Ticket #3634: 3634-gf2e-minpoly.patch

a/sage/libs/ntl/decl.pxi

@@ -828 +828 @@
-
-    GF2X_c GF2XModulus_GF2X "GF2X" (GF2XModulus_c m)
-
+    GF2X_c GF2X_IrredPolyMod "IrredPolyMod" (GF2X_c g, GF2XModulus_c F)
+
-    #### GF2EContext_c
-
-    ctypedef struct GF2EContext_c "struct GF2EContext":

a/sage/rings/finite_field_ntl_gf2e.pyx

@@ -31 +31 @@
-include "../ext/stdsage.pxi"
-
-from sage.structure.sage_object cimport SageObject
-
-
-from sage.structure.parent  cimport Parent
-from sage.structure.parent_base cimport ParentWithBase
@@ -67 +66 @@
-cdef object Polynomial
-cdef object FreeModuleElement
-cdef object GF
+cdef object GF2, GF2_0, GF2_1
-
-cdef void late_import():
-    """
@@ -84 +84 @@
-           MPolynomial, \
-           Polynomial, \
-           FreeModuleElement, \
-
+
+
-
-    if is_IntegerMod is not None:
-        return
@@ -121 +122 @@
-
-    import sage.rings.finite_field
-    GF = sage.rings.finite_field.FiniteField
+    GF2 = GF(2)
+    GF2_0 = GF2(0)
+    GF2_1 = GF2(1)
-
-cdef extern from "arpa/inet.h":
-    unsigned int htonl(unsigned int)
@@ -337 +341 @@
-            if e.parent() == self:
-                res.x = (<FiniteField_ntl_gf2eElement>e).x
-                return res
-(2) or e.parent() == GF(2)
+2 or e.parent() == GF2
-                GF2E_conv_long(res.x,int(e))
-                return res
-
@@ -466 +470 @@
-            sage: F.prime_subfield()
-            Finite Field of size 2
-        """
-(2)
-
+2
+
-    def fetch_int(FiniteField_ntl_gf2e self, number):
-        r"""
-        Given an integer $n$ return a finite field element in self
@@ -1108 +1112 @@
-        for i from 0 <= i <= GF2X_deg(r):
-            C.append(GF2_conv_to_long(GF2X_coeff(r,i)))
-        return self._parent.polynomial_ring(name)(C)
+
+    def minpoly(self, var='x'):
+        """
+        Return the minimal polynomial of self, which is
+        the smallest degree polynomial $f \in \F_{2}[x]$
+        such that $f(self) = 0$. 
+        
+        EXAMPLES:
+            sage: K.<a> = GF(2^100)
+            sage: f = a.minpoly(); f
+            x^100 + x^57 + x^56 + x^55 + x^52 + x^48 + x^47 + x^46 + x^45 + x^44 + x^43 + x^41 + x^37 + x^36 + x^35 + x^34 + x^31 + x^30 + x^27 + x^25 + x^24 + x^22 + x^20 + x^19 + x^16 + x^15 + x^11 + x^9 + x^8 + x^6 + x^5 + x^3 + 1
+            sage: f(a)
+            0
+            sage: g = K.random_element()
+            sage: g.minpoly()(g)
+            0
+        """
+        cdef GF2X_c r = GF2X_IrredPolyMod(GF2E_rep(self.x), GF2E_modulus())
+        cdef int i
+        C = []
+        for i from 0 <= i <= GF2X_deg(r):
+            C.append(GF2_conv_to_long(GF2X_coeff(r,i)))
+        return self._parent.polynomial_ring(var)(C)
+
+    def trace(self):
+        """
+        Return the trace of self.
+
+        EXAMPLES:
+            sage: K.<a> = GF(2^25)
+            sage: a.trace()
+            0
+            sage: a.charpoly()
+            x^25 + x^8 + x^6 + x^2 + 1
+            sage: parent(a.trace())
+            Finite Field of size 2
+            
+            sage: b = a+1
+            sage: b.trace()
+            1
+            sage: b.charpoly()[1]
+            1
+        """
+        if GF2_IsOne(GF2E_trace(self.x)):
+            return GF2_1
+        else:
+            return GF2_0
+
+    def weight(self):
+        """
+        Returns the number of non-zero coefficients in the polynomial
+        representation of self.
+
+        EXAMPLES:
+            sage: K.<a> = GF(2^21)
+            sage: a.weight()
+            1
+            sage: (a^5+a^2+1).weight()
+            3
+            sage: b = 1/(a+1); b
+            a^20 + a^19 + a^18 + a^17 + a^16 + a^15 + a^14 + a^13 + a^12 + a^11 + a^10 + a^9 + a^8 + a^7 + a^6 + a^4 + a^3 + a^2 
+            sage: b.weight()
+            18
+        """
+        return GF2X_weight(GF2E_rep(self.x))
-            
-    def _finite_field_ext_pari_element(FiniteField_ntl_gf2eElement self, k=None):
-        r"""

a/sage/rings/polynomial/polynomial_element.pyx

@@ -3909 +3909 @@
-        
-        return RR(sum([abs(i)**p for i in coeffs]))**(1/p)
-
+    def hamming_weight(self):
+        """
+        Returns the number of non-zero coefficients of self.
+
+        EXAMPLES:
+            sage: R.<x> = ZZ[]
+            sage: f = x^3 - x
+            sage: f.hamming_weight()
+            2
+            sage: R(0).hamming_weight()
+            0
+            sage: f = (x+1)^100
+            sage: f.hamming_weight()
+            101
+            sage: S = GF(5)['y']
+            sage: S(f).hamming_weight()
+            5
+            sage: cyclotomic_polynomial(105).hamming_weight()
+            33   
+        """
+        cdef long w = 0
+        for a in self.coeffs():
+            if a:
+                w += 1
+        return w
+
-# ----------------- inner functions -------------
-# Sagex can't handle function definitions inside other function
-