Analysis of a Modified RC4 Algorithm

Transcription

1 International Journal of Computer Appliations ( ) Analysis of a Modified RC4 Algorithm T.D.B Weerasinghe MS.Eng, BS.Eng (Hons), MIEEE, AMIE (SL) Software Engineer IFS R&D International, 363, Udugama, Kandy, Sri Lanka ABSTRACT In this paper, analysis of a simply modified RC4 algorithm is presented. RC4 is the most widely used stream ipher and it is not onsidered as a ipher that is strong in seurity. Many alternatives have been proposed to improve RC4 key generation and pseudo random number generation but the thoughts behind this work is to try out a simple modifiation of RC4 s PRGA, where we an mention like this: Output = M XOR GeneratedKey XOR j After having done the modifiation the modified algorithm is tested for its serey and performane and analyzed over the variable key length with respet to those of the original RC4. The results show that the modified algorithm is better than the original RC4 in the aspets of serey and performane. General Terms Symmetri Key Algorithms, Stream Cipher, RC4 KSG, RC4 PRGA Keywords Serey of RC4, Modified RC4 1. INTRODUCTION RC4 is the most widely used stream ipher. In open literature there are a lot of modifiations done in-order to improve the seurity and performane level of the partiular algorithm. In this researh the fous was to slightly alter the Output generation by adding one parameter into the XOR operation. Intension behind the onept was to hek the serey level and performane by doing the partiular hange and see the results. Sine, Shannon s theories of serey of iphers are not used quite often in the area in Cryptography in open literature; the obvious thought was to selet them as the serey measurement riteria. And the performane is related to the enryption time. Serey and performane are analyzed over the variable key lengths whih varied from 64 bits to 1856 bits. (Key lengths of RC4 an vary from 40 bits to 048 bits). All the algorithms were written in Java as well as the time alulation (in miroseonds) and serey measurements are also done using two separate Java programs. To generate a random alpha numeri harater key, random harater generation in Java is used!. RC4 ALGORITHM RC4 is the widely used stream ipher. In this setion, the original RC4 is desribed in a nutshell. The following desription is illustrated from a researh work done by assir Nawaz et. al. [1] The RC4 algorithm has of two major parts: The key sheduling algorithm (KSA) and the pseudo-random generation algorithm (PRGA). l - Length of the initial key in bytes N - Size of the array S or the S-box in words. Normally RC4 is used with a n = 8 and array size N = 8. In the first phase of RC4 operation an identity permutation (0, 1..., N-1) is loaded in the array S. A seret key K (initial key) is then used to initialize S to a random permutation by shuffling the words in S. During the seond phase, PRGA produes random words from the permutation in S. An iteration of the PRGA loop produes one output word that onstruts the running key stream (generated key stream). The keystream is bit-wise XORed with the plaintext to obtain the iphertext. [1] for i = 0 to (N-1) j = 0; for i = 0 to (N-1) i = 0, j =0; j = (j + S[i] + K[i mod l]) mod N swap (S[i], S[j]) Output Generation Loop i = (i+1) mod N J = (j+s[i]) mod N Swap(S[i], S[j]) Output = S[(S[i] + S[j]) mod N] The general strutures of KSA and PRGA are shown in the above figrue. The original RC4 an be illustrated as follows beause n = 8 and N = 8 (56) KSA: j=0 1

