness%

population size=40 91 90 89 88 87 86 85 84 100

200

300 400 500 600 700 number of generations

800

900 1000

fitness%

population size=60 90 88 86 84 82 80 78 100 200 300 400 500 600 700 800 900 1000 number of generations

300 400 500 600 700 number of generation

800

fitness%

RNCIYFJEBOSZDGAPMKUXLHVQTX RNKIYUHGTMJFVDAECLPZWOQSBX RNKIYBUELQHZWOASXFPDGTVCJM RNKIYJUELQHZGOASMFPDWTVCBX OSTIYBUELQKZWRAHXPFDGNVCJM RNTIYBUELQKZWOAHXPFDGSVCJM OHTIYBUELQNZWRASMPFDKGVCJX RHTIYBKELQAZWOGSXPFDVNUCJM RNTIYBKELQAZWOGSXPFDVHUCJM RNCIYFJEBOSZDGAPMKUXLHVQTW RNCIYBJEDGSZFKAOMLPTUHVQXW RNJVYULSFQKZICMTHEPDWOAGBX RNBIYUHGTMJFVDAECLPZWOQSKX RSJGYUKDLNXIWEAHMPFOVTZQBC RNSVYUJOLCGIWAEHMKPDZQTFBX AGKMWFJDLSCIUEYTNOPRZHVQBX RNSVYUEDPCJIWKLHMTFOZQGABX RNTIYJKELQAZWOGSXPFDVHUCBM ONTIYJKELQAZWRGSXPFDVHUCBM RNTIYBKELQAZGOWSXPFDVHUCJM

14 11 11 17 7 10 9 7 9 13 13 11 10 7 10 9 8 10 8 10

In this paper a genetic algorithm attack on a simple cryptographic cipher, called mono-alphabetic substitution, was implemented successfully. The algorithm was implemented using the MATLAB program. Different parameters were tested such as the number of population and the time required finishing the algorithm for different number of generations. It is apparent from the results that increasing the number of population above 20 was not helpful in retrieving the original key. This is evident in Fig.8 where the highest fitness was attained after 400 generations disregarding the number of the population. This can be

87 86 85 84 83 82 81 200

95 89 94 94 85 94 94 89 89 90 91 88 89 87 89 88 88 89 89 89

VIII. CONCLUSION

population size=80

100

Population

Correct letter

Table 1: The best solution for fitness value and correct no. of letters when the population is 20 and after 400 generations

96 94 92 90 88 86 84

Fitness

fitness%

population size=20

900 1000

Fig.8 Fitness value for different values of population

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REFERENCES

explained by the great number of probable keys i.e. 26! that makes any population minor. Using GA to attack a mono-alphabetic substitution cipher proved to be an efficient method of cryptanalysis based on the aspect of comparing the frequency of letter occurrence in the model text.

[1]

[2] [3]

elapsed time(sec)

population size=20 [4]

400 350 300 250 200 150 100 50 0

[5] [6] 100 200 300 400 500 600 700 800 900 1000

[7]

number of generations

[8]

elapsed time(sec)

population size=40 [9]

1400 1200 1000 800 600 400 200 0

[10] [11] [12] 100 200 300 400 500 600 700 800 900 1000 [13]

number of generations

population size=60

elapsed time(sec)

2000

[14]

1500

[15]

1000

[16]

500

[17]

0 100 200 300 400 500 600 700 800 900 1000 number of generations

[18]

[19]

elapsed time(sec)

population size=80 3000 2500 2000 1500 1000 500 0

[20]

100 200 300 400 500 600 700 800 900 1000 number of genertaion

Fig.9 Elapsed time for different no. of population

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