Hardness of approximating the Minimum Solutions of Linear Diophantine Equations

Let 1≤p<∞ be any fixed real. We show that assuming P≠NP, it is hard to approximate the Minimum Solutions of Linear Diophantine Equations in ℓp norm within any constant factor and it is also hard to approximate the Minimum Solutions of Linear Diophantine Equations in ℓp norm within the factor nc/loglogn for some constant c>0 where n is the number of variables in the equations.