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INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH Volume 1, No2, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 Use of Chinese Remainder Theorem to generate random numbers for cryptography Saurabh Singh 1 , Gaurav Agarwal 2 1­P.G student, UPTU, Lucknow 2­P.G student, KSOU, Mysore, Karnataka [email protected] ABSTRACT Random numbers are the numbers, which play an important role for various network security applications. hers are some techniques that are used to generating random numbers such as” pseudorandom number generator” and “linear congruent generator” also cryptographically generated random numbers” etc. but here we are using the Chinese Reminder Theorem for the purpose of generating Random numbers. In essence, CRT says it is possible to reconstruct integers in a certain range from their residues modulo a set of pair wise relatively prime modulo. Keywords: Random number, Chinese reminder theorem, number generator 1. Introduction 1.1 Overview Chinese Remainder Theorem, CRT, is one of the main theorems of mathematics. This can be used in the field of cryptography. It is a perfect combination of beauty and utility, CRT continues to present itself in new contexts and open vistas for new types of applications. So far, its usefulness has been obvious within the realm of “three C's”. Computing was its original field of application, and continues to be important as regards various aspects of algorithmic and modular computations. Theory of codes and cryptography are two more recent fields of applications. 1.2 CRT History The Chinese Remainder Theorem was first presented as problem 26 of the last volume of Master Sun’s Mathematical Manual, which divides into three volumes, sometime before Joseph Lagrange presented his interpolation formula, which is described by him as a short version of Isaac Newton’s (1642­1727) interpolation formula in his Lectures at the Ecole Normale in 1795. According to , many of the Chinese findings n mathematics ultimately made their way to Europe via India and Arabia. The Chinese Remainder Theorem became known in Europe through article, “Jottings on the science of Chinese arithmetic”, by Alexander Wylie in 1853 . Furthermore, says that J.L. Lagrange worked on problems on Indeterminate Analysis around 1767­68.Whether there was any direct transmission of mathematical knowledge from China to the West remains a matter of conjecture. However, the possibility should not be dismissed out of hand, as many historians of mathematics are inclined to do – either because they find the idea unpalatable or because there is insufficient
168 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH Volume 1, No2, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 documentary evidence. The fact remains that, as early as the third century B.C. Chinese silk and fine ironware were to be found in the markets of Imperial Rome. And a few centuries later a whole range of technological innovations found their way slowly to Europe. It is not unreasonable to argue that some of China’s intellectual products, including mathematical knowledge, were also carried westwards to Europe, there perhaps to remain dormant during Europe’s intellectual Dark Ages, but coming to life once more with the cultural awakening of the Renaissance [No implications can be made (at this point) on whether Lagrange was able to recognize the connection between The Chinese Remainder Theorem for polynomials and his proof on the Interpolation Formula. 1.3 CRT Formula And Algorithm According to this theorem, integers can be represented by their residues modulo a set of pair wise relatively prime module. To compute A (Mod m) First compute ai= A mod mi separately Then determine constants c i below, where Mi=M/mi Then combine results to get answer using: k A≡ (∑ aici )(mod M) ­­­­­­­­­­­1 i=1 (This formula is used for generating relatively prime modulo) For calculating the value of ci, we will use the given formula: C i= M i* (M i ­1 mod m i) for 1<= i<= k ­­­­­­­­­­­­­­­­­­­­ 2 (used for generating the constant value C i ) Where (Proof) A is a solution since, Mi = m 1 * m 2………* m i­1* m i+1 …..* m k C i = M i* (M i mod m i) = 1 mod m i 0 mod m j (For any j ≠i) Therefore
169 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH Volume 1, No2, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 k A≡ (∑ aici )(mod M) i=1 = c 1a 1 + c 2a 2 +…….c ka k + r * m = a i mod m i A is unique in Zm If A is not the unique answer, there must exist another answer A’ ≡ a i mod m i in Zm Then A ≡ A’ mod m i A­A’= r 1 * m 1 = r 2 * m 2 =. ...rk * mk Ri|mj where i ≠ j (since mi’s are relatively prime) R i * m i = r i’* m 1 * m 2 *…..mk = r i’ * m m|A­A’ A ≡ A’ mod m, proving uniqueness. (Uniqueness define that in every calculation unique number will generate) 2. Random Number Generation Using CRT As we know, the CRT is an algorithm with so many applications in mathmatics, computing is the main area of its application and moreover, recently it is being used in cryptography also. But in the field of cryptosystem, the algorithm is used for functionality for modular computation. Algorithm may provide the random numbers Firstly suppose we have Zm :={ 0, 1…m­1}, With addition modulo m is a group. p, q large random primes Now let us take Z15 i.e. (table 1) Table 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
