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Unformatted text preview: Questions/Complaints About Homework? Heres the procedure for homework questions/complaints: 1. Read the solutions first. 2. Talk to the person who graded it (check initials) 3. If (1) and (2) dont work, talk to me. Further comments: Theres no statute of limitations on grade changes although asking questions right away is a good strategy Remember that 10/12 homeworks count. Each one is roughly worth 50 points, and homework is 35% of your final grade. 16 homework points = 1% on your final grade Remember were grading about 80 homeworks and graders are not expected to be mind readers. Its your problem to write clearly. Dont forget to staple your homework pages together, add the cover sheet, and put your name on clearly. Ill deduct 2 points if thats not the case 1 Algorithmic number theory Number theory used to be viewed as the purest branch of pure mathematics. Now its the basis for most modern cryptography. Absolutely critical for e-commerce How do you know your credit card number is safe? Goal: To give you a basic understanding of the mathematics behind the RSA cryptosystem Need to understand how prime numbers work 2 Division For a,b Z , a negationslash = 0, a divides b if there is some c Z such that b = ac . Notation: a | b Examples: 3 | 9, 3 negationslash | 7 If a | b , then a is a factor of b , b is a multiple of a . Theorem 1: If a,b,c Z , then 1. if a | b and a | c then a | ( b + c ). 2. If a | b then a | ( bc ) 3. If a | b and b | c then a | c (divisibility is transitive). Proof: How do you prove this? Use the definition! E.g., if a | b and a | c , then, for some d 1 and d 2 , c = ad 1 and c = ad 2 . That means b + c = a ( d 1 + d 2 ) So a | ( b + c ). Other parts: homework. Corollary 1: If a | b and a | c , then a | ( mb + nc ) for any integers m and n . 3 The division algorithm Theorem 2: For a Z and d N , d > 0, there exist unique q,r Z such that a = q d + r and 0 r < d . r is the remainder when a is divided by d Notation: r a (mod d ); a mod d = r Examples: Dividing 101 by 11 gives a quotient of 9 and a remain- der of 2 (101 2 (mod 11); 101 mod 11 = 2). Dividing 18 by 6 gives a quotient of 3 and a remainder of 0 (18 0 (mod 6); 18 mod 6 = 0). Proof: Let q = floorleft a/d floorright and define r = a- q d . So a = q d + r with q Z and 0 r < d (since q d a ). But why are q and d unique? Suppose q d + r = q prime d + r prime with q prime ,r prime Z and r prime < d . Then ( q prime- q ) d = ( r- r prime ) with- d < r- r prime < d . The lhs is divisible by d so r = r prime and were done....
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This note was uploaded on 10/09/2008 for the course COM S 280 taught by Professor Kleinberg during the Spring '05 term at Cornell University (Engineering School).
- Spring '05