Merge branch 'u/arpitdm/skew_polynomials' of git://trac.sagemath.org/sage into 13215_skew_polynomials

Author: johanrosenkilde

src/doc/en/reference/polynomial_rings/index.rst

@@ -29,6 +29,16 @@ Multivariate Polynomials
    sage/rings/invariant_theory
    polynomial_rings_toy_implementations
 
+Skew Polynomials
+----------------
+
+.. toctree::
+   :maxdepth: 2
+
+   sage/rings/polynomial/skew_polynomial_element
+   sage/rings/polynomial/skew_polynomial_ring_constructor
+   sage/rings/polynomial/skew_polynomial_ring
+
 Rational Functions
 ------------------
 

src/module_list.py

@@ -1799,6 +1799,9 @@ def uname_specific(name, value, alternative):
 
     Extension('*', ['sage/symbolic/*.pyx']),
 
+    Extension('sage.rings.polynomial.skew_polynomial_element',
+              sources = ['sage/rings/polynomial/skew_polynomial_element.pyx']),
+
     ################################
     ##
     ## sage.tests

src/sage/categories/map.pyx

@@ -1411,6 +1411,23 @@ cdef class Section(Map):
         """
         return "Section"
 
+    def inverse(self):
+        """
+        Return inverse of ``self``.
+
+        TEST::
+
+            sage: from sage.categories.map import Section
+            sage: R.<x,y> = QQ[]
+            sage: f = R.hom([x+y, x-y], R)
+            sage: sf = Section(f)
+            sage: sf.inverse()
+            Ring endomorphism of Multivariate Polynomial Ring in x, y over Rational Field
+              Defn: x |--> x + y
+                    y |--> x - y
+        """
+        return self._inverse
+
 cdef class FormalCompositeMap(Map):
     """
     Formal composite maps.

src/sage/categories/rings.py

@@ -737,6 +737,13 @@ def __getitem__(self, arg):
                 sage: GF(17)['a']['b']
                 Univariate Polynomial Ring in b over Univariate Polynomial Ring in a over Finite Field of size 17
 
+            We can create skew polynomial rings::
+
+                sage: k.<t> = GF(5^3)
+                sage: Frob = k.frobenius_endomorphism()
+                sage: k['x',Frob]
+                Skew Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5
+
             We can also create power series rings by using double brackets::
 
                 sage: QQ[['t']]
@@ -849,6 +856,12 @@ def normalize_arg(arg):
                 from sage.rings.power_series_ring import PowerSeriesRing
                 return PowerSeriesRing(self, elts)
 
+            if isinstance(arg, tuple):
+                from sage.categories.morphism import Morphism
+                if len(arg) == 2 and isinstance(arg[1], Morphism):
+                   from sage.rings.polynomial.skew_polynomial_ring_constructor import SkewPolynomialRing
+                   return SkewPolynomialRing(self, arg[1], names=arg[0])
+
             # 2. Otherwise, if all specified elements are algebraic, try to
             #    return an algebraic extension
 

src/sage/rings/polynomial/all.py

@@ -44,5 +44,8 @@
 # Infinite Polynomial Rings
 from sage.rings.polynomial.infinite_polynomial_ring import InfinitePolynomialRing
 
+# Skew Polynomial Rings
+from sage.rings.polynomial.skew_polynomial_ring_constructor import SkewPolynomialRing
+
 # Evaluation of cyclotomic polynomials
 from sage.rings.polynomial.cyclotomic import cyclotomic_value

src/sage/rings/polynomial/multi_polynomial_sequence.py

@@ -150,6 +150,10 @@
    in Topics in Cryptology -- CT RSA'06; LNCS 3860; pp. 313--331; Springer Verlag 2006;
    pre-print available at http://eprint.iacr.org/2005/200
 
+.. [CMR05] C. Cid, S. Murphy, M. Robshaw Small Scale Variants of the AES;
+   in Proceedings of Fast Software Encryption 2005; LNCS 3557;
+   Springer Verlag 2005; available at http://www.isg.rhul.ac.uk/~sean/smallAES-fse05.pdf
+
 Classes
 -------
 """

src/sage/rings/polynomial/skew_polynomial_element.pxd

@@ -0,0 +1,39 @@
+include "cysignals/signals.pxi"
+include "../../ext/cdefs.pxi"
+include '../../ext/stdsage.pxi'
+
+from sage.rings.integer cimport Integer
+
+from sage.structure.element cimport Element, AlgebraElement, ModuleElement
+from sage.structure.parent cimport Parent
+from polynomial_compiled cimport CompiledPolynomialFunction
+
+from sage.rings.morphism cimport RingHomomorphism
+from sage.structure.element cimport RingElement
+
+from sage.rings.polynomial.polynomial_element cimport Polynomial_generic_dense
+
+cdef class SkewPolynomial(AlgebraElement):
+    cdef _is_gen
+
+    cdef long _hash_c(self)
+    cdef SkewPolynomial _new_c(self,list coeffs,Parent P,char check=*)
+    cpdef SkewPolynomial _new_constant_poly(self,RingElement a,Parent P,char check=*)
+    cpdef _neg_(self)
+    cpdef list list(self)
+
+    cdef list _coeffs
+    cdef void __normalize(self)
+    cdef void _inplace_add(self, SkewPolynomial_generic_dense right)
+    cdef void _inplace_sub(self, SkewPolynomial_generic_dense right)
+    cdef void _inplace_rmul(self, SkewPolynomial_generic_dense right)
+    cdef void _inplace_lmul(self, SkewPolynomial_generic_dense right)
+    cdef void _inplace_pow(self, Py_ssize_t n)
+
+cdef class SkewPolynomial_generic_dense(SkewPolynomial):
+    cpdef _rightpow_(self,right,modulus=*)
+    cpdef _leftpow_(self,right,modulus=*)
+
+cdef class SkewPolynomialBaseringInjection(RingHomomorphism):
+    cdef RingElement _an_element
+    cdef object _new_constant_poly_