170 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH Volume 1, No2, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 Now use the CRT as For n=pq (where p and q are prime) a function X: Zn—Zp * Zq Table 2 i 0 1 i mod 5 0 1 i mod 3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 2 3 4 0 1 2 3 4 0 1 2 3 4 2 0 1 2 0 1 2 0 1 2 0 1 2 Defined as X(i):= (i mod p,i mod q) [Chiness Remainder Theorem] For n=pq (where p and q are prime) a function X: Zn—Zp*Zq Defined as X(i):= (i mod p,i mod q) If X (i) = X (j) then Because p and q are prime i mod p = j mod p ­­ p divides i­j n divides i­j i mod q = j mod q ­­ q divides i­j i.e.(i = j mod n) For example: Z15 (i mod 5) and (i mod 3) Now what will happen with CRT? Z15: (table 2) Table 3 i mod 5 i mod3 0 1 2 3 4 0
Now simply, plotting the values in matrix……… 171 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH Volume 1, No2, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 First see one by one. Here comes the value on (0, 0) in the matrix….. i.e. 0 (table 3) Same procedure will be done by each and every value as (1,1) 1,(2,2) 2,(3,0) 3,(4,1)…..4,(0,2)…..5,(1,0)….6 etc. So final matrix will be as follows (table 4) i mod 5 i mod 3 Table 4 0 1 2 3 4 0 6 12 3 9 10 5 1 11 7 2 13 8 4 14 0 1 2 It is clear that on every point a number is being generated for every value…. Now we should check it for another matrix, on different mod values. Suppose we take the prime numbers p = 4 and q = 6 i.e. (i mod 4) and ( i mod 6) Now for this condition, the matrix will be plotted like that— And it’ll also prove that it will not same ever. Consider p = 4 and q = 6 i mod 6 0 1 2 3 4 5 i mod 4 0,12 8,20 1,13 6,18 4,16 9,21 2,14 7,19 0 5,17 10,22 3,15 11,23
1 2 3 3. Practical Application and Use Of Random Numbers Random number generators have application in gambling, statistical sampling, computer simulation, cryptography, and other areas where a random number is useful in producing an unpredictable result. The random numbers also is useful for the prevention of reply attack also for counter measures. 4. Result 172 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH Volume 1, No2, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 In the area of cryptography and network security random numbers can be used as the Nonce and it can be attached with the message packets on the sender end as the identification of each and every packet. In the above technique the CRT generate unique random numbers for every key value which can be attached with the message as follows. Example: A draft is sent by me in favor of my organization. is our message and we take the key value 12 .for the key value 12 the value will be entered between 0 and 11.Then the theorem will as follows and will attach the random numbers with each packet for their uniquely identification. As follows… 1 1 1 2 2 2 3 0 3 4 1 0 5 2 1 6 0 2 7 1 3 8 2 0 9 0 1 10 1 2 11 2 3 Matrix will as follows 0 9 6 3 4 1 10 7 8 5 2 11 And the message will turn in the form of packets with unique random numbers . 0 0 A 1 9 Draft is 2 6 Sent by m 3 3 e in favo 4 4 ur of my 5 1 organiz. 6 10 . 5. Conclusion Chiness reminder theorem, provide benefits in computing, mathematics and also in the field of cryptography, where the algorithm provides relief in case of modular computation and also in case of generating the random numbers. But we had made a complex study of this theorem to develop a new concept of producing random numbers. As we know that random numbers has a wide application in cryptography to make security of the network communication more and stronger so that the intruder can hot hamper the network security.
173 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH Volume 1, No2, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 We have also these random to show how these random numbers can be used in secure message transmission preventing packet reply which is an attack in which an intruder captures a message packet and reply it again to gain access to information or system application. We have used random numbers which are unique for a session of message transmission. At the receiver side these random numbers are checked and if the number has been received earlier also it means that the packet is replayed and discarded. 6. References 1. Junod, P. "Cryptographic Secure Pseudo­Random Bits Generation: The Blum­Blum­ Shub Generator." August 1999. http://crypto.junod.info/bbs.pdf 2. Gentle, J. E. Random Number Generation and Monte Carlo Methods, 2nd ed. Springer­Verlag, 2003 3. Donald knuth.The Art of Computer programming, volume 2:seminumerical Algorithms,Third Edition. Section­Wesley,1997.
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