Can be transmitted with the message but must be unpredictable Encryption Cipher

Can be transmitted with the message but must be

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Can be transmitted with the message but must be unpredictable. Encryption
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Cipher Block Chaining (CBC) Decryption plaintext key ciphertext + + + IV
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Error Recovery An error in ciphertext affects one block and several bits of plaintext key ciphertext plaintext + + Error extension
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Potential Problems With CBC Mallory can: Add blocks Drop blocks Introduce bit errors Bit loss/addition is not recoverable
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Public Key Cryptography Everyone has two keys: Public key K1 that everyone knows Private key K2 that only he knows Encryption algorithm and key properties ensure that M M E D K K )) ( ( 1 2
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Modular Arithmetic Galois Field GF(n) All operations are on numbers 0,…n-1 Observe all operations in Galois Field GF(n) a = b mod n if k, a = k*n + b e.g. 26 mod 16 = 10 so 26 = 10 mod 16 Modulo operation (modular reduction) can be performed at any point, e.g. (a b) mod n ((a mod n) (b mod n)) mod n
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Modular Exponentiation Key step in asymmetric crypto How many operations are needed to calculate a x ? Exponentiation can be performed very efficiently ( addition chaining ): We want to calculate a x mod n Write x as a binary number, result=1 Traverse x from left to right If digit is 1, result=result 2 *a If digit is 0, result=result 2 Perform modular reduction often to keep result small. This is cheap if n =2 m
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Example 9 23 mod 676 23 mod 4 * 13 result : 1 is bit step5 13 23 mod 36 23 mod 4 * 3 result : 1 is bit step4 3 23 mod 256 23 mod 16 result : 0 is bit step3 16 23 mod 4 result : 0 is bit step2 4 4 * 1 result : 1 is bit step1 23 mod 4 10011 19 23 mod 4 2 2 2 2 2 19 19
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Prime Numbers A number n is prime if it is only divisible by 1 and itself Numbers x and y are relatively prime if they share no factors greater than 1 E.g. 7 and 15 are relatively prime, 9 and 15 are not because they have 3 as common factor
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Inverses Modulo a Number Multiplicative inverse y for x is a number that satisfies: In GF(n) inverse y for x modulo n is a number that satisfies: Inverse y in GF( n ) can be found uniquely if x and n are relatively prime, otherwise it cannot be found If n is prime then it is relatively prime to all numbers { 0, n-1 } and each number has its inverse in GF(n) x and y will be our public/private key pair 1 y * x 1 n mod y * x
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Extended Euclidean Algorithm How to find an inverse y for x mod n Extended Euclidean algorithm will find y and k given x and n Generate one key starting from the other (e.g., have public key, generate private key) 1 n * k - y * x n * k 1 y * x 1 n mod y * x
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Factorization of Large Numbers We have seen that exponentiation in GF( n ) can be performed efficiently On the other hand, it is hard to factor large numbers (find possible x and y , given x*y ) Computationally intensive, must use brute-force search Generally factoring time of a large number n increases exponentially with each binary digit added to n
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RSA Algorithm Created by Ron Rivest, Adi Shamir, sand Leonard Adleman Choose two prime numbers p and q of equal length Compute n=p*q , and Euler Totient function (n)= (p-1)*(q-1) We will work in GF(n) Choose public key e relatively prime to ( n )
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  • Spring '17
  • Cryptography, Alice, Plaintext, Eve, Block cipher example, • Mallory

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