# Help for generator.
sage: mq.SR?
# We construct SR(1,1,1,4) over GF(2^4).
sage: sr = mq.SR(1,1,1,4)
SR(1,1,1,4)
# The constructor may fail due to zero inversions.
sage: F,s = sr.polynomial_system()
---------------------------------------------------------------------------
<type 'exceptions.ZeroDivisionError'>     Traceback (most recent call last)
...
<type 'exceptions.ZeroDivisionError'>: A zero inversion occurred during an 
encryption or key schedule.
# So we try again.
sage: F,s = sr.polynomial_system(); F
Polynomial System with 40 Polynomials in 20 Variables
# The object F is a polynomial system and the object s a solution
# dictionary. Help about F can be found via tab completion.
sage: F.<tab>

# F can be exported to Magma.
sage: sage: F._magma_()
Ideal of Polynomial ring of rank 20 over GF(2^4)
Graded Reverse Lexicographical Order
Variables: k100, k101, k102, k103, x100, x101, x102, x103, w100, w101, w102, 
w103, s000, s001, s002, s003, k000, k001, k002, k003
Basis:
[
w100 + k000 + $.1^4,
w101 + k001 + $.1^8,
w102 + k002 + $.1,
w103 + k003 + $.1^2,
k000^2 + k001,
....
....
....

# F can be exported to Singular.
sage: F._singular_()
w100+k000+(a+1),
w101+k001+(a^2+1),
w102+k002+(a),
w103+k003+(a^2),
k000^2+k001,
k001^2+k002,
k002^2+k003,
...

# Or we can use those systems transparently in the background.
sage: F.groebner_basis() # Singular in the background
[k002 + (a^3 + 1)*k003 + (a^2), k001 + (a^3 + a^2)*k003 + (a^3), k000 + 
(a^2)*k003 + (a^3 + a^2), s003 + (a^3 + a)*k003 + (a^3 + a^2 + a), s002 + 
(a^3 + a^2 + a)*k003 + (a^2), s001 + (a^3 + a^2 + a + 1)*k003 + (a + 1), s000 
+ (a^2 + a)*k003 + 1, w103 + k003 + (a^2), w102 + (a^3 + 1)*k003 + (a^2 + a), 
w101 + (a^3 + a^2)*k003 + (a^3 + a^2 + 1), w100 + (a^2)*k003 + (a^3 + a^2 + a 
+ 1), x103 + (a^3 + a)*k003, x102 + (a^3 + a^2 + a)*k003 + (a^3 + 1), x101 + 
(a^3 + a^2 + a + 1)*k003 + (a^3 + a), x100 + (a^2 + a)*k003 + (a^3 + a), k103 
+ (a^3 + a + 1)*k003 + 1, k102 + (a^2 + a + 1)*k003 + (a^3 + a^2), k101 + (a 
+ 1)*k003 + (a + 1), k100 + (a)*k003 + (a^2 + a + 1), k003^2 + (a^2)*k003 + 
(a^3 + a^2)]
sage: F.groebner_basis(algorithm='magma:GroebnerBasis') # Magma
[k003^2 + (a^2)*k003 + (a^3 + a^2), k100 + (a)*k003 + (a^2 + a + 1), k101 + (a 
+ 1)*k003 + (a + 1), k102 + (a^2 + a + 1)*k003 + (a^3 + a^2), k103 + (a^3 + a 
+ 1)*k003 + 1, x100 + (a^2 + a)*k003 + (a^3 + a), x101 + (a^3 + a^2 + a + 
1)*k003 + (a^3 + a), x102 + (a^3 + a^2 + a)*k003 + (a^3 + 1), x103 + (a^3 + 
a)*k003, w100 + (a^2)*k003 + (a^3 + a^2 + a + 1), w101 + (a^3 + a^2)*k003 + 
(a^3 + a^2 + 1), w102 + (a^3 + 1)*k003 + (a^2 + a), w103 + k003 + (a^2), s000 
+ (a^2 + a)*k003 + 1, s001 + (a^3 + a^2 + a + 1)*k003 + (a + 1), s002 + (a^3 
+ a^2 + a)*k003 + (a^2), s003 + (a^3 + a)*k003 + (a^3 + a^2 + a), k000 + 
(a^2)*k003 + (a^3 + a^2), k001 + (a^3 + a^2)*k003 + (a^3), k002 + (a^3 + 
1)*k003 + (a^2)]

# We can also construct equation systems over GF(2).
sage: sr = mq.SR(2,1,1,4,gf2=True)
sage: F,s = sr.polynomial_system()

# For those we can use PolyBoRi to compute the Groebner basis.
sage: R= F.ring()
sage: B = BooleanPolynomialRing(R.ngens(), R.variable_names(), order="lex")
sage: F2 = B.ideal([B(f) for f in F]) 
sage: F2.groebner_basis()
[k200 + k001 + k003 + 1, k201 + k001, k202 + 1, k203 + k000, x200 + k003, x201 
+ k000 + k001, x202 + k000 + k001 + k003, x203 + k000 + k003, w200 + k000 + 
k003 + 1, w201 + k001 + k003 + 1, w202 + k001 + 1, w203, s100 + k003 + 1, 
s101 + k000 + k001, s102 + k000 + k001 + k003, s103 + k000 + k003, k100 + 1, 
k101 + k001 + k003, k102 + k000 + 1, k103 + k003, x100 + k001 + k003 + 1, 
x101 + k000 + k001 + k003 + 1, x102 + k001 + 1, x103 + k000, w100 + k000 + 1, 
w101 + k001 + 1, w102 + k003, w103 + k003 + 1, s000 + k000 + k001 + k003, 
s001 + k000 + k001, s002 + k000 + k003, s003 + k001 + 1, k000*k001 + k000 + 
k001 + 1, k000*k003 + k000 + k003 + 1, k001*k003 + k001, k002 + k003]