280wk4_x4 - Questions/Complaints About Homework? Heres the...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Questions/Complaints About Homework? Here’s the procedure for homework questions/complaints: 1. Read the solutions Frst. 2. Talk to the person who graded it (check initials) 3. If (1) and (2) don’t work, talk to me. ±urther comments: There’s 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 Fnal grade. 16 homework points = 1% on your Fnal grade Remember we’re grading about 100 homeworks and graders are not expected to be mind readers. It’s your problem to write clearly. Don’t forget to staple your homework pages together, add the cover sheet, and put your name on clearly. I’ll deduct 2 points if that’s not the case 1 Algorithmic number theory Number theory used to be viewed as the purest branch of pure mathematics. Now it’s 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 ±or a, b Z , a 6 = 0, a divides b if there is some c Z such that b = ac . Notation: a | b Examples: 3 | 9, 3 6 | 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 deFnition! E.g., if a | b and a | c , then, for some d 1 and d 2 , b = 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: ±or 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 = b a/d c and deFne 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 0 · d + r 0 with q 0 , r 0 Z and 0 r 0 < d . Then ( q 0 - q ) d = ( r - r 0 ) with - d < r - r 0 < d . The lhs is divisible by d so r = r 0 and we’re done. 4
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Primes If p N , p > 1 is prime if its only positive factors are 1 and p . n N is composite if n > 1 and n is not prime. If n is composite then a | n for some a N with 1 < a < n Can assume that a n . * Proof: By contradiction: Suppose n = bc , b > n , c > n . But then bc > n , a contradiction. Primes: 2 , 3 , 5 , 7 , 11 , 13 , . . .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

280wk4_x4 - Questions/Complaints About Homework? Heres the...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online