sagemathgh-40571: Add lattice utility functions
    
This PR adds two lattice utility functions, namely, `gaussian_heuristic`
and `hadamard_ratio`. These are useful when evaluating how good the
approx CVP/SVP solutions are to the exact CVP/SVP solutions.

### 📝 Checklist

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- [x] The title is concise and informative.
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preview.
    
URL: sagemath#40571
Reported by: Sahil Jain
Reviewer(s): Sahil Jain, user202729

Author: Release Manager

src/sage/modules/free_module_integer.py

@@ -37,6 +37,10 @@
 from sage.misc.cachefunc import cached_method
 from sage.modules.free_module import FreeModule_submodule_with_basis_pid, FreeModule_ambient_pid
 from sage.modules.free_module_element import vector
+from math import prod
+from sage.functions.all import gamma
+from sage.symbolic.constants import pi, e
+
 
 try:
     from sage.rings.number_field.number_field_element import OrderElement_absolute
@@ -892,3 +896,90 @@ def babai(self, *args, **kwargs):
         Alias for :meth:`approximate_closest_vector`.
         """
         return self.approximate_closest_vector(*args, **kwargs)
+
+    def hadamard_ratio(self, use_reduced_basis=True):
+        r"""
+        Computes the normalized Hadamard ratio of the given basis.
+
+        The normalized Hadamard ratio of the basis `B = \left{ v_1, v_2, \dots, v_n \right}` is defined as
+
+        .. MATH::
+
+            \mathcal{H}(B) = \left( \dfrac{\det L}{\|v_1\| \|v_2\| \cdots \|v_n\|} \right)^{\frac{1}{n}}
+
+        The closer this ratio is to 1, the more orthogonal the basis is.
+
+        INPUT:
+
+        - ``use_reduced_basis`` -- boolean (default: ``True``); uses reduced basis for computing the ratio
+
+        OUTPUT: the ratio described above.
+
+        EXAMPLES::
+
+            sage: from sage.modules.free_module_integer import IntegerLattice
+            sage: L = IntegerLattice([[101, 0, 0, 0], [0, 101, 0, 0], [0, 0, 101, 0], [-28, 39, 45, 1]], lll_reduce=False)
+            sage: L.hadamard_ratio()
+            4331^(1/8)*(1/4331)^(1/4)
+            sage: float(L.hadamard_ratio()) # rel tol 1e-13
+            0.351096481348176
+            sage: L.LLL()
+            [  1  -5   2  18]
+            [ -5  25 -10  11]
+            [-17 -16 -34  -3]
+            [-39  -7  23   5]
+            sage: float(L.hadamard_ratio()) # rel tol 1e-13
+            0.9933322263147489
+        """
+        if use_reduced_basis:
+            basis = self.reduced_basis
+        else:
+            basis = self.basis_matrix()
+
+        n = basis.nrows()
+        r = self.rank()
+        assert r == n
+
+        ratio = (self.discriminant().sqrt() / prod([v.norm() for v in basis]))**(1/r)
+        assert 0 < ratio <= 1
+        return ratio
+
+    def gaussian_heuristic(self, exact_form=False):
+        r"""
+        Computes the Gaussian expected shortest length, also known as the Gaussian
+        heuristic. This estimates the expected norm of the shortest non-zero vector
+        in the lattice. The heuristic is independent of the chosen basis.
+
+        INPUT:
+
+        - ``exact_form`` -- boolean (default: ``False``); uses exact formulation
+          based on gamma function, instead of using stirling's approximation
+
+        OUTPUT: The Gaussian heuristic described above.
+
+        EXAMPLES::
+
+            sage: from sage.modules.free_module_integer import IntegerLattice
+            sage: L = IntegerLattice([[101, 0, 0, 0], [0, 101, 0, 0], [0, 0, 101, 0], [-28, 39, 45, 1]])
+            sage: L.gaussian_heuristic()
+            1030301^(1/4)*sqrt(2)*e^(-1/2)/sqrt(pi)
+            sage: float(L.gaussian_heuristic()) # rel tol 1e-13
+            15.418206247181422
+
+        For small `n`, the exact and approximate forms differ significantly::
+
+            sage: float(L.gaussian_heuristic(exact_form=True)) # rel tol 1e-13
+            21.375859827168494
+        """
+        basis = self.basis_matrix()
+
+        n = basis.nrows()
+        r = self.rank()
+        assert r == n
+
+        D = self.discriminant().sqrt()
+
+        if exact_form:
+            return (D * gamma(1 + (r/2)))**(1/r) / pi.sqrt()
+        else:
+            return D**(1/r) * (r/(2*pi*e)).sqrt()