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- 1. RSA Public-Key Cryptosystem- The development of public key cryptosystems is the greatest and perhaps the onlytrue revolution in the entire history of cryptography.- Symmetric encryption encrypts and decrypts with the same key- Public key cryptosystems is asymmetric which use two keys one for encryption andthe other for decryption.- Public key cryptosystems depend on mathematical functions and number theoryrather than substitution.- Public key cryptosystems have five ingredients. 1. Plaintext is the readable message or text before encryption. 2. Encryption algorithm performs various transformations on the plaintext. 3. Public and private keys one for encryption and one for decryption the algorithm depends on these keys for transforming text. 4. Ciphertext the encrypted message (the text after encryption). 5. Decryption algorithm retrieves the original message from the ciphertext.- Public key cryptosystems applications. 1. Encryption/Decryption. 2. Digital Signature the sender signs a message with its private key. Signing is achieved by a cryptographic algorithm applied to the message or to a small block of data that is a function of the message. 3. Key exchange. Two sides cooperate to exchange a session key.- Prime number is the number that accepts division by itself or one only. ex., 1, 2, 3, 5, 7, 11…………- Composite number. Is the number that accepts division by at least a number that is not one or itself. Ex. 4 accept division by 2, 9 accept division by 3, 12 accept division by 2,3,4,6 And so on. Relatively prime Two numbers x1, x2 are relatively prime if and only if gcd(x1, x2) = 1. Ex. 12, 25 are relatively prime since gcd (12, 25) = 1. 12, 15 are not relatively prime since gcd (12, 15) = 3. Prime number factorizationAny composite number consists of a unique factorization of prime numbers.a = (p1 ^ e1) * (p2 ^ e2) * ……. * (pr ^ er)Where a is a composite number and p1, p2… are prime number where p1<p2<...<pr Ex. 4 = 1 * 2^2. ^ stands for power 6 =1*2*3 8 = 1 * 2^3 10 = 1 * 2 * 5 12 = 1 * 2 ^2 * 3 26 = 1 * 2 * 13 60 = 1 * 2^2 * 3 * 5 and so on
- 2. Modular Arithmeticwe use modular arithmetic to reduce calculating modular powers# (a + b) % m = [ (a % m) + (b % m) ] % m# (a * b) % m = [ (a % m) * (b % m) ] % mLet we formalize the previous notes. (a + b) % m = [a]m +m [b]m (a * b) % m = [a]m *m [b]mExamples(7 + 6 ) % 4 = 13 % 4 = 1(7 + 6 ) % 4 = [ (7 % 4) + (6 % 4) ] % 4 = [ 3 + 2 ] % 4 = [ 5 ] % 4 = 1(7 * 6 ) % 4 = 42 % 4 = 2(7 * 6 ) % 4 = [ (7 % 4) * (6 % 4) ] % 4 = [ 3 * 2 ] % 4 = [ 6 ] % 4 = 2(3 ^ 8) % 7 = [ { (3 ^ 2) % 7 } * { (3 ^ 2) % 7 } * { (3 ^ 4) % 7 } ] % 7(3 ^ 8) % 7 = [ 2 * 2 * 4 ] % 7 = [ 16 ] % 7 = 2(11 ^ 23) % 187 = [(11^1) % 187 * (11^2) % 187 * (11^4) % 187 * (11^8) % 187 * (11^8) % 187] % 187(11 ^ 23) % 187 = [11 * 121 * 55 * 33 * 33] % 187 = 79720245 % 187 = 88 Note that (((M ^ e) % n) ^ d) % n = (M ^ ed) % nExample[((5 ^ 2) % 7) ^ 3] % 7 = (5 ^ (2*3)) % 7 (4 ^ 3) % 7 = (5 ^ 6) % 7 64 % 7 = (15625) % 7 ---- 7 * 2232 = 15624 1 = 1 Modulo InverseThe identity of additive modulo is [0]mThe additive inverse of [m]n is [n – m]mEx.The additive inverse of [1]5 is [5 – 1]5 = [4]5 [1]5 + [4]5 = [0]5 " The identity"The identity of multiplicative modulo m is [1]m[m]n have a multiplicative inverse [k]n where [m]n × [k]n = [1]n.Ex.The multiplicative inverse modulo of [5]9 is [k]9 where[5]9 * [k]9 = [1] 9 = [5 * k] 9 = [1] 9 = [5 * 2] 9 = [1] 9Then the multiplicative inverse of [5]9 is [2]9CorollaryIf m, k are multiplicative inverses modulo nThen (m * k) % n = 1 = (m * k) = (z * n + 1) where z is positive integerNote from the previous example (5 * 2) % 9 = 1