2 International Journal of Computer Appliations ( ) j = (j+s[i]+k[i]) mod 56; sawp S[i] and S[j]; PRGA: i = 0, j=0; for x = 0 to M-1 i = (i+1) mod 56; j = (j+s[i]) mod 56; swap S[i] and S[j]; GeneratedKey = S[ (S[i] + S[j]) mod 56] Output = M XOR GeneratedKey Where M is the plain text message length. [4] 3. MODIFIED RC4 Here is the pseudo ode of the modified RC4: KSA: j=0 j = (j+s[i]+k[i]) mod 56; sawp S[i] and S[j]; PRGA: i = 0, j=0; for x = 0 to M-1 i = (i+1) mod 56; j = (j+s[i]) mod 56; swap S[i] and S[j]; GeneratedKey = S[ (S[i] + S[j]) mod 56] Output = M XOR GeneratedKey XOR j Where M is the plain text message length. 4. SECREC OF CIPHERS 4.1 Definitions related to serey Entropy: Entropy of a message X, alled H(X), is the minimum number of bits needed to enode all possible ourrenes (meanings) of the message, assuming all messages are equally likely. [5] Entropy of a given message X is defined by the weighted average: H( X ) Unertainty: n 1 x )log x ) i Unertainty of a message is the number of plaintext bits that must be reovered when the message is srambled in ipher text in order to learn the plaintext. The unertainty of a message is measured by its entropy. [5] Higher the number of bits, higher the unertainty. i Equivoation: Equivoation is the unertainty of a message that an be redued by given additional information. [5] Equivoation is the onditional entropy of X given : H H ( X ) ( X ) Serey of iphers: X, X, )log [ P ) X P ( X )] ( X )log [ P ( X )] Serey of a ipher is alulated in terms of the key equivoation H (K) of a key K for a given ipher text C; that is the amount of unertainty in K given C: [5] H ( K) C C) K P ( K)log [ ( )] P K Note: This is the equation used in the serey alulation in this researh and it was used in some of my previous work [], [3] and all the above definitions were derived from theories of Shannon related to entropy and serey. Claude Elwood Shannon [April 30, 1916 February 4, 001] is alled the Father of Information Theory. All the above equations/definitions are illustrated from the leture notes of Dr.Issa Traore of the University of Vitoria, British Columbia, Canada. URL: 4. Calulation of the serey of iphers H ( K) C C) K P ( K)log [ ( )] P K How the alulation is done in the program: Consider the highlighted part first: That is the entropy of K given the relevant ipher. (This ipher has ome due to this key) o Calulate how often eah key byte has appeared in the key. o And then alulate the probability of eah byte appears (given the ipher) in the key and get the summation of K) * log P (K). Then onsider the other part: Calulating C) and the summation. o Calulate how often eah ipher byte has appeared in the ipher text. o And then alulate the probability of eah byte appears in the key and get the summation (for all possibilities of the ipher bytes). This ipher is related to the above key; i.e. this ipher is obtained by enrypting plain text with the above key. Then get the multipliation of the highlighted part and C) is alulated and finally the summation of all possibilities is alulated. 13

3 International Journal of Computer Appliations ( ) 5. RESEARCH METHOD 5.1 Method for analyzing serey For 100 random keys serey values are observed and the average value is alulated. By omparing the average serey values of both RC4 and modified RC4 over the length of the random key, the serey analysis is done! Important: 5. Method for analyzing performane For 100 random keys enryption times (in miroseonds) are alulated and the average value is onsidered. By omparing the average enryption times of both RC4 and modified RC4 over the length of the random key, the performane analysis is done! Important: Input message (When the input is onsidered to be a password): #THARINDU_WEERASINGHE# Input message (When the input is onsidered to be a password): #THARINDU_WEERASINGHE# To ompare the serey values of different data sizes, the input is given as text file ontaining alpha numeri values Java method to alulate serey: [6] publi stati double alulateserey(byte[] key, byte[] ipher, int start) double entropy = 0; double serey = 0; final int[] ountedkey = ountbytedistribution(key, start, key.length- 1); final int[] ountedcipher = ountbytedistribution(ipher, start, ipher.length-1); for (int i=0;i<56;i++) final double p_k = 1.0 * ountedkey[i] / key.length; final double p_ = 1.0 * ountedcipher[i] / ipher.length; if (p_k > 0) return serey; entropy += p_k * log(p_k); serey += -p_ * entropy; To ompare the enryption time of different data sizes, the input is given as text file ontaining alpha numeri values 5.3 Method of generating the initial key With the help of java.seurity.seurerandom and java.math.biginteger, streams of random alphanumeri haraters are generated. Example ode: new BigInteger(40, random).tostring(3); Need to hange the first parameter of the BigInteger aording to the required key length in bytes. E.g. If you need a random key having a length of 40bytes then you need to input 00 there. See below for the example: new BigInteger(00, random).tostring(3); 5.4 Method of analyzing the overall results The idea behind the researh was to modify the existing RC4 algorithm in order to hek how it responds towards the serey of the ipher generated as well as the overall performane of the algorithm. Sine the serey alulation is hardly used to evaluate seurity level of a ipher in open literature, the fous was to hek the serey of the generated ipher text in order to evaluate the modified algorithm over the original one! The variation of the serey over the different key sizes is studied to give the result of the researh with respet to the serey. Initial thought (prior to the outome) was the serey level of the modified ipher should be higher than that of the original ipher beause the modifiation leads to an additional operation. The average serey is alulated by running the Performane is measured by alulating the average enryption time. All the tests were arried out in a mahine whih has the following onfiguration: Intel Core i3 CPU, GHz.40 GHz 1.86 GB usable RAM MS Windows 7 Home Basi (3 bit) 14

