Multiplication inverse element in a finite field

It is widely used in cryptography and security fields. Including the well-known Euclidean Algorithm for the most common factor.

This article describes how to calculate the inverse element of multiplication in a finite field. Including integers and polynomials. The extended Euclid algorithm is used. There is a great Gartner proposal.

1. Multiplication against RMB W': Any w belongs to ZP, W! = 0. The existence of Z is ZP so that w * z = 1 (mod P );

Example 5: Calculate the multiplication inverse element of MOD 14. 5*3 = 1 (mod 14); so the modulo 14 multiplication inverse element of 5 is 3;

2. Multiplication of integers inverse element

Extended Euclid algorithm flow:

1. (A1, A2, A3) = (1, 0, m); (B1, B2, B3) = (0, 1, B)

2. If B3 = 0 return no inverse;

3. If B3 = 1 return B2;

4. q = a2/B3;

5. (T1, T2, T3) = (A1-Q * B1, A2-Q * B2, A3-Q * B3 );

6. (A1, A2, A3) = (B1, B2, B3)

7. (B1, B2, B3) = (T1, T2, T3)

8. goto2

Code:

Int gcd (int A, int B) {if (a <B) {A ^ = B; B ^ = A; A ^ = B;} while (B! = 0) {int temp = A; A = B; B = TEMP % B;} return a;} // extend Euclidean to evaluate B in GF (m) only the mutual quality of B and M has the inverse element. Int Euclid (int m, int B) {int A [4]; A [1] = 1; A [2] = 0; A [3] = m; int B [4]; B [1] = 0; B [2] = 1; B [3] = B; while (true) {If (0 = B [3]) {cout <"no reverse element" <Endl; return gcd (M, B );} else if (1 = B [3]) {cout <"reverse meta:" <B [2] <Endl; return B [2];} int quotient = A [3]/B [3]; int T [4]; t [1] = A [1]-Quotient * B [1]; T [2] = A [2]-Quotient * B [2]; t [3] = A [3]-Quotient * B [3]; A [1] = B [1]; A [2] = B [2]; A [3] = B [3]; B [1] = T [1]; B [2] = T [2]; B [3] = T [3];} return 0;} int main () {Euclid (14, 5 ); euclid (8, 4 );}

3. polynomial multiplication inverse element

The inverse principle of Polynomial multiplication is the same as that of integer multiplication. It is only a polynomial.

Method 1: polynomial addition, subtraction, multiplication, division

Method 2: Use an integer (Binary bit) to simulate the addition, subtraction, multiplication, and division of polynomials. This method is also the simplest and most appropriate method.

For example, x ^ 8 + x ^ 4 + x ^ 3 + x + 1 is described in binary format. 100011011 can be stored as an integer of 283.

Let's take a closer look. For the multiplication of two polynomials, the polynomial addition is not directly to this integer or multiplication or addition. Division is a little more complex. The first time we found that we could use Integers to simulate polynomial calculation. Very knowledgeable! .

Reference: http://blog.csdn.net/wjh200821/article/details/7570573