Computer Lab 2
Friday, September 30

Reminders

• The first lab is here. It has some information about how to use the computers and Maple.
• You can download crypto.mws, or find it in the directory My computer &rarr Math on Pop G: &rarr m360. In either case, you'll need to copy it to My computer &rarr local disk c &rarr temp.

Warming up

Handing in your work

It's sometimes nice to be able to include prose with your Maple worksheet. If you either hit CTRL-T, or go to Insert &rarr Text, you'll be able to type text, as opposed to Maple commands.

To resume your interaction with Maple, either hit CTRL-M, or go to Insert &rarr Maple input.

Problems

1. Do Problem 6.9.1.
2. In this problem, we'll work with the numbers
   • e₁ = 3301078408713960082274773523312923571950795102213
   • e₂ = 4044075324761191450682605260790697460884437355197
   • N = 2981087123707774106343687168904733579319894764661937
   • c₁=1156578537257780771382742263720412405961756805641460
   • c₂=1227132066583063095718313311605287814919255534195858
   To get these in to Maple, it's probably easiest to copy and paste the following fragment:

      N := 2981087123707774106343687168904733579319894764661937;
      e1 := 3301078408713960082274773523312923571950795102213;
      e2 := 4044075324761191450682605260790697460884437355197;
      c1 := 1156578537257780771382742263720412405961756805641460;
      c2 := 1227132066583063095718313311605287814919255534195858;
   

   Essentially, N is too big to factor; you're welcome to try to do so, but you'll want to locate the stop button...
   • Find numbers a and b so that a e₁ + b e₂ = 1.
   • A secret message M is encrypted using RSA with modulus N and exponent e₁. The result is c₁.

     The same message M is encrypted again using RSA with modulus N but, this time, exponent e₂. The result is c₂.

     What is M?

     If you like, here's a hint on how to approach this.

3. In this problem, we'll write a short program and use it to form a conjecture.

   For a natural number N let &tau(N) denote the number of divisors of N.

   • Write a program to compute the function &tau(N).

     Before you start, you may want to type

     unprotect('tau');
     

     in order to remove any other definition of tau.

     Programming this sort of thing in Maple isn't too bad. For instance, the following is a (very inefficient) way to compute the sum of all odd numbers between 1 and N:

     sillyoddsum := proc(N)
       local i, total;
       total := 0;
       for i from 1 to N do
         if ( (i mod 2) = 1) then
           total := total +i;
         end;
       end;
       return( total );
     end;
     

   • Compute &tau(N) for all N between 1 and 30.
   • Formulate a conjecture about &tau(p^e) for prime numbers p.
   • Formulate a conjecture relating &tau(mn) to &tau(m) and &tau(n)