Cliff Bergman on Thu, 6 Apr 2000 11:47:27 -0500 |
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arithmetic over finite fields |
Hello. I am teaching a course in cryptography, and have the students using pari to do some of the computations. So far it has been working well, since we have been working mostly over Z_n. However, we are now on to finite fields, and I could use a few pointers about how best to use pari/gp in this context. A couple of questions: 1. (Well, this question is really about Z_n) In cryptography, one frequently has to compute a^k%n where a, k, n are large integers. I have been using the definition powermod(a,k,n)=lift(Mod(a,n)^k) This has worked nicely, but I wonder if there is a built-in function that would do the same thing. 2. I assume the only way to represent an element of a finite field is as a polmod. Correct? This means the coefficients have to be integermods. This is kind of a hassle to type all of the time. My approach has been to make the inderminate a "variablemod". Thus I define: y=Mod(1,p)*x [where p is a predefined prime] a=y^3+3*y^2+2 [to define a particular polynomial a over Z_p, for example] This isn't so bad, except that the output format for a is in terms of the variable x. Then the elements of a finite field are of the form Mod(a,b), where a and b are defined as above. Now the output format gets pretty hard to read, since it has nested 'Mods'. One can get a more manageable output by asking for lift(lift(%)). Is there any way to get this output automatically? Is there a better way to do the whole thing that I am missing? 3. I haven't quite gotten to elliptic curves (over a finite field) yet. I suppose questions similar to #2 will apply to those. I do realize that there are lots of built-in commands for elliptic curves. I suppose the only complication will be in representing the components as elements of the finite field. Thanks for any advice that anybody has to offer. cliff b.