/// <summary>
        /// Computes the inverse mod 2. 
        /// <para>The algorithm is described in <a href="http://www.securityinnovation.com/uploads/Crypto/NTRUTech014.pdf">
        /// Almost Inverses and Fast NTRU Key Generation</a>.</para>
        /// </summary>
        /// 
        /// <returns>Returns <c>null</c> if the polynomial is not invertible.</returns>
        private IntegerPolynomial InvertF2()
        {
            int N = Coeffs.Length;
            int k = 0;
            IntegerPolynomial b = new IntegerPolynomial(N + 1);
            b.Coeffs[0] = 1;

            IntegerPolynomial c = new IntegerPolynomial(N + 1);
            IntegerPolynomial f = new IntegerPolynomial(Coeffs.CopyOf(N + 1));
            f.Mod2();

            // set g(x) = x^N − 1
            IntegerPolynomial g = new IntegerPolynomial(N + 1);
            g.Coeffs[0] = 1;
            g.Coeffs[N] = 1;

            while (true)
            {
                while (f.Coeffs[0] == 0)
                {
                    for (int i = 1; i <= N; i++)
                    {
                        f.Coeffs[i - 1] = f.Coeffs[i];          // f(x) = f(x) / x
                        c.Coeffs[N + 1 - i] = c.Coeffs[N - i];  // c(x) = c(x) * x
                    }

                    f.Coeffs[N] = 0;
                    c.Coeffs[0] = 0;
                    k++;

                    if (f.EqualsZero())
                        return null;    // not invertible
                }

                if (f.EqualsOne())
                    break;

                if (f.Degree() < g.Degree())
                {
                    // exchange f and g
                    IntegerPolynomial temp = f;
                    f = g;
                    g = temp;
                    // exchange b and c
                    temp = b;
                    b = c;
                    c = temp;
                }

                f.Add(g);
                f.Mod2();
                b.Add(c);
                b.Mod2();
            }

            if (b.Coeffs[N] != 0)
                return null;

            // Fq(x) = x^(N-k) * b(x)
            IntegerPolynomial Fq = new IntegerPolynomial(N);
            int j = 0;
            k %= N;

            for (int i = N - 1; i >= 0; i--)
            {
                j = i - k;
                if (j < 0)
                    j += N;
                Fq.Coeffs[j] = b.Coeffs[i];
            }

            return Fq;
        }