diff --git a/src/sage/libs/ntl/ntl_ZZ_p.pyx b/src/sage/libs/ntl/ntl_ZZ_p.pyx
index 82ab0ff1fd1..d97ca5734f8 100644
--- a/src/sage/libs/ntl/ntl_ZZ_p.pyx
+++ b/src/sage/libs/ntl/ntl_ZZ_p.pyx
@@ -414,6 +414,34 @@ cdef class ntl_ZZ_p:
         ZZ_p_modulus( &r.x, &self.x )
         return r
 
+    def lift_centered(self):
+        """
+        Compute a representative of ``self`` in `(-n/2 , n/2]` as an
+        ``ntl.ZZ object``.
+
+        OUTPUT:
+
+        - An ``ntl.ZZ`` object `r` such that  `-n/2 < r <= n/2` and `Mod(r, n) == self`.
+
+        EXAMPLES::
+
+            sage: x = ntl.ZZ_p(8, 18)
+            sage: x.lift_centered()
+            8
+            sage: type(x.lift())
+            <type 'sage.libs.ntl.ntl_ZZ.ntl_ZZ'>
+            sage: x = ntl.ZZ_p(12, 18)
+            sage: x.lift_centered()
+            -6
+            sage: type(x.lift())
+            <type 'sage.libs.ntl.ntl_ZZ.ntl_ZZ'>
+        """
+        cdef ntl_ZZ r = self.lift()
+        cdef ntl_ZZ m = self.modulus()
+        if r*2 > m:
+            r -= m
+        return r
+
     def _integer_(self, ZZ=None):
         """
         Return a lift of self as a Sage integer.
diff --git a/src/sage/matrix/matrix2.pyx b/src/sage/matrix/matrix2.pyx
index 984c1896d8c..6148b2fab16 100644
--- a/src/sage/matrix/matrix2.pyx
+++ b/src/sage/matrix/matrix2.pyx
@@ -1174,7 +1174,7 @@ cdef class Matrix(matrix1.Matrix):
                 d = R(self._pari_().matdet())
             else:
                 # Lift to ZZ and compute there.
-                d = R(self.apply_map(lambda x : x.centerlift()).det())
+                d = R(self.apply_map(lambda x : x.lift_centered()).det())
             self.cache('det', d)
             return d
 
diff --git a/src/sage/rings/arith.py b/src/sage/rings/arith.py
index 63a6b34ecfe..50964562af9 100644
--- a/src/sage/rings/arith.py
+++ b/src/sage/rings/arith.py
@@ -2184,6 +2184,38 @@ def rational_reconstruction(a, m, algorithm='fast'):
     else:
         raise ValueError("unknown algorithm")
 
+def lift_centered(p):
+    r"""
+    Compute a representative of the element `p` mod `n` in `(-n/2 , n/2]`
+
+    INPUT:
+
+    - ``p`` -- an integer mod `n`
+
+    OUTPUT:
+
+    - An integer `r` in `\ZZ` such that  `-n/2 < r \leq n/2`
+
+    For specific classes, check the `p.lift_centered` attribute.
+
+    EXAMPLES::
+
+        sage: p = Mod(2,4)
+        sage: lift_centered(p)
+        2
+        sage: p = Mod(3,4)
+        sage: lift_centered(p)
+        -1
+    """
+    try:
+        return ZZ(p.lift_centered())
+    except(AttributeError):
+        pass
+    r = p.lift()
+    if r*2 > p.modulus():
+        r -= p.modulus()
+    return ZZ(r)
+
 def _rational_reconstruction_python(a,m):
     """
     Internal fallback function for rational_reconstruction; see
diff --git a/src/sage/rings/finite_rings/integer_mod.pyx b/src/sage/rings/finite_rings/integer_mod.pyx
index ebc2af42c76..27750477b31 100644
--- a/src/sage/rings/finite_rings/integer_mod.pyx
+++ b/src/sage/rings/finite_rings/integer_mod.pyx
@@ -723,28 +723,28 @@ cdef class IntegerMod_abstract(FiniteRingElement):
         """
         return self
 
-    def centerlift(self):
+    def lift_centered(self):
         r"""
         Lift ``self`` to an integer `i` such that `n/2 < i <= n/2`
         (where `n` denotes the modulus).
 
         EXAMPLES::
 
-            sage: Mod(0,5).centerlift()
+            sage: Mod(0,5).lift_centered()
             0
-            sage: Mod(1,5).centerlift()
+            sage: Mod(1,5).lift_centered()
             1
-            sage: Mod(2,5).centerlift()
+            sage: Mod(2,5).lift_centered()
             2
-            sage: Mod(3,5).centerlift()
+            sage: Mod(3,5).lift_centered()
             -2
-            sage: Mod(4,5).centerlift()
+            sage: Mod(4,5).lift_centered()
             -1
-            sage: Mod(50,100).centerlift()
+            sage: Mod(50,100).lift_centered()
             50
-            sage: Mod(51,100).centerlift()
+            sage: Mod(51,100).lift_centered()
             -49
-            sage: Mod(-1,3^100).centerlift()
+            sage: Mod(-1,3^100).lift_centered()
             -1
         """
         n = self.modulus()