Techniques & Theorem for Achieving Perfect Security

1. Prof. (Dr.) C. T. Bhunia Director National Institute of Technology, Arunachal Pradesh Techniques & Theorem for Achieving Perfect Security

2. 1. One Key Pad:EDK The Vernum Theory is known as the one time (as it would require change of key from session to session) secret key(key must be secret and made known to only transmitter and intended receiver) technique. This technique, however, has one major problem. The secret key would be made known only to the two communicating parties and no one else. If a third party somehow gets a copy of the secret key, the very purpose of coding will be defeated. Shannon proved in his original work of 1949 in connecting cryptography with information theory that if Vernum theory is applied, data will be absolutely secured. It is said that in 1967, Fridel Castro of Cuba used the Vernum technique for defense communication. It is believed that the hot line communication between Moscow-Washington was done via Vernum code. For successful communication under this method, the receiver must be informed of the secret key used by the transmitter, every time a block of message is transmitted. The secret key is usually transmitted over conventional channels like the telephone line.In other words, unconditionally secure algorithm is the one time key algorithm. Vernum code falls in this class. But that has also flaws: an eavesdropper can see the two plain texts by overlying the two cipher texts. Proof is as below when the algorithm is XOR operation:

3. Basic principle for one time key with C, P and K as cipher text, Plain text and key respectively: C = P  K; P = C  K = P  K  K Attacker may reveal:C1  C2 = P1  K  P2  K = P1  P2. Making guess or sense out of two overlying plaintexts may make cryptanalysis possible. Only when key changes from cipher to cipher, the security is unconditional; and exactly that is what is Vernum code. Plain text of original message 0010110100011111 Random number or secret key 0101010101010101 Encrypted message after XOR 0111100001011010 Fig: Illustration of Vernum Code

4. Superiority of Vernum variable key over conventional one-time key pad Consider P number sessions. Each session takes a time of T seconds for completion on average. Assume key size of N bits. We now apply brute force attacks for getting the session keys. Under one-time key The eavesdropper may try on average 2 N-1 trials over a period of PT seconds. The required time for analysis of a pattern will be ( PT / 2 N-1 ) trials. Under Vernum variable key The eavesdropper has to try on average 2 N-1 trials over a period of single session i.e. T seconds. This is because the key will change from session to session. Thus the required time of analysis will be ( T / 2 N-1 ) trials. Hence in case of one-time key the attack is more effective by an order of P. In other words , the Vernum code is more secured over one-time key by an order of P. But if p=1, both are same.

5. 1.1 The Challenges The one-time pad is a secure cryptosystem. Represent all messages as a sequence of 0's and 1's. The key is a random sequence of 0's and 1's and is the same length as the message. To get the ciphertext, XOR the message with the key. The one time pad can be written mathematically in the following way: Assume that a parameter k, the length of the message and key, is known to everyone beforehand, and that M draws a message, m, from {0, 1}k; Let s be a random number from the uniform distribution of {0, 1}k;. We can define the ciphertext as: c = m XOR s In order to show that the one-time pad is secure we want to show that: for all m and c; Pr[m =M c = C] = Pr[m = M] This theorem states that the probability that m is the message given that c is the ciphertext is equal to the probability that m is the message. This means that the message being chosen given the ciphertext are independent. This implies security, because if the message and ciphertext are totally independent, then no function whatsoever can take the ciphertext to anything related to its corresponding plaintext. This is a very strong notion of security, but one we can actually prove for the one-time pad.

6. This proves that the ciphertext is independent of the message. Thus the one-time pad issecure. But in order to ensure that the method is secure, the key, s can only be used once, since if it was used more than once Eve would have some information that may allow her todecrypt the next message. For instance, Eve would know the bits where the two messages agree.

