# Most efficient way to solve a system of 36 polynomial equations of degree 2 and 36 variables over integers

I need to solve a system of 36 polynomial equations of degree 2 with integer coefficients in 36 variables over integers. (I do not need rational, real or complex solutions)

I know that the system has at least one solution in integers.

I can use different methods such as eliminating variables one by one, using resultants or using Gröbner basis. None of these methods seems to be efficient for 36 variables and 36 equations.

Do you know a better method? How long should it take to solve it? Can I take as an advantage of knowing that one integer solution exist somehow?

• It is a special case of the system of polynomial equations? – kotomord Jan 31 '17 at 21:01
• It is a general type of system of polynomial equations. What is "special" about it is only that it is of degree 2 and that it has an integer solution. – azerbajdzan Jan 31 '17 at 21:06
• There are several other methods, using number theory, but unless there is something special about the equations, it is hopeless. Having an integer solution, or degree $2$ is not special. It is just a system of Diophantine equations. – Dietrich Burde Jan 31 '17 at 21:30
• For 3 equation and 3 variables it could be for example:$$\left\{3 x^2+4 y^2+5 z^2+2 x y-x z-2 y z+11 x+12 y+7 z-109=0,2 x^2-5 y^2+4 z^2+6 x y+2 x z+8 y z+9 x+4 y-z-98=0,3 x^2+8 y^2+7 z^2+4 x y-4 x z+2 y z+5 x+6 y+2 z-129=0\right\}$$ which has solution x=1,y=2,z=3 – azerbajdzan Jan 31 '17 at 21:38
• If you wonder why modulus 13... it is because more of the character of the cipher than for some mathematical reason. – azerbajdzan Jul 24 '17 at 6:51