Lecture: Modern Cryptography and Number Theory

Background Material

• Proof techniques, Induction and Recursion

Reading

• Whitfield Diffie and Martin E. Hellman, "New Directions in Cryptography", IEEE Transactions on Information Theory, Vol. IT-22, No. 6, Nov. 1976.
• Chapter 1 sections 1.1-1.2 of the text.

Topics

• Theorem: p|ab => p|a or p|b, p a prime number
• Proof that sqrt(2) is irrational.
• Unique Factorization (the fundamental theorem of arithmetic) - the existence of the factorization of an integer into primes follows from the correctness proof of a simple algorithm to compute the prime factorization of an integer, and the uniqueness of the factorization follows from the previous theorem - see ufd.mw for worksheet containing the algorithm.
• Example where unique factorization does not hold: Z[sqrt(10)].

Lecture Slides

• (NewDirections.ppt,NewDirections.pdf) - Slides on New Directions in Cryptography

Maple worksheets and documentation

• ufd.mw - Maple worksheet illustrating unique factorization and the Euclidean algorithm.

Resources

• www.maplesoft.com (Maplesoft Web site)
• www.mapleapps.com (Maple Application Center Web site)
• www.maple4students.com (Maple Student Center Web site)
  

Assignments

• Review induction. Use induction to prove that sum(x^i,i=0..n) = (x^(n+1)-1)/(x-1)
• Load and run the examples from the worksheet from this lecture.
• Create a simple Maple worksheet and perform several computations
  1. Use evalf to approximate sqrt(3) to 100 decimal places.
  2. Use the plot command to plot x^2-3 over the interval [0,2]
  3. Use the solve command to solve the equation x^2-3=0.
  4. Use the fsolve command to solve the equation x^2-3=0.
  5. Use Maple to determine the number of primes less than 100.
  6. How many primes are there less than 1000?
• Create a title and some text explaining your computations.
• Add a section to your worksheet with the title "Bisection"