7. 2. Shannon's Theorem Claude Shannon, one of the founders of information theory, realized that if that the set of keys is maller than the set of messages than the encryption method is not secure. Assume M is such that Pr[m] ≥0; for all m Э M, C be the set of all ciphertexts, and let K be the set of all keys. Shannon's Theorem: If |M|> |K|; Эm; c : Pr[m = M] ≠Pr[m = M |c = C]. Proof: Let C be the set of all possible ciphertexts. If |C|< |M|we can't always decrypt correctly, since there would only be |C| results of decryption which is less than the number of possible messages, so some messages would be decrypted incorrectly. Since this must not happen, we know |C| ¸ |M|. If |c| ¸ |M| > |K|; then for any message m, there are only |K| ciphertexts that correspond to m. Thus, since there are more than |K| ciphertexts, there is some c such that Pr[m = M|c = C] = 0. However, Pr[m = M] ≠ 0 since m Э M which is defined to be the set of all possible messages. Since either we can't decrypt the message or the probability of the message being chosen give the ciphertext is 0, we know that the method is not secure unless the length of the key is at least as long as the length of the message. Thus, in a sense, the one-time pad is optimal, at least for this definition of security.

8. 3. Key Exhaustion Algorithm

9. 3.1 Perfect Security Scheme of Shannon “Perfect Secrecy” is defined by requiring of a system that after a cryptogram is attacked by an intruder the a posteriori probabilities of this cryptogram representing various messages be identically the same as the a priori probabilities of the same messages before the interception. It reveals that perfect secrecy is achievable but requires , if the number of messages is finite , the same number of possible keys. If the message is thought of as being constantly generated at a given rate , the key must be generated at the same or a greater rate. Secured transport of information over network is a pertaining research challenge in today’s context. The problem continues to aggregate with increasing volume of network traffic , which is evident from several researches.

10. Security basically refers to the protection of the data against intentional modification , loss or damage and fabrication of data , and / or deliberate disclosure of data to unauthorized persons or miscreants. The two basic approaches of information security are Symmetric Cryptography and Asymmetric Cryptography. Both the systems use Key as the main parameter of security. In his classical paper on Information Security , Shannon, the father of Information Theory established that perfect security can be achieved only when key is made variable from session to session and/or data to data . If Mi be the ith message for 1<= i <= n and the respective cipher be Eithe same being operated upon the encryption function Tjusing ith key of n keys as TjMj = Ei where j is transformation function with jthkey operated on ithmessage. In this case we see that PE(M) = 1/n = P(E) and we have the perfect security as because a necessary and sufficient condition for perfect security is that PM (E) = P (E) for all M and E ; where

11. PE (M) = conditional probability of getting message M if cryptogram E is intercepted. PM (E) = conditional probability of getting cryptogram E if message M is chosen i.e. the sum of probabilities of all keys , which produce the cryptogram E from message M P (E) = probability of obtaining cryptogram E from any cause . The superiority of time variant key in achieving perfect security is studied elsewhere. The famous Vernum Code was the first attempt in the direction of achieving perfect secrecy but no effective variable key has yet been applied neither any concrete theory has been established . For this reason , recently an approach AVK (Automatic Variable Key) has been proposed where key has been made as a function of previously transmitted secret data .The need of automatic variability of key can be explained suitably. If P be a matrix containing plaintext ( say n x m data) , K be another matrix containing fixed key k . Then if C be the cipher matrix , then C I J = f ( K , A I J ) , I and J respectively indicating row number and column number and A I J is an element of message matrix. Using automatic variable key , the matrix K will also hold (n x m )values and in that case C I J = f ( K I J , A I J ). Here the security level is increased as the attacker has to deal with much more number of keys in order to crack it. The superiority of time variant , AVK over fixed key in terms of brute force attack and differential frequency attack is illustrated below:

12. ( i )Fixed key P0 C0 P1 C1 : : (each of K => (each of : n bits) (n bits) : n bits) : encryption : Pm-1 function Cm-1 Plain text message space Key space Cipher space Under Brute Force Attack , average number of trials to break a key = 2 n-1 . The trial is required to be completed in full message encryption time ,T.…………………….(1) Under differential frequency attack , if there are R repetitions in plain text then there will be exactly R repetitions in the cipher. This will provide a cipher breaking probability by differential frequency attack = R/m ………………………………...(2) , m being the number of messages

13. (ii) Under AVK P0 K0 C0 P1 K1 C1 : : : (each of (each of => (each of : n bits) : n bits) : n bits) : : encryption : Pm-1 Km-1 function Cm-1 Plain text message space Key Space Cipher spacer

