week3 - SIT 281 Introduction to Cryptography WEEK 3...

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7/27/2010 1 SIT 281 Introduction to Cryptography WEEK 3 Objectives > We cover sections ± 3.1, ± 3.2, ± 3.3 and ± 3.4 > of Chapter 3 this week.
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7/27/2010 2 Modular arithmetic Of course, we saw that the Caesar cipher and other modular arithmetics are cyclical. They repeat once you reach the modulus So clock They repeat once you reach the modulus. So clock arithmetic repeats after you get to 12 and the Caesar cipher after you reach 26. Divisibility In order to understand arithmetic modulo any fixed value, we take a step back and look at ordinary divisibility in numbers. Example: 17 divides 51, 51 is divisible by 17. Example: Find all numbers which divide 28. To do this, we usually factor So 1, 2, 4, 7, 14 and 28 are all (positive) divisors of 28. . 7 2 7 2 2 28 2 = =
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7/27/2010 3 Divisibility cont’d Definition. Let a and b be integers with We say that a divides b if there is an integer k such that b=ak . Notation. We write a|b if a divides b . We write a|b if a does not divide b . . 0 a We also say that b is a multiple of a , or b is divisible by a . Mental arithmetic: Which of the following is divisible by 3: 291, 374, 921? Some properties 1. Every integer divides 0. [Look at 0|0 more closely; we try to avoid this!] 2. The numbers divide every integer. 3. Every integer divides both plus and minus itself. 4. If a|b and b|c , then a|c . WRITE A PROOF! 1 ± 5. If a|b and a|c , then for all integers and . WRITE A PROOF! ) ( | c b a β α +
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7/27/2010 4 Prime Numbers Recall that a positive integer is prime if its only integer divisors are itself and . Otherwise, it is called composite . Examples? There are lots of primes. It fact, this was proved in 1896: Prime Number Theorem Let be the number of primes ± 1 ± π Prime Number Theorem. Let be the number of primes in the range 2 to x . Then This is an ‘asymptotic’ result. It is good for very large x . x x x ln ) ( ) ( x The number of primes up to 10000000000 > (There are 10 zeros in the number) > The previous formula tells us that it is about natural logarithm of this number divided by the number itself. > A scientific calculator will compute this for you to get: 434294481.9 to one decimal place. TRY: (a) The number 12 (b) the number 100000000000000000000 (20 zeros).
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7/27/2010 5 Large primes The following number has 149 decimal digits. >Fa c to r i t : 103652740928683365687456059316479333608983340154540 595567624209291514551376406240206652791535419489259 61617220570528383159719295364041108999475987291 > Hint: It factors into two large primes each with 75 digits. Factoring large numbers > The basis of many sophisticated cryptosystems is the > The basis of many sophisticated cryptosystems is the principle that factoring is hard.
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This note was uploaded on 08/12/2010 for the course CRYOPTOGRA 3232 taught by Professor Jhhgjhgjh during the Spring '10 term at Trinity College, Hartford.

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week3 - SIT 281 Introduction to Cryptography WEEK 3...

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