20110417, 12:43  #1 
Apr 2011
111_{2} Posts 
modulo operation for polynomials?
I know , how to calculate " x mod p " for a given very large "x" value and some p value. Here , x and p are integers
But how can we calculate f(x) mod g(x) , for which f(x) has higher degree(or order) than g(x). And that degree turns out to be a very large number. Ex:[ (x+1)^1729 mod ((x^5)1) ] or [ (x+1)^1729 mod (1729,((x^5)1)) ] Is there any method or algorithm to calculate it at faster speeds 
20110417, 13:21  #2  
"Forget I exist"
Jul 2009
Dumbassville
10000011000000_{2} Posts 
Quote:


20110417, 14:13  #3 
Tribal Bullet
Oct 2004
2·29·61 Posts 
Polynomial division works just like integer division, except that each term is treated independently of all the others, so there are no carries from one term to the next. So modular exponentiation of polynomials can use the same algorithms as exponentiation of integers.
If the denominator polynomial is monic (i.e. leading coefficient is 1) then the remainder will always have degree less than the denominator. When an intermediate polynomial gets a coefficient of x^5, subtract that coefficient times the denominator polynomial. If the numerator has a leading coefficient x^k larger than x^5, pretend you're dividing integers and subtract off (coefficient of x^k) * x^(k5) * denominator(x) to kill off the x^k term, then repeat with lower order terms until the remainder is less than the denominator. 
20110418, 17:18  #4  
Nov 2003
7460_{10} Posts 
Quote:
arithmetic or algebra be required to read Knuth Vol II before posting. 

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