- 3. Note that nonprimes may dont have multiplicative inverse modulo mEx.[6]9 * [k]9 = [1] 9 [6*k] 9 = [1] 9We couldnt find k that make the equation (6*k) = z * 9 + 1 true. Note also that if m and k have multiplicative inverses modulo nthen both m and k must be relatively prime to nin the previous example both 5 and 2 are relatively prime to nEx.[3]9 * [k]9 = [1] 9We couldnt find k that make the equation (3*k) = z * 9 + 1 true because 3 isnot relatively prime to 9 since gcd(3 , 9) = 3. Eulers Totient functionEulers totient function is denoted by ΦΦ(N) = how many numbers between 1 and (N – 1) which are relatively prime to N.And is given by the following rule.Φ(N) = N * ∏p|n (1 – (1 / p) ) where p runs over all primes that divide N including Nif it is primeEx.Φ(4) = 4 * ( 1 – (1 / 2) ) = 4 – 2 = 2 --- relative prime numbers to 4 is { 1 , 3 }Φ(5) = 5 * ( 1 – (1 / 5) ) = 5 – 1 = 4 --- { 1 , 2 , 3 , 4 }Φ(6) = 6 * ( 1 – (1 / 2) ) * ( 1 – (1 / 3) ) = 3 – 1 = 2 --- { 1 , 5 }Φ(7) = 7 * ( 1 – (1 / 7) ) = 7 – 1 = 6 --- { 1 , 2 , 3 , 4 , 5 , 6 }Let N be a prime number then its factors is 1, N thenΦ(N) = N * ( 1 – (1 / N ) = N – 1Also we note from the previous example that prime numbers has and advantage whichare Φ(N) = N – 1 when N is prime.There is another amazing fact that Φ(N) is also easy to calculate when N has twoprime Numbers. For example if N = p * q where p, q are two prime numbersΦ(N) = (p – 1) * (q – 1)Proof:Since p, q are all the prime factors of N then by applying Eulers Totient functionΦ(N) = Φ(pq) = pq * (1 – (1 / p)) * (1 – (1 / q)) = [p * (1 – (1 / p))] * [q * (1 – (1 / q))] = [p – 1] * [q – 1] RSA algorithm requirementsIf we have the message is MThen C = (M ^ e) % n (C is the encrypted message)And M = (C ^ d) % n
- 4. Both sender and receiver must know the value of n and the sender knows the value ofe (e may known to any one) and the receiver only must know the value of d.PK = {e , n} and PK = {d , n}For the algorithm to be satisfactory as a public key encryption the followingrequirements must be met 1. It is possible to find values for e, d, n such that (M ^ ed) % n = M for all M < n 2. It is relatively easy to calculate (M ^ e) % n and (C ^ d) % n. (Modular Arithmetic) 3. It is infeasible to determine d given e, n.For (M ^ ed) % n = M to be true e, d must be multiplicative inverses modulo Φ(n)Then the relation between d, e can be expressed as(e * d) % Φ(n) = 1 ==== (e * d) = (z * Φ(n)) + 1 == d = (z * Φ(n) + 1) / eand this is true if and only if e, d are relatively prime to Φ(n) Why prime numbers in RSA 1. Prime numbers have the property of multiplicative inverses modulo 2. Factoring of the product of two prime numbers is harder than any other numbers. 3. Φ(n) have a direct rule for the product of two primes RSA AlgorithmRivest–Shamir-Adleman algorithm developed at MIT in 1978.The algorithm Key generationSelect two large prime numbers p,q and p ≠ q. ----- p,q (private, chosen)Calculate n = p * q ----- n (public, calculated)Calculate Φ(n) = (p - 1) * (q - 1) -----Select integer e which is relatively prime with Φ (n) gcd(e, Φ(n)) = 1; 1<e< Φ(n) ------ e (public, chosen)calculate d where de % Φ(n) = 1 i.e. d = (z * Φ(n) + 1) / e -- d (private, calculated)Public key {e, n}Private Key {d, n} EncryptionLet M to be the plain text given M<nCiphertext C = (M ^ e) % n ----- public key {e, n} DecryptionCiphertext CPlaintext M = (C ^ d) % n ------- private key {d, n}
- 5. Example Let the plain text is 88 encrypt it with RSA 1. Select two primes p = 17 , q = 11. 2. Calculate n = pq = 17 * 11 = 187. 3. Calculate Φ(n) = (p - 1)*(q - 1) = 16*10 = 160. 4. Select e which is relatively prime with Φ(n) and e < Φ(n) let we choose e = 7. 5. Calculate d = ( 1 + z * Φ(n) ) / e = ( 1 + 160 ) / 7 = 23 let z = 1 6. Public key { 7,187 } 7. Private key { 23, 187 } Encryption with public key {7, 187} Cipher text = (88^7) % 187 = 11Decryption with private key {23, 187}Given the ciphertext is 11Plaintext M = ( 11 ^ 23 ) % 187 = 88 CryptanalysisReferencesCryptography and Network Security Principles and Practices,Fourth Edition By William Stallings Good Luck With my best wishes Farag Zakaria Safy Saad farag_cs2005@yahoo.com

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