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Trac #16162: Cantor-Zassenhaus may enter infinite loop over GF(2**k) …
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…and cannot be interrupted

A solution is to use random polynomials just as for odd characteristic.

This also replaces calls to `power_mod` by calls to `pow` with three
arguments which uses implementation optimized code for modular
exponentiation.

URL: http://trac.sagemath.org/16162
Reported by: jpflori
Ticket author(s): Jean-Pierre Flori
Reviewer(s): Peter Bruin
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@vbraun
Release Manager authored and vbraun committed May 21, 2014
2 parents 9a4b0b8 + 80db6aa commit 5159542
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Showing 2 changed files with 48 additions and 12 deletions.
2 changes: 2 additions & 0 deletions src/sage/libs/ntl/ntl_ZZ_pEX_linkage.pxi
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Original file line number Diff line number Diff line change
Expand Up @@ -327,7 +327,9 @@ cdef inline int celement_pow(ZZ_pEX_c* res, ZZ_pEX_c* x, long e, ZZ_pEX_c *modul
cdef ZZ_pEX_c y
if modulus == NULL:
if ZZ_pEX_IsX(x[0]):
sig_on()
ZZ_pEX_LeftShift(res[0], x[0], e - 1)
sig_off()
else:
sig_on()
ZZ_pEX_power(res[0], x[0], e)
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58 changes: 46 additions & 12 deletions src/sage/rings/polynomial/polynomial_element.pyx
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Original file line number Diff line number Diff line change
Expand Up @@ -1478,6 +1478,39 @@ cdef class Polynomial(CommutativeAlgebraElement):
sage: for _ in range(40):
....: f = R.random_element(degree=4)
....: assert f(f.any_root(K)) == 0
Check that our Cantor-Zassenhaus implementation does not loop
over finite fields of even characteristic (see :trac:`16162`)::
sage: K.<a> = GF(2**8)
sage: x = polygen(K)
sage: (x**2+x+1).any_root() # used to loop
Traceback (most recent call last):
...
ValueError: no roots A 1
sage: (x**2+a+1).any_root()
a^7 + a^2
Also check that such computations can be interrupted::
sage: K.<a> = GF(2**8)
sage: x = polygen(K)
sage: alarm(1)
sage: (x**1000000+x+a).any_root()
Traceback (most recent call last):
...
AlarmInterrupt
Check root computation over large finite fields::
sage: K.<a> = GF(2**50)
sage: x = polygen(K)
sage: (x**10+x+a).any_root()
a^49 + a^47 + a^44 + a^42 + a^41 + a^39 + a^38 + a^37 + a^36 + a^34 + a^33 + a^29 + a^27 + a^26 + a^25 + a^23 + a^18 + a^13 + a^7 + a^5 + a^4 + a^3 + a^2 + a
sage: K.<a> = GF(2**150)
sage: x = polygen(K)
sage: (x**10+x+a).any_root()
a^149 + a^148 + a^146 + a^144 + a^143 + a^140 + a^138 + a^136 + a^134 + a^132 + a^131 + a^130 + a^129 + a^127 + a^123 + a^120 + a^118 + a^114 + a^113 + a^112 + a^111 + a^108 + a^104 + a^103 + a^102 + a^99 + a^98 + a^94 + a^91 + a^90 + a^88 + a^79 + a^78 + a^75 + a^73 + a^72 + a^67 + a^65 + a^64 + a^63 + a^62 + a^61 + a^59 + a^57 + a^52 + a^50 + a^48 + a^47 + a^46 + a^45 + a^43 + a^41 + a^39 + a^37 + a^34 + a^31 + a^29 + a^27 + a^25 + a^23 + a^22 + a^20 + a^18 + a^16 + a^14 + a^11 + a^10 + a^8 + a^6 + a^5 + a^4 + a + 1
"""
if self.base_ring().is_finite() and self.base_ring().is_field():
if self.degree() < 0:
Expand All @@ -1501,7 +1534,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
if ring is None:
allowed_deg_mult = Integer(1)
else:
if not self.base_ring().is_prime_field():
if not (self.base_ring().is_field() and self.base_ring().is_finite()):
raise NotImplementedError
if ring.characteristic() != self.base_ring().characteristic():
raise ValueError, "ring must be an extension of the base ring"
Expand All @@ -1511,7 +1544,8 @@ cdef class Polynomial(CommutativeAlgebraElement):
if degree is None:
x = self.parent().gen()
if allowed_deg_mult == 1:
self = self.gcd(x**q-x)
xq = pow(x,q,self)
self = self.gcd(xq-x)
degree = -1
if self.degree() == 0:
raise ValueError, "no roots B %s"%self
Expand All @@ -1528,7 +1562,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
break
while d < allowed_deg_mult:
d = d+1
xq = xq**q % self
xq = pow(xq,q,self)
if d.divides(allowed_deg_mult):
break
A = self.gcd(xq-x)
Expand All @@ -1553,7 +1587,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
while True:
# we waste a little effort here in computing the xq again.
d = d+1
xq = xq**q % self
xq = pow(xq,q,self)
if allowed_deg_mult.divides(d):
break
A = self.gcd(xq-x)
Expand All @@ -1575,7 +1609,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
raise ValueError, "no roots D %s"%self
break
d = d+1
xq = xq**q % self
xq = pow(xq,q,self)
if d == degree:
break
A = self.gcd(xq-x)
Expand All @@ -1601,28 +1635,28 @@ cdef class Polynomial(CommutativeAlgebraElement):
except IndexError:
raise ValueError, "no roots F %s"%self
if q % 2 == 0:
T = x
while True:
T = R.random_element(2*degree-1)
if T == 0:
continue
T = T.monic()
C = T
for i in range(degree-1):
C = T + C**q % self
C = T + pow(C,q,self)
h = self.gcd(C)
hd = h.degree()
if hd == 0 or hd == self.degree():
T = T * x**q
else:
if hd != 0 or hd != self.degree():
if 2*hd <= self.degree():
return h.any_root(ring, -degree, True)
else:
return (self//h).any_root(ring, -degree, True)
else:
from sage.rings.arith import power_mod
while True:
T = R.random_element(2*degree-1)
if T == 0:
continue
T = T.monic()
h = self.gcd(power_mod(T, Integer((q**degree-1)/2), self)-1)
h = self.gcd(pow(T, Integer((q**degree-1)/2), self)-1)
hd = h.degree()
if hd != 0 and hd != self.degree():
if 2*hd <= self.degree():
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