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COMP4690 Tutorial

COMP4690 Tutorial. Cryptography & Number Theory. Outline. DES Example Number Theory RSA Example Diffie-Hellman Example. DES. Some remarks DES works on bits DES works by encrypting groups of 64 bits, which is the same as 16 hexadecimal numbers

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COMP4690 Tutorial

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  1. COMP4690 Tutorial Cryptography & Number Theory

  2. Outline • DES Example • Number Theory • RSA Example • Diffie-Hellman Example

  3. DES • Some remarks • DES works on bits • DES works by encrypting groups of 64 bits, which is the same as 16 hexadecimal numbers • DES uses keys which are also apparently 64 bits long. However, every 8th key bit is ignored in the DES algorithm, so the effective key size is 56 bits. • If the length of the message to be encrypted is not a multiple of 64 bits, it must be padded. E.g.: • The plaintext message "Your lips are smoother than vaseline" is, in hexadecimal,"596F7572206C6970 732061726520736D 6F6F74686572207468616E2076617365 6C696E650D0A". • We then pad this message with some 0s on the end, to get a total of 80 hexadecimal digits: "596F7572206C6970 732061726520736D 6F6F746865722074 68616E2076617365 6C696E650D0A0000". • Then apply DES.

  4. Key generation example • Let K be the hexadecimal key K = 133457799BBCDFF1. This gives us as the binary key : • K = 00010011 00110100 01010111 01111001 10011011 10111100 11011111 11110001 • 16 subkeys (48-bit) will be generated from K.

  5. Key generation example • Based on table PC-1 (Permuted Choice 1), we get the 56-bit permutation • K+ = 1111000 0110011 0010101 0101111 0101010 1011001 1001111 0001111 • Next, split this key into left and right halves, C0 and D0, where each half has 28 bits. • C0 = 1111000 0110011 0010101 0101111 D0 = 0101010 1011001 1001111 0001111

  6. Key generation example • we now create sixteen blocks Cn and Dn, 1<=n<=16. Each pair of blocks Cn and Dn is formed from the previous pair Cn-1 and Dn-1, respectively, for n = 1, 2, ..., 16, using a“schedule of left shifts".

  7. Key generation example • C0 = 1111000011001100101010101111D0 = 0101010101100110011110001111 • C1 = 1110000110011001010101011111D1 = 1010101011001100111100011110 • C2 = 1100001100110010101010111111D2 = 0101010110011001111000111101 • C3 = 0000110011001010101011111111D3 = 0101011001100111100011110101 • ……

  8. Key generation example • We now form the subkeys Kn, for 1<=n<=16, by applying the table PC-2 (Permutation Choice Two) to each of the concatenated pairs CnDn. • For the first subkey, we have • C1D1 = 1110000 1100110 0101010 1011111 1010101 0110011 0011110 0011110 • After we apply the permutation PC-2: • K1 = 000110 110000 001011 101111 111111 000111 000001 110010

  9. Modular Arithmetic • Two integers a and b are said to be congruent modulo n, if : (a mod n) = (b mod n) • This is written as a≡b modn • Define Zn as the set of nonnegative integers less than n: Zn={0,1,…,(n-1)}

  10. Modular Arithmetic • Properties of modular arithmetic

  11. Modular Arithmetic • Define Zp as the set of nonnegative integers less than a given prime number p: Zp={0,1,…,(p-1)} • Because p is prime, all of the nonzero integers in Zp are relatively prime to p. • There exists a multiplicative inverse for all of the nonzero integers in Zp : • For each nonzero w in Zp, there exists a z in Zp such that w x z ≡ 1 modp. z is called the multiplicative inverse of w. Or, z = w-1.

  12. Number Theory • Fermat’s Little Theorem: • ap-1≡ 1 modp • where p is prime and gcd(a,p)=1 • E.g. • a = 7, p = 19 • 72=49≡11 mod 19 • 74≡121≡7 mod 19 • 78≡49≡11 mod 19 • 716≡121≡7 mod 19 • ap-1=718=716x72≡7x11=77≡1 mod 19

  13. Number Theory • An alternative form of Fermat’s Little Theorem: • ap≡amod p • where p is prime and ais any positive integer • E.g. • p=5,a=3,35=243≡3 mod 5 • p=5,a=10,105=100000≡10 mod 5≡0

