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Avalanche Effect in DES key desirable property of encryption algo where a change of one input or key bit results in changing approx half output bits making attempts to “home-in” by guessing keys impossible DES exhibits strong avalanche

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Avalanche Effect in DES DES exhibits a strong avalanche effect. Table shows some results. In Table (a), two plaintexts that differ by one bit were used: with the key

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The Table (a) shows that after just three rounds, 21 bits differ between the two blocks. On completion, the two ciphertexts differ in 34 bit positions. Table (b) shows a similar test in which a single plaintext is input: with two keys that differ in only one bit position: Again, the results show that about half of the bits in the ciphertext differ and that the avalanche effect is pronounced after just a few rounds.

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Strength of DES – Key Size 56-bit keys have 2 56 = 7.2 x values brute force search looks hard recent advances have shown is possible – in 1997 on Internet in a few months – in 1998 on dedicated h/w (EFF) in a few days – in 1999 above combined in 22hrs! still must be able to recognize plaintext must now consider alternatives to DES

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Key Size (bits)Number of Alternative Keys Time required at 1 decryption/µs Time required at 10 6 decryptions/µs = 4.3 µs= 35.8 minutes 2.15 milliseconds = 7.2 µs= 1142 years hours = 3.4 µs= 5.4 years 5.4 years = 3.7 µs= 5.9 years 5.9 years 26 characters (permutation) 26! = 4 µs= 6.4 years 6.4 10 6 years Brute Force Search

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Chapter 4

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Modular Arithmetic define modulo operator “ a mod n” to be remainder when a is divided by n use the term congruence for: a = b mod n – when divided by n, a & b have same remainder – eg. 100 = 34 mod 11 b is called a residue of a mod n – since with integers can always write: a = qn + b – usually chose smallest positive remainder as residue ie. 0 <= b <= n-1 – process is known as modulo reduction eg. -12 mod 7 = -5 mod 7 = 2 mod 7 = 9 mod 7

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Divisors say a non-zero number b divides a if for some m have a=mb ( a,b,m all integers) that is b divides into a with no remainder denote this b|a and say that b is a divisor of a eg. all of 1,2,3,4,6,8,12,24 divide 24

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Modular Arithmetic Operations is 'clock arithmetic' uses a finite number of values, and loops back from either end modular arithmetic is when do addition & multiplication and modulo reduce answer can do reduction at any point, ie – a+b mod n = [a mod n + b mod n] mod n

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Modular Arithmetic can do modular arithmetic with any group of integers: Z n = {0, 1, …, n-1} form a commutative ring for addition with a multiplicative identity note some peculiarities – if (a+b)=(a+c) mod n then b=c mod n – but if (a.b)=(a.c) mod n then b=c mod n only if a is relatively prime to n

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Modular arithmetic Properties Modular arithmetic exhibits the following properties: [(a mod n) + (b mod n)] mod n = (a + b) mod n [(a mod n) (b mod n)] mod n = (a b) mod n [(a mod n) x (b mod n)] mod n = (a x b) mod

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Example 11 mod 8 = 3; 15 mod 8 = 7 [(11 mod 8) + (15 mod 8)] mod 8 = 10 mod 8 = 2 ( ) mod 8 = 26 mod 8 = 2 [(11 mod 8) (15 mod 8)] mod 8 = 4 mod 8 = 4 (11 15) mod 8 = 4 mod 8 = 4 [(11 mod 8) x (15 mod 8)] mod 8 = 21 mod 8 = 5 (11 x 15) mod 8 = 165 mod 8 = 5

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Exponentiation is performed by repeated multiplication, as in ordinary arithmetic. (We have more to say about exponentiation in Chapter 8.) To find 11 7 mod 13, we can proceed as follows: 11 2 = 121 Ξ 4 (mod 13) 11 4 = (11 2 ) 2 Ξ 4 2 Ξ 3 (mod 13) 11 7 Ξ 11 x 4 x 3 Ξ 132 Ξ 2 (mod 13)

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Modulo 8 Addition Example

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