rsa algorithm full report
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 computer science technology Active In SP Posts: 740 Joined: Jan 2010 24-01-2010, 10:11 PM   rsa algorithm full report.doc (Size: 54.5 KB / Downloads: 415) INTRODUCTION RSA Algorithm was discovered by a group of three scientists namely Ron Rivest,Adi Shamir and Len Adleman and was first published in 1978. The RSA scheme is a block cipher in which the plain text and cipher text are integers between 0 and n-1 for some n. A Typical size of n is 1024 bits or 309 decimal digits. This is a public key encryption scheme. In this scheme two pairs of integers {e, n} and {d, n} are used. First of them i.e. {e.n} is called the RSA public key and the other one i.e. {d, n} is called the RSA secret key. The sender uses the public key and encrypts the message say M into cipher text as â€œ C = M^e mod n. Where C is the cipher text and M is the message or the plane text At the receiving end the receiver accept the cipher text C and decrypt the C into M using secret key {d, n}- M = C^d mod n. Example: Let , e=3, d=7, n=33. Suppose the message is ËœSUNâ„¢ and we use the numeric values of the characters according to their serial in alphabets. Plaintext Ciphertext© after decryption Sym num M^3 M^3 mod33 C^7 C^7mod33 sym S 19 6859 28 13492928512 19 S U 21 9261 21 1801088541 21 U N 14 2744 5 78125 14 N KEY GENERATION The process of Key Generation contain the following steps 1- Select two prime numbers say p and q randomly Where p q. 2- Calculate- n = p *q. 3- Calculate ÃƒËœ(n) = (p-1) (q-1) Note- What is ÃƒËœ(n) ÃƒËœ(n) is called the Eulerâ„¢s Totient function. It is the no. of positive integers that are relative prime to n and are less then n. For example: - to determine ÃƒËœ(35), we list all the positive integers less then 35 that are relatively prime to it: 1,2,3,4,6,8,9,11,12,13,16,17,18,19,22,23,24,26,27,29,31, 32,33,34. There are 24 no on the list , so ÃƒËœ(35) =24. One thing which is important is that the value of ÃƒËœ(1) is without meaning but is defined to have the value 1. Â¢ It should be clear that for a prime no p , ÃƒËœ(p) = p-1. Now suppose that we have two prime no p and q , with p does not equal to q .Then for n = pq ÃƒËœ (n) = ÃƒËœ (pq) = ÃƒËœ (p)* ÃƒËœ (q) = (p-1)*(q-1). Two integers are said to be relatively prime if there only common positive integer factor is one. 4- Select any integer e such that gcd (ÃƒËœ(n),e)=1; 1 < e < ÃƒËœ (n). Note:- Gcd means greatest common divisor. The gcd of any two positive integers can be calculated with the help of Euclidâ„¢s algorithm which is as under â€œ EUCLID (a, b) 1. A a; B b 2. if B=0 return A = gcd (a, b) 3. R = A mod B 4. A B 5. B R 6. goto 2 5- calculate the value of d â€œ de = 1 mod ÃƒËœ(n) or d = e^-1 mod ÃƒËœ(n) In calculation of Ëœdâ„¢ we need the multiplicative inverse of Ëœeâ„¢ modulo ÃƒËœ(n) . We know that if gcd (m, b)=1, then b has a multiplicative inverse modulo m. That is, for positive integer b
 project report helper Active In SP Posts: 2,270 Joined: Sep 2010 14-10-2010, 12:57 PM   RSA-Cryptosystem.rar (Size: 306.03 KB / Downloads: 159) The RSA Cryptosystem Orhan K AKYILDIZ COMP5703 - Advanced Algorithms oka@kayra.ca Abstract In this seminar and presentation report, we review public key cryptography, and RSA public cyryptosystem in particular. RSA public cryptosystem is a asymmetrical cryptosystem: it uses a pair of keys, one of which is used to encrypt the data in such a way that it can only be decrypted with the other key. The keys are generated by a common process, but they can not be feasably generated from each other. The relation between these keys relies on a NP-hard problem: in virtually all cases, the function is a one-way function. There are a number of such one-way functions that various cryptographic algorithms rely on, including integer factorization, discrete logarithm, etc. 1 Introduction A cryptosystem is a sytem that allows secure exchange of information over an insecure channel [2],[3]. A crypto-system is typically modelled as a three-party system in which, a sender (historically called `Alice' ) sends a message to a recipient (`Bob' ) over an insecure channel, in the existance of an adversary (Eve), who tries to extract information o , or potentially change, the message. Figure 1 on page 3 depicts basic elements of a symmetric cryptosystem. Figure 2 on page 3 represents a newer class of cryptosystems, called asymmetric cryptosystems. A crypto-system is symmetrical when both encryption (the E(M; x) box in gure 2) and decryption (the D(C; x) box) use the same key; in other words, e and d are the same. Note that, it would not be wise to send the key on an insecure channel in this case, as the adversary will then be able to capture the key and will be able to gain control on the message.
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