14. Under Brute Force Attack , average number of trials to break a key = m * 2 n-1 ,that is required in time T/m ……………………………………………...(3) Even if there are R repetitions, there will be no repetition in the cipher. The cipher breakage probability under differential attack is then 0. …………..(4) Comparison of (1) with (3) and that of (2) with (4) is conclusive evidence of superiority of AVK over fixed key. With brevity the basic idea is illustrated with an example. Suppose n bit key will be used for transmission of m messages from a source, A to a destination, B (fig 9.1). We assume n=2c.We propose that many or all keys of the whole key space of 2n shall be used in the session , unlike a single key as in DES and AES (or a single key pair as in RSA). For this, whole key space will be divided into several groups each of n bits. Total number of groups will be then 2n / n equals to say 2p , where p+c = n. At the beginning of the session, A will select any data of n bits (except a data made of all 0s) called Key for Key Selection, KKS (Say KKS0). It will be sent to B under existing RSA encryption. n bits KKS0 is made of p group bits and another c bits. Keys selection for subsequent messages will be indicated by the position of 1’s in KKS0 in the group defined by p. The positional indication is as follows: 1 in RMB (Right Most Bit) position, 1 in one but RMB position … and 1 in LMB (Left Most Bit) position will refer respectively to first, second ….and last key of the group. If there are l such keys, first (l-1) keys will be used for first (l-1) messages and last key for transmission of second KKS, (say KKS1). The process will continue till the transmission of the all messages is made. The following Fig. llustrates the scheme with n=4.. When n=4, c=2, key space (2n) =16, number of groups = 2n / n (= 22) and p=2. Session is supposed to transmit messages, m0, m1, …, …mi… As the KKSs are secret, the secret keys, ks exchanged under the protocol will remain secret and only known to sender and receiver. The proposed protocol is a typical key agreement protocol for time variant key

15. 4. AVK The AVK is illustrated in the table (1) for a session between Alice and Bob whereby they respectively exchange data 126 and 598. Table 1: Illustration of AVK for exchange of data 126 and 598 An Illustration of AVK with an Example:

16. Experimental Results: Plain text: A message is encrypted. Key is 1101. Another message is encrypted. Key is 1101. Results using Normal RSA : 8e 4c 83 54 9d 9d 5c 89 54 4c 60 9d 4c 54 42 b0 7e 4d 49 4a 54 90 7 4c 72 54 4d 4c 60 9d 4c 19 19 9f 19 7 8e 42 9b 4a b3 54 7e 4c 83 54 9d 9d 5c 89 54 4c 60 9d 4c 54 42 b0 7e 4d 49 4a 54 90 7 72 54 4d 4c 60 9d 4c 19 19 9f 19 7 af User time + System time = 0.001 + 0.001 = 0.002 Results using RSA with AVK ( Method 1): 8e 51 24 8e 6e a0 b8 9 30 87 11 15 1a 97 69 af 22 1f 9b 23 3c b b5 8c b1 b9 92 75 ad 12 76 71 4 23 ab 17 67 46 2e 5f 84 2f a8 aa b1 b9 5d 7 77 87 81 53 61 ab 1c 4d 12 14 77 6c ad 7b 7b 7d 8 43 2f 67 71 1d 42 15 9 4e 50 5c 7e 4a User time + System time = 0.001 + 0.002 = 0.003 Results using RSA with AVK ( Method 2): 8e b6 1a 38 8f ab 24 4f 7c 9f 33 51 a0 96 b9 46 4e 82 87 ab 33 9e 99 68 68 24 60 16 65 6b 81 a9 63 8e b2 3 b4 31 66 16 8f 4c 76 61 81 68 61 8e 85 b1 f 43 12 a5 7 8 a7 85 4 3d 4a 0 32 2 92 b5 2f 15 a6 58 af 74 83 9a 92 1c 7c 1d User time + System time = 0.002 + 0.003 = 0.005 Conclusion: In the present work, we have been reported different approaches of using AVK in RSA. It has been verified that it is time saving if parallelism is implemented. The work confirms the superiority of AVK application in RSA.

