Galois Field multiplier

[max420]

Is it possible to implement a 5 bit Galois Field multiplier?I found the equations for 8bit in the net which is given below:

module gf256mult(a, b, z);
input [7:0] a;
input [7:0] b;
output [7:0] z;
assign z[0] = b[0]&a[0]^b[1]&a[7]^b[2]&a[6]^b[3]&a[5]^b[4]&a[4]^b[5]&a[3]^b[5]&a[7]^b[6]&a[2]^b[6]&a[6]^b[6]&a[7]^b[7]&a[1]^b[7]&a[5]^b[7]&a[6]^b[7]&a[7];
assign z[1] = b[0]&a[1]^b[1]&a[0]^b[2]&a[7]^b[3]&a[6]^b[4]&a[5]^b[5]&a[4]^b[6]&a[3]^b[6]&a[7]^b[7]&a[2]^b[7]&a[6]^b[7]&a[7];
assign z[2] = b[0]&a[2]^b[1]&a[1]^b[1]&a[7]^b[2]&a[0]^b[2]&a[6]^b[3]&a[5]^b[3]&a[7]^b[4]&a[4]^b[4]&a[6]^b[5]&a[3]^b[5]&a[5]^b[5]&a[7]^b[6]&a[2]^b[6]&a[4]^b[6]&a[6]^b[6]&a[7]^b[7]&a[1]^b[7]&a[3]^b[7]&a[5]^b[7]&a[6];
assign z[3] = b[0]&a[3]^b[1]&a[2]^b[1]&a[7]^b[2]&a[1]^b[2]&a[6]^b[2]&a[7]^b[3]&a[0]^b[3]&a[5]^b[3]&a[6]^b[4]&a[4]^b[4]&a[5]^b[4]&a[7]^b[5]&a[3]^b[5]&a[4]^b[5]&a[6]^b[5]&a[7]^b[6]&a[2]^b[6]&a[3]^b[6]&a[5]^b[6]&a[6]^b[7]&a[1]^b[7]&a[2]^b[7]&a[4]^b[7]&a[5];
assign z[4] = b[0]&a[4]^b[1]&a[3]^b[1]&a[7]^b[2]&a[2]^b[2]&a[6]^b[2]&a[7]^b[3]&a[1]^b[3]&a[5]^b[3]&a[6]^b[3]&a[7]^b[4]&a[0]^b[4]&a[4]^b[4]&a[5]^b[4]&a[6]^b[5]&a[3]^b[5]&a[4]^b[5]&a[5]^b[6]&a[2]^b[6]&a[3]^b[6]&a[4]^b[7]&a[1]^b[7]&a[2]^b[7]&a[3]^b[7]&a[7];
assign z[5] = b[0]&a[5]^b[1]&a[4]^b[2]&a[3]^b[2]&a[7]^b[3]&a[2]^b[3]&a[6]^b[3]&a[7]^b[4]&a[1]^b[4]&a[5]^b[4]&a[6]^b[4]&a[7]^b[5]&a[0]^b[5]&a[4]^b[5]&a[5]^b[5]&a[6]^b[6]&a[3]^b[6]&a[4]^b[6]&a[5]^b[7]&a[2]^b[7]&a[3]^b[7]&a[4];
assign z[6] = b[0]&a[6]^b[1]&a[5]^b[2]&a[4]^b[3]&a[3]^b[3]&a[7]^b[4]&a[2]^b[4]&a[6]^b[4]&a[7]^b[5]&a[1]^b[5]&a[5]^b[5]&a[6]^b[5]&a[7]^b[6]&a[0]^b[6]&a[4]^b[6]&a[5]^b[6]&a[6]^b[7]&a[3]^b[7]&a[4]^b[7]&a[5];
assign z[7] = b[0]&a[7]^b[1]&a[6]^b[2]&a[5]^b[3]&a[4]^b[4]&a[3]^b[4]&a[7]^b[5]&a[2]^b[5]&a[6]^b[5]&a[7]^b[6]&a[1]^b[6]&a[5]^b[6]&a[6]^b[6]&a[7]^b[7]&a[0]^b[7]&a[4]^b[7]&a[5]^b[7]&a[6];
endmodule



can anyone say me the equations for 5 bit?thank you.

 

[permute]

It clearly depends on the choice of connection polynomial.

For reference, a Galois Field is formed by creating a "primitive" element. The actual value of which is essentially unimportant -- it acts like the imaginary number does. It is the root of an equation that has neither 1, nor 0 as a root (an irreducible polynomial). In this case, a rule can be written similar to the rule for i^2 = -1. For your case, one possible case would be a^5 = a^4 + a^2 + 1. again, it is very important not to become overly concerned with the "value" of "a". (I might have the connection polynomial incorrect above, I haven't check it rigorously)

now, all GF32 elements can be written as b4*a^4 + b4*a^4 + ... b0*a^0. again, the value of "a" is unimportant, what is important is the rule that a^5 = a^4 + a^2 + 1.

From here, it is a task of multiplication and reduction. a second GF32 element might have an equation c4*a^4 + ... + c0*a^0.

the product of the two elements is c4*b4*a^8 + (c3*b4 + c4*b3)*a^7 + ... a0*b0, where the terms for each power of "a" is the terms of c/b which add up to that power. (4,4; 4,3 3,4; 3,3 4,2 2,4; ...)

Now reduction is done. if a^5 = a^4 + a^2 + 1, then a^6 = a*a^5. likewise a^7 = a^2 * a^5, etc... using this reduction rule will allow the 8th order polynomial to be reduced back to the 4th order one desired.

other methods exist. the above is the easiest to explain.

 

[tariq786]

Hi,
can you please tell me the internet source of galois field equations?

Thanks

 

[permute]

as mentioned above, there is not a single galois field equation. not a single one per size of galois field. wikipedia has some. you can extract them from the taps given **broken link removed** for fibbonacci LFSRs as well.