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function [x, sflag] = gflineq(a, b, p)
// This function finds a solution for linear equation Ax = b over a prime Galois field.
//
// Calling Sequence
// [X, SFLAG] = GFLINEQ(A, B)
// [X, SFLAG]= GFLINEQ(A, B, P)
//
// Description
// [X, SFLAG] = GFLINEQ(A, B) returns a particular solution (X) of AX=B in GF(2).
// If the equation has no solution, then X is empty and SFLAG = 0 else SFLAG = 1.
//
// [X, SFLAG]= GFLINEQ(A, B, P) returns a particular solution of the linear
// equation A X = B in GF(P) and SFLAG=1.
// If the equation has no solution, then X is empty and SFLAG = 0.
//
// Examples
// A=[1 0 1; 1 1 0; 1 1 1]
// p=3
// [x,vld] = gflineq(A,[1;0;1],p)
// disp(A,'A=')
// disp(x,'x=');
// if(vld)
// disp('Linear equation has solution x')
// else
// disp('Linear equation has no solution and x is empty')
// end
// disp( pmodulo(A*x,p),'B =')
// See also
// gfadd, gfconv, gfdiv, gfrank, gfroots
// Authors
// Pola Lakshmi Priyanka, IIT Bombay//
//*************************************************************************************************************************************//
// Check number of input arguments
[out_a,inp_a]=argn(0)
if inp_a >3 | out_a> 2 | inp_a <2 then
error('comm:gflineq: Invalid number of arguments')
end
// Error checking .
if inp_a < 3
p = 2;
elseif ( isempty(p) | length(p)~=1 | abs(p)~=p | ceil(p)~=p | length(factor(p))~=1 )
error('comm:gflineq:Input argument 3 must be a positive prime integer.');
end;
[row_a, col_a] = size(a);
[row_b, col_b] = size(b);
// Error checking - A & B.
if ( isempty(a) | ndims(a) > 2 )
error('comm:gflineq:Input argument 1 must be a two-dimensional matrix.');
end
if ( isempty(b) | ndims(b) > 2 | col_b > 1 )
error('comm:gflineq:Invalid dimensions of input argument 2 .');
end
if ( row_a ~= row_b )
error('comm:gflineq:Dimensions of A and B are not compatible');
end
if (( or( abs(a)~=a | floor(a)~=a | a>=p )) | ( or( abs(b)~=b | floor(b)~=b | b>=p )) )
error('comm:gflineq:Elements of input matrices should be integers between 0 and P-1.');
end
// Solution is found by using row reduction (Reducing it to echelon form)
ab = [a b]; // Composite matrix
[row_ab, col_ab] = size(ab);
row_i = 1;
col_i = 1;
row = [];
col = [];
while (row_i <= row_ab) & (col_i < col_ab)
// Search for a non zero element in current column
while (ab(row_i,col_i) == 0) & (col_i < col_ab)
idx = find( ab(row_i:row_ab, col_i) ~= 0 );
if isempty(idx)
col_i = col_i + 1; // No non zero element
else
// Swap the current row with a row containing a non zero element
// (preferably with the row with value 1).
idx = [ find(ab(row_i:row_ab, col_i) == 1) idx ];
idx = idx(1);
temp_row = ab(row_i,:)
ab(row_i,:) = ab(row_i+idx-1,:)
ab(row_i+idx-1,:) = temp_row
end
end
if ( ( ab(row_i,col_i) ~= 0 ) & ( col_i < col_ab ) )
// Set major element to 1.
if (ab(row_i,col_i) ~= 1)
ab(row_i,:) = pmodulo( field_inv( ab(row_i,col_i),p ) * ab(row_i,:), p );
end
// The current element is a major element.
row = [row row_i];
col = [col col_i];
// Find the other elements in the column that must be cleared,
idx = find(ab(:,col_i)~=0);
for i = idx
if i ~= row_i
ab(i,:) = pmodulo( ab(i,:) + ab(row_i,:) * (p - ab(i,col_i)), p );
end
end
col_i = col_i + 1;
end
row_i = row_i + 1;
end
if ( rank(ab) > rank( ab(:,1:col_a) ) )
disp('comm:gflineq:Solution does not exist');
x = [];
sflag = 0;
else
x = zeros(col_a, 1);
x(col,1) = ab(row,col_ab);
sflag = 1;
end
endfunction
function [x] = field_inv(a,n)
t = 0;
newt = 1;
r = n;
newr = a;
while newr ~= 0
quotient = floor(r / newr);
temp = t;
t = newt;
newt = temp -quotient*newt;
temp = r;
r = newr;
newr = temp - quotient*newr;
end
if r>1
[x c] = find( pmodulo( (1:(p-1)).' * (1:(p-1)) , p ) == 1 );
end
if t<0
t = t + n;
end
x = t;
endfunction
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