# Reduction of the coefficients of a polynomial using the LLL algorithm

Given the two polynomials in two variables x and y

```
A(y,x)=((a1)*y+(a2))*((a3)*x+(a4))
B(y,x)=((b4)-(b3)*x)*((b2)-(b1)*y)
```

both congruent to zero mod a semiprimal number N

Using the LLL algorithm I should find m and n such that:

```
m*(a1)*(a3)+n*(b1)*(b3) = N*t +T
m*(a1)*(a4)-n*(b1)*(b4) = N*s + S
m*(a2)*(a3)-n*(b2)*(b3) = N*w + W
64 < T <= 64 *j
0 < S <= sqrt(N)
0 < W <= sqrt(N)
```

Where j is an integer greater than 1

Is anyone kind enough to show me the implementation in sagemath?