  14. Number Theory • Euler’s Totient Function ø(n) • The number of positive integers less thannand relatively prime ton • For prime numberp, • ø(n)= p – 1 • For n = pqwherepandqare two different prime numbers • ø(n)= (p – 1) (q – 1)

  15. Number Theory • Example: ø(21) • From 1 to 21, totally 21 numbers • 21 = 3x7, 3 and 7 are prime • 3’s multiples: • 3, 6, 9, 12, 15, 18, 21 • 7’s multiples: • 7, 14, 21 • Other numbers are all relatively prime to 21 • 21-7-3+1 = (3-1)x(7-1)

  16. Number Theory • Euler’s Theorem • aø(n)≡ 1 mod n • where gcd(a,n)=1 • E.g. • a=3;n=10; ø(10)=4; • hence 34 = 81 ≡ 1 mod 10 • a=2;n=11; ø(11)=10; • hence 210 = 1024 ≡ 1 mod 11

  17. Number Theory • The powers of an integer a, modulo n • a, a2, a3, … (mod n) • If a and n are relatively prime, based on Euler’s theorem, we have aø(n)≡ 1 mod n • a, a2, a3, … will have a repeated pattern • E.g., ø(5)=4, 3ø(5)=81≡1 mod 5 • 3, 4, 2, 1, 3, 4, 2, 1, … • There may exist lots of m such that am ≡ 1 mod n • The least positive exponent m such that am ≡ 1 mod n is referred to as • the order of a (mod n) • the exponent to which a belongs (mod n) • the length of the period generated by a

  18. Number Theory

  19. Number Theory • Primitive root • If a number’s order (mod n) is ø(n), this number is called a primitive root of n • Property of primitive root • If a is a primitive root of n, then its powers a,a2, a3,…,aø(n)are distinct (mod n), and are all relatively prime to n. • In particular, for a prime number p, if a is a primitive root of p, then a,a2, a3,…,ap-1are distinct (mod p). • From the previous table, we can see that prime number 19’s primitive roots are 2, 3, 10, 13, 14, and 15.

  20. RSA Example • Select primes: p=17 & q=11 • Computen = pq =17×11=187 • Compute ø(n)=(p–1)(q-1)=16×10=160 • Select e: gcd(e,160)=1; choose e=7 • Determine d: de=1 mod 160 and d < 160 • d=23 since 23×7=161= 10×160+1 • Publish public key KU={7,187} • Keep secret private key KR={23,17,11}

  21. RSA Example • given message M = 88 • encryption: C = 887 mod 187 = 11 • decryption: M = 1123 mod 187 = 88

  22. RSA Example • Fast Modular Exponentiation • To calculate 887 mod 187 • 881 mod 187 = 88 • 882 mod 187 = 7744 mod 187 = 77 • 884 mod 187 = 772 mod 187 = 132 • 887 mod 187 = 884+2+1 mod 187 = 132x77x88 mod 187 = 894,432 mod 187 = 11 • To calculate 1123 mod 187 • 111 mod 187 = 11 • 112 mod 187 = 121 • 114 mod 187 = 14,641 mod 187 = 55 • 118 mod 187 = 552 mod 187 = 33 • 1116 mod 187 = 332 mod 187 = 154 • 1123 mod 187 = 1116+4+2+1 mod187 = 154x55x121x11 mod 187 = 11,273,570 mod 187 = 88

  23. Diffie-Hellman Key Exchange

  24. Diffie-Hellman Key Exchange • users Alice & Bob who wish to swap keys: • agree on prime q=7 and α=5 • select random secret keys: • A chooses xA=3, B chooses xB=2 • compute public keys: • yA=53 mod 7 = 6(Alice) • yB=52 mod 7 = 4(Bob) • compute shared session key as: Alice: KAB= yBxA mod 7 = 43 mod 7 = 1 Bob: KAB= yAxB mod 7 = 62mod 7 = 1

  25. Diffie-Hellman Key Exchange • users Alice & Bob who wish to swap keys: • agree on prime q=353 and α=3 • select random secret keys: • A chooses xA=97, B chooses xB=233 • compute public keys: • yA=397 mod 353 = 40 (Alice) • yB=3233 mod 353 = 248 (Bob) • compute shared session key as: Alice: KAB= yBxA mod 353 = 24897mod 353= 160 Bob: KAB= yAxB mod 353 = 40233mod 353 = 160

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