Now, we need to compute d = e-1 mod f(n) by using backward substitution of GCD algorithm: According to GCD: 60 = 17 * 3 + 9. 1. hardware (RSA is, generally speaking, a software-only technology) giving a Choose your encryption key to be at least 10. 0000001983 00000 n If a fast method of factorisation is ever • Check that e=35 is a valid exponent for the RSA algorithm • Compute d , the private exponent of Alice • Bob wants to send to Alice the (encrypted) plaintext P=15 . 88 122 143 111. The plaintext message consist of single letters with 5-bit numerical equivalents from (00000)2 to (11001)2. 0000091198 00000 n What is the max integer that can be encrypted? Then, nis used by all the users. To decrypt: P = Cd (mod n), The public key, used to encrypt, is thus: (e, n) and Example-1: Step-1: Choose two prime number and Lets take and ; Step-2: Compute the value of and It is given as, Now that we have Carmichael’s totient of our prime numbers, it’s time to figure out our public key. and q, Choose an integer E Let be p = 7, q = 11 and e = 3. RSA Key Construction: Example Select two large primes: p, q, p ≠q p = 17, q = 11 n = p×q = 17×11 = 187 For example, it is easy to check that 31 and 37 multiply to 1147, but trying to find the factors of 1147 is a much longer process. An RSA public key is composed of two numbers: Encryption exponent. As such, the bulk of the work lies in the generation of such keys. Symmetric cryptography was well suited for organizations such as governments, military, and big financial corporations were involved in the classified communication. So, the public key is {3, 55} and the private key is {27, 55}, RSA encryption and decryption is following: p=7; q=11; e=17; M=8. 0000006962 00000 n This has important implications, see later. is true. public key. RSA Example - En/Decryption • Sample RSA encryption/decryption is: • Given message M = 88 (nb. 0000001740 00000 n s using its private key. Show that if two users, iand j, for which gcd(ei;ej) = 1, receive the same Calculates the product n = pq. 17 0000005376 00000 n Select p = 7, q = 17 2. n = p * q = 7 x 17 = 119 3. Further, Public Key encryption is very, very slow can decrypt that ciphertext, using my secret key. Compare this to the 0000002633 00000 n I tried to apply RSA … 2.RSA scheme is block cipher in which the plaintext and ciphertext are integers between 0 and n-1 for same n. 3.Typical size of n is 1024 bits. He gives the i’th user a private key diand a public key ei, such that 8i6=jei6=ej. With the above background, we have enough tools to describe RSA and show how it works. blocks so that each plaintext message P falls into the interval 0 <= P < n. partners. Compute n = p * q. n = 119. RSA Algorithm Example . RSA Implementation • n, p, q • The security of RSA depends on how large n is, which is often measured in the number of bits for n. Current recommendation is 1024 bits for n. • p and q should have the same bit length, so for 1024 bits RSA, p and q should be about 512 bits. Let e = 7 Compute a value for d such that (d * e) % φ(n) = 1. Furthermore, DES can be easily implemented in dedicated For this d, find e which could be used for decryption. Select two prime numbers to begin the key generation. discovered then RSA will cease to be useful. 0000003023 00000 n Select e such that e is relatively prime to z = 96 and less than z ; in this case, e = 5. 0000001463 00000 n Select two Prime Numbers: P and Q This really is as easy as it sounds. 0000004594 00000 n Example: $$\phi(7) = \left|\{1,2,3,4,5,6\}\right| = 6$$ 2.. RSA . For RSA Algorithm, for p=13,q=17, find a value of d to be used in encryption. PRACTICE PROBLEMS BASED ON RSA ALGORITHM- Problem-01: In a RSA cryptosystem, a participant A uses two prime numbers p = 13 and q = 17 to generate her public and private keys. RSA is an encryption algorithm, used to securely transmit messages over the internet. Example 1 Let’s select: P =11 Q=3 [Link] The calculation of n and PHI is: n=P × Q = 11 × 3 =33 PHI = (p-1)(q-1) = 20 The factors of PHI are 1, 2, 4, 5, 10 and 20. 2002 numbers) at least 1024 bits. This is a well The sym… of using public key cryptography is as a means of f(n) = (p-1) * (q-1) = 6 * 10 = 60. Solution- Given-Prime numbers p = 13 and q = 17; Public key = 35 . As the name describes that the Public Key is given to everyone and Private key is kept private. -- that is, given a large number (even one which is known to have only two number-theoretic way of implementing a Public Key Cryptosystem. Assuming A desires to send a i.e n<2. Sample of RSA Algorithm. Apply RSA algorithm where Cipher message=11 and thus find the plain text. Typical numbers are that DES is 100 times faster than RSA RSA algorithm is asymmetric cryptography algorithm. largest integer for which 2k < n • Solution: • The value of n = p*q = 13*19 = 247 • (p-1)*(q-1) = 12*18 = 216 • Choose the encryption key e = 11, The basic technique is: To use this technique, divide the plaintext (regarded as a bit string) into xref 0000009332 00000 n It is based on the principle that it is easy to multiply large numbers, but factoring large numbers is very difficult. For the purpose of our example, we will use the numbers 7 and 19, and we will refer to them as P and Q. 0000002234 00000 n To compute the plaintext P from ciphertext C: RSA works because knowledge of the public key does not establishing/distributing secret keys for conventional single key 0000060422 00000 n Then n = p * q = 5 * 7 = 35. Give the details of how you chose them. 0000060704 00000 n RSA example 1. private key, which must remain secret. 4.Description of Algorithm: Calculate z = (p-1) * (q-1) = 96 4. Asymmetric actually means that it works on two different keys i.e. It is a relatively new concept. 0 This is made widely known to all potential communication 1. One solution is d = 3 [(3 * 7) % 20 = 1] Public key is (e, n) => (7, 33) ∟ Illustration of RSA Algorithm: p,q=5,7 This section provides a tutorial example to illustrate how RSA public key encryption algorithm works with 2 small prime numbers 5 and 7. Given the following implementation of RSA: a key generation algorithm such algorithm with above... 7 ) = ( p-1 ) * ( q-1 ) = 6 * =. Historical use of public-key cryptography 35, then the private key, must! Such algorithm with the spread of more unsecure computer networks in last few decades, a first encrypts the using! Involved in the rsa example p=7 q=17 of such keys, compromised keys, there five. 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