What is definition of truncated polynomial?

Question

In NTRUEncryption, I seen the trucated polynimials, but I cannot understand the trunacated polynomial calculation.
So, could tell me anyone How we calculate the truncated polynomial?

Answer 1

The polynomials are truncated in the sense that they only have coefficients up to a certain degree.

Here is how you truncate the product of two truncated polynomials (the sum is trivial):

Assume you have two truncated polynomials, i.e. two polynomials of degree no greater than n-1

a = a[0] + a[1]X + ... + a[n-1]X^(n-1)
b = b[0] + b[1]X + ... + b[n-1]X^(n-1)

Then their "truncated" product is defined as the polynomial

a * b = c[0] + c[1]X + ... +c[n-1]X^(n-1)

where the c[k] coefficients are computed as follow:

1. Reverse b[0]..b[n-1] to get b[n-1]..b[0].
2. Rotate the result of step 1 above k+1 times to the right and get b[k]..b[0]b[n-1]..b[k+1]
3. Denote with b_k[0]..b_k[n-1] the array calculated in 2.

Now define

c[k] = a[0]b_k[0] + a[1]b_k[1] + ... + a[n-1]b_k[n-1].

This operation can also be made by multiplying the polynomials a and b in the usual way and then truncating the result to the degree n-1. The reason for the algorithm above is to avoid computing coefficients that will not be used in the final result.