4 International Journal of Computer Appliations ( ) 6. RESULTS AND ANALSIS Table 1. Average Serey Value Vs Key Length Key Average Serey of Average Serey of Length/Bits RC4 modified RC Fig. : Average Enryption Time Vs Key Length Table 3. Average Serey Value Vs Data Size Data Size/KB Average Serey Average Serey of RC4 of modified RC Fig 1: Average Serey Value Vs Key Length Table. Average Enryption Time Vs Key Length Average Enryption Key Average Enryption Time of modified Length/Bits Time of RC4/ µs RC4/ µs Fig. 3: Average Serey Value Vs Data Size 15

5 International Journal of Computer Appliations ( ) Table 4. Average Enryption Time Vs Data Size Data Size/KB Average Enryption Time of RC4/ µs Average Enryption Time of modified RC4/ µs Fig. 4: Average Enryption Time Vs Data Size 7. CONCLUSIONS AND FUTURE WORK Conlusion with respet to password type inputs: As shown by the Fig 1, modified RC4 has better average serey than that of the original RC4. Both urves does not have smooth variation beause of the limited number of samples, 100 to be préised (sample keys) taken for the experiment. But if we an test this for larger number of sample then the urve will have a smooth variation. If this is further explained if we take one instane of the experiment: onsider the initial key length is 40 bits then we do have ^40 number of possible keys. If we are to take the average serey of the instane where we have the key length of 40 bits then we need to get the serey values for all ^40 keys whih is unrealisti. So, in this researh 100 sample keys are onsidered. (For measuring both serey and performane) As far as the performane is onerned, the modified RC4 has the upper hand of the original RC4. By looking at the graph shown in the Fig, it is proved that the modifiation has lead to derease the enryption time; hene the higher performane is ahieved. Conlusion with respet to the input messages that have larger data sizes: As depited in Fig. 3 the serey of the modified RC4 against the data size is more often than not, higher than that of original RC4. But again the measurements and alulations are done for 100 different keys (note: in this experiment the key size is taken as the most ommon size used 18 bits) and the urve ould have been smoother if the numbers of samples are higher. (E.g samples) When it omes to the performane, it is obvious that the modifiation has improved the performane (refer: Fig.4) General onlusion: Thus, a onlusion an be made upon the evident results, as the simple modifiation has done an enormous improvement of the RC4 (regardless of the limited number of samples taken even for the limited number of keys modified RC4 gave good results, thus it is learly obvious that if more samples were taken the results ould have been muh better. So, the simple modifiation in made RC4 to give better serey and performane simultaneously. Suggested future work: Same known plaintext attak for the modified RC4 and original RC4 and evaluate the tolerane levels. REFERENCES [1] assir Nawaz and Kishan Chand Gupta and Guang Gong A 3-bit RC4-like Keystream Generator. In Cryptology eprint Arhive: Report 005/175. [] T.D.B Weerasinghe. 01. Analysis of a Hybrid Cipher Algorithm for Data Enryption. In IFRSA s International Journal of Computing, VOL. No., [3]T.D.B Weerasinghe. 01. Serey and Performane Analysis of Symmetri Key Enryption Algorithms. In IAES International Journal of Information and Network Seurity, VOL.1 No., [4] Allam Mousa and Ahmad Hamad Evaluation of the RC4 Algorithm for Data Enryption. In International Journal of Computer Siene and Appliations, VOL.3, No., [5] Leture notes of Dr.Issa Traore of the University of Vitoria, British Columbia, Canada, related to serey of iphers. URL: [6] 16