Assignment 3

Integer Multiplication, Primality Testing, Interpolation

CS 300 Applied Symbolic Computation
Instructor: Jeremy Johnson 
Due Tuesday May 20 by class time.

Problems to hand in

1. [20 points] Count the number of multiplications used by the binary powering algorithm, Powerrec (see below), presented in Lecture 5.  Plot the number of multiplications for exponent 0..128.  Can you determine a formula for the number of multiplications?  
• > Powerrec := proc(M,e)  if e = 0 then return 1; fi;
  >   if e = 1 then return M; fi;
  >   return Powerrec(M,floor(e/2))^2*M^(e mod 2);
  > end;
  
  
2. [30 points]
   
• Using Maple, implement and experiment with the primality test described in the RSA paper. I.E. For an integer n, generate random integers between 1 and n-1 and check to see if (i) gcd(a,n) = 1, and if (ii) J(a,n) = a^((n-1)/2) mod n, where J(a,n) is the Jacobi symbol of a and n. If n is prime these two conditions will always be true. If n is composite, the condition will be false with atleast probability 1/2. Your procedure should allow the number of trials to be passed as a parameter. You may use Maple's jacobi function in the numtheory package. You may also want to use ifactor, and isprime for testing purposes.
• Implement a Maple procedure to compute compute the Jacobi symbol (as indicated on page 124 of the RSA paper. A discussion of the Jacobi symbol is provided in chapter 3 of the Niven and Zuckerman handout.
  
3. [30 points]
   
• Implement a Maple procedure that uses Karatsuba's algorithm to multiply integers. Time your version of Karatsuba's algorithm for a range of input sizes (i.e. number of digits) and empirically determine the asymptotic computing time rate.
• Reimplement Karatsuba's algorithm using the internal representation of integers. You must perform the extraction of the high and low order digits and multiplication by powers of ten in linear time. You can get the internal representation of a maple integer using the disassemble instruction. Similarly you can reassemble an integer in the internal format using Maple's assemble instruction. See internal.mws for an example of how to use these functions. You should also refer to the online documentation.
4. [20 points] Implement a Maple procedure that multiplies two polynomials with coefficients in Z_p using the evaluation, pointwise product, interpolation algorithm.