17. The key is made variable ,after every transmission it changes dynamically such that K0 = initial secret key Ki = Ki-1 XOR Di-1 for all i >0 where Di-1 andKi-1 are data and key of (i-1)th session respectively. In fact, other than XOR operation as illustrated in table 0.1, any other logic function like AND, OR, NOT, XNOR, etc. may be used. As XOR is operational logic function, we have used the same for this research. In subsequent chapters ,we have implemented several means of combining XOR with AVK. Any new scheme of key for cryptosystem needs to be evaluated by the scheme’s ability to challenge the eavesdropper’s ability to break key or cipher. Out of several breaking attacks two important attacks are brute force attack and differential attack. This thesis investigates the issues and challenges of brute force attack and differential frequency attack in the light of application of AVK in different forms of encryption. Repetitions of data and characters in messages of plain text result in repetitions of codes in cipher when a single key is used. Repeated codes in cipher are the source of differential frequency attack. It can be shown that use of different keys reduces this source of differential frequency attack in cipher, thereby making the AVK scheme superior to existing scheme. This is due to use of different keys for making different cipher for the repeated data or characters of plain text. We assume a plain text as “The beauty is appreciated when it is good”. When the text is encrypted with a single key k , we may get a cipher like “A B C D E F C G”. The repetition of “is” in plain text has resulted in repetition of corresponding character “C” in cipher. Attack in repeated character in cipher is source of differential attack. An illustration is given below:

18. Single Key The beauty is appreciated when it is good E {K K K K K K K K } A B C D E F C G Repetition Repetition Variable Key The beauty is appreciated when it is good E {K1 K2 K3 K4 K5 K6 K7 K8 } A B1 C1 D1 E 1 F1 C2 G1 ( No Repetition )

19. 5. Selection On Key Space Out of several attacks, two important attacks are brute force attack and differential frequency attack. We analyze the proposed scheme under these attacks. Brute force attack When a single key is used, as in existing technique, the probability of success of brute force attack, P0 = m / 2n When multiple keys are used as in proposed scheme, the probability of success of brute force attack: P1 = (m/k) / 2n when k < 2n <= m != 1 so long (m/k)< 2n where k = number of different keys used in the protocol It is seen that P1 < P0 when k>1.

20. KKS0 = 1011 sent under RSA Key SelectionKey Space 10 1 1 0000 group 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 m0 is ciphered under k0 (k0 =1000) m1 is ciphered under k1 (k1 =1001) KKS1 is ciphered under k2(k2 =1011) ……… ………

21. An estimate of k will be as follows: We assume KKSs are selected with equal probability. This is a reasonable assumption as KKSs are independent to each other, and there is no preference for choice. Under the assumption, different time variant keys equal to the number of 1s present in whole key space, N: 2n * n N = = 21n-1 * n ………………….(1) 2 as total number of bits equals to 2n * n out of which 50% are 1s. Out of total N, number of keys used for transport of KKS except the first one is 2n - 1. Thus the total number, k of time variant key used in the proposed protocol is given as: k = (2n-1 *n) – (2n – 1) ………………. (2) where k is not necessarily the different keys alone, but may also include some keys of the key space used in repetition. As a proof of equ(2), Table 2 is referred to.

22. Table 2: Proof of equ.(2)

23. The value of k increases with n, and as such superiority of the proposed scheme will enhance with higher size of n. Table 2 predicts this promising picture of the proposed technique. For example when n=8, number of keys (including repeated key over messages/ time out of total different key of 256) is as high as 769. The superiority of the proposed scheme over existing scheme is also established by comparing the average number of trials required to break a key under the schemes using key exhaustion algorithm. In existing single key scheme the average trials required is 2n-1, whereas in the proposed scheme it is (2n-1) * (2& -1) that is much higher than that of existing scheme When k=1, both becomes same and one as it must be.

24. 6. Differential Frequency Attack Repetitions of data and characters in messages of plain text result in repetitions of codes in cipher when a single key is used. Repeated codes in cipher are the source of differential frequency attack. Use of different many keys reduces this source of differential frequency attack in cipher, thereby making the proposed scheme superior to existing scheme. This is due to use of different keys in the proposed scheme for making cipher for the repeated data or characters of plain text. We perform experiments on two sets of messages having repeated characters in plain text. For experimental purpose we took each character as a message. Cipher was generated using (i) DES with single key under existing scheme and (ii) DES with multiple keys under proposed scheme. We measure and compare the repetitions of codes in cipher in the two schemes with a parameter of weighted frequency. The results obtained are shown in the Table 9.2. It is found that in both data sets, the proposed technique is superior to existing technique. As weighted frequency of plain text increases the superiority of the proposed scheme enhances.

25. Table 3: Comparison of weighted frequency in different schemes

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33. Thank You