gh-38359: Homogenise .log() api across implementations of finite fi…

…eld elements

    
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Fixes #38350 by adding `order=` and `check=` arguments to the `.log()`
method of finite field elements, under all implementations.
- `element_pari_ffelt.pyx`: done by @yyyyx4 in #37329.
- `element_ntl_gf2e.pyx`: if provided, does not compute `base_order`.
- `element_givaro.pyx`: if provided, passes `order` to the underlying
`discrete_log` call.
- `integer_mod.pyx`: the argument is discarded (unless `check=True`, in
which case the order is checked).

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- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

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URL: #38359
Reported by: Justin Carel
Reviewer(s): Lorenz Panny

src/sage/rings/finite_rings/element_givaro.pyx

@@ -1411,11 +1411,18 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
         """
         return Integer(self._cache.log_to_int(self.element))
 
-    def log(FiniteField_givaroElement self, base):
+    def log(FiniteField_givaroElement self, base, order=None, *, check=False):
         """
         Return the log to the base `b` of ``self``, i.e., an integer `n`
         such that `b^n =` ``self``.
 
+        INPUT:
+
+        - ``base`` -- non-zero field element
+        - ``order`` -- integer (optional), multiple of order of ``base``
+        - ``check`` -- boolean (default: ``False``): If set,
+          test whether the given ``order`` is correct.
+
         .. WARNING::
 
             TODO -- This is currently implemented by solving the discrete
@@ -1429,9 +1436,22 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
             sage: a = b^7
             sage: a.log(b)
             7
+
+        TESTS:
+
+        An example for ``check=True``::
+
+            sage: F.<t> = GF(3^5, impl='givaro')
+            sage: t.log(t, 3^4, check=True)
+            Traceback (most recent call last):
+            ...
+            ValueError: 81 is not a multiple of the order of the base
         """
         b = self.parent()(base)
-        return sage.groups.generic.discrete_log(self, b)
+        if (order is not None) and check and not (b**order).is_one():
+            raise ValueError(f"{order} is not a multiple of the order of the base")
+
+        return sage.groups.generic.discrete_log(self, b, ord=order)
 
     def _int_repr(FiniteField_givaroElement self):
         r"""

src/sage/rings/finite_rings/element_ntl_gf2e.pyx

@@ -1233,14 +1233,17 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
         """
         return unpickleFiniteField_ntl_gf2eElement, (self._parent, str(self))
 
-    def log(self, base):
+    def log(self, base, order=None, *, check=False):
         """
         Compute an integer `x` such that `b^x = a`, where `a` is ``self``
         and `b` is ``base``.
 
         INPUT:
 
         - ``base`` -- finite field element
+        - ``order`` -- integer (optional), the order of the base
+        - ``check`` -- boolean (default: ``False``): If set,
+          test whether the given ``order`` is correct.
 
         OUTPUT:
 
@@ -1278,16 +1281,34 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
             ...
             ValueError: no logarithm of z50 exists to base z50^49 + z50^46 + z50^45 + z50^44 + z50^41 + z50^34 + z50^33 + z50^32 + z50^27 + z50^25 + z50^24 + z50^21 + z50^18 + z50^17 + z50^16 + z50^15 + z50^12 + z50^11 + z50^10 + z50^8 + z50^7 + z50^3 + z50^2
 
+        An example for ``check=True``::
+
+            sage: F.<t> = GF(2^5, impl='ntl')
+            sage: t.log(t, 3^4, check=True)
+            Traceback (most recent call last):
+            ...
+            ValueError: base does not have the provided order
+
         AUTHORS:
 
         - David Joyner and William Stein (2005-11)
         - Lorenz Panny (2021-11): use PARI's :pari:`fflog` instead of :func:`sage.groups.generic.discrete_log`
         """
+        cdef Integer base_order
+
         base = self.parent()(base)
 
         # The result is undefined if the input to fflog() is invalid.
         # Since the unit group is cyclic, it suffices to check orders.
-        cdef Integer base_order = base.multiplicative_order()
+        if order is None:
+            base_order = base.multiplicative_order()
+        else:
+            if check:
+                from sage.groups.generic import has_order
+                if not has_order(base, order, '*'):
+                    raise ValueError('base does not have the provided order')
+            base_order = order
+
         cdef Integer self_order = self.multiplicative_order()
         if not self_order.divides(base_order):
             raise ValueError(f'no logarithm of {self} exists to base {base}')

src/sage/rings/finite_rings/integer_mod.pyx

@@ -638,7 +638,7 @@ cdef class IntegerMod_abstract(FiniteRingElement):
         else:
             return sib(self.parent())(v)
 
-    def log(self, b=None):
+    def log(self, b=None, order=None, check=False):
         r"""
         Compute the discrete logarithm of this element to base `b`,
         that is,
@@ -654,6 +654,13 @@ cdef class IntegerMod_abstract(FiniteRingElement):
            ``R.multiplicative_generator()`` is used, where
            ``R`` is the parent of ``self``.
 
+        -  ``order`` -- integer (unused), the order of ``b``.
+           This argument is normally unused, only there for
+           coherence of apis with finite field elements.
+
+        - ``check`` -- boolean (default: ``False``). If set,
+           test whether the given ``order`` is correct.
+
 
         OUTPUT:
 
@@ -760,6 +767,15 @@ cdef class IntegerMod_abstract(FiniteRingElement):
             sage: R(1).factor()
             1
 
+        An example for ``check=True``::
+
+            sage: F = GF(127, impl='modn')
+            sage: t = F.primitive_element()
+            sage: t.log(t, 57, check=True)
+            Traceback (most recent call last):
+            ...
+            ValueError: base does not have the provided order
+
         AUTHORS:
 
         - David Joyner and William Stein (2005-11)
@@ -781,6 +797,11 @@ cdef class IntegerMod_abstract(FiniteRingElement):
             if not b.is_unit():
                 raise ValueError(f"logarithm with base {b} is not defined since it is not a unit modulo {b.modulus()}")
 
+        if check:
+            from sage.groups.generic import has_order
+            if not has_order(b, order, '*'):
+                raise ValueError('base does not have the provided order')
+
         cdef Integer n = Integer()
         cdef Integer m = one_Z
         cdef Integer q, na, nb