Approximate closest vector solver. Despite running the whole of LLL it's way faster than the common Gram-Schmidt version since Sage rationals are so slow...

cvp.py

def cvp_coords(B, t, reduce=True):
    '''
    Returns both the (approximate) closest vector
    to t and its coordinates in the lattice B
    '''

    t = vector(ZZ, t)

    if reduce: B, R = B.LLL(transformation=True)
    else: R = identity_matrix(ZZ, B.nrows())

    # an LLL reduced basis is ordered 
    # by increasing norm
    S = B[-1].norm().round()+1

    L = block_matrix([
        [B,         0],
        [matrix(t), S]
    ])
    
    L, U = L.LLL(transformation=True)
    for u, v in zip(U, L):
        if abs(u[-1]) == 1:
            # *u[-1] cancels the sign to be positive
            # just in case
            return t - v[:-1]*u[-1], -u[:-1]*u[-1]*R
    raise ValueError("babai failed? plz msg @blupper on discord (unless you didn't reduce?)")

def cvp(B, t, reduce=True):
    return cvp_coords(B, t, reduce)[0]