Polynomial factorization modulo p

This calculator finds irreducible factors of a given polynomial modulo p using the Elwyn Berlekamp factorization algorithm. The algorithm description is just below the calculator.

Berlekamp polynomial factorization

Integer polynomial coefficients

Modulus

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Input polynomial

 

Solution

 

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Berlekamp factorization algorithm

The algorithm described here is a compact compilation of the factoring algorithm described in TAOCP vol 2 by D.Knuth ¹.

Initial data

• u(x) - n-degree polynomial, n>=2
• p - prime number modulus

Preparation steps

• Check the polynomial is monic. If not, then divide all the polynomial coefficients by the highest-degree coefficient u_n
• Check the polynomial is square-free using Square free polynomial factoring in finite field
• For each square-free polynomial factor of degree 2 or higher, run the algorithm below

The algorithm

• Find Q matrix (n * n ), where n is a polynomial degree by the procedure below:
  • Initialize a vector A (a₀, a₁ ... a_n-1) = 1,0...0
  • Initialize the first row of the Q matirx (q_0,0, q_0,1 ... q_0,n-1) with 0,0...0
  • Loop on i = 1..n-1 do the following
    • Loop on k = 1..n-1 do the following
      • Set t = a_n-1
      • Loop on j = n-1 .. 0 do the following
        • a_j=a_j-1-t*u_j, assume a_-1 = 0
    • Set i row of matrix Q to A vector
    • Subtract 1 from q_i,i element of matrix Q
• Find v^[1] ... v^[r] linearly independent vectors, such as v^[1] Q = v^[2] Q = ... v^[r] Q = (0,0...0)
  • Set all elements of n-element vector C to -1 : c₀ = c₁ = .. = c_{n-1 = -1}
  • Set r = 0
  • Loop on k = 0 ... n-1 do the following
    • Loop on j = 0 ... n-1 do the following
      • if q_k,j ≠ 0 and c_j<0
        • Set a = q_k,j
        • Multiply column j of matrix Q by -1/a
        • Add to each other columns (i ≠ j) column j times q_k,i
      • else (if q_k,j=0 or c_j equal to 0 or greater than 0)
        • Set r = r + 1
        • Set every i element of a new n-element vector v^[r] to one of the following three:
          • a_k,s, if found s-element of C vector, such as c_s = i
          • 1, if i = k
          • 0 - otherwise
• Find r factors of u(x) polynomial using vectors v^[2] ... v^[r]
  • Find all w_i = gcd(u(x),v^[2]-s) ≠ 1 for each s = 0 ... p
  • if the count of w < r do the following
  • Loop on j=3 ... r until the count of w < r
    • Replace w_i with factors found by gcd(v^[j]-s,w_i) ≠ 1 for each s = 0 ... p