commit 6b80453f9f7cd302899ee62eb068d1bfe0bc9ba5
parent 31f855d9a13e237e81c46d1c363a9c74e443af43
Author: Sebastiano Tronto <sebastiano.tronto@gmail.com>
Date: Tue, 17 Sep 2019 10:29:15 +0200
Removed some commented code, changed README.md
Diffstat:
2 files changed, 0 insertions(+), 183 deletions(-)
diff --git a/parameters_Q4_new.sage b/parameters_Q4_new.sage
@@ -1,60 +0,0 @@
-# Returns the l-dvisibility parameters of G over Q, given a good basis gb of G,
-# as a list of pairs (di,hi).
-def parameters_Q( gb, l ):
- ret = []
- for i in range( len( gb ) ):
- for j in gb[i]:
- ret.append( (i,0) )
- return ret
-
-# Computes the l-divisibility parameters of G over Q4, given a good basis gb
-# over Q for G. Returns a list of pairs (di,hi).
-# If l is odd it just uses the good basis given to compute the parameters.
-# If l=2, it uses the results of PST-2.
-def parameters_Q4_new( gb, l ):
- if l != 2:
- return parameters_Q( gb, l )
- else:
- # Converts from "good basis format" to simple list
- b = []
- for x in gb:
- b += x
-
- M = exponent_matrix( b )
-
- # Thanks to my terrible notation, the elements of b are what are called
- # g_i in the article, while the elements of bb will be the b_i's.
- bb = [ abs(b[i]) ^ (2^(-divisibility(M[i],2))) for i in range(len(b)) ]
-
- # Computing a combination of bb elements of the form 2 * square.
- MM = exponent_matrix( bb + [2] ).change_ring( GF( 2 ) )
-
- for a in MM.kernel().basis():
- if a[-1] != 0:
- # the vector a[0:-1] gives the coefficients for a combination
- # of the b_i's of the form 2 * square
-
- # Basis elements that actually appear in the combination
- c = [ b[i] for i in range(len(a[0:-1])) if a[i] != 0 ]
-
- # Return the parameters, changing only the ones fo the element
- # of highest divisibility that appears in the combination.
- ret = []
- div_max = -1
- ind_max = -1
- for i in range(len(b)):
- ret.append( (divisibility(M[i],2), 1-max(0,sgn(b[i]))) )
- if a[i] != 0 and ret[i][0] > div_max:
- div_max = ret[i][0]
- ind_max = i
- d1, h1 = ret[ind_max]
- if d1 == 1:
- h1 = 1 - h1
- elif d1 == 0:
- h1 = 2
- ret[ind_max] = ( d1+1, h1 )
-
- return ret
-
- return parameters_Q( gb, 2 )
-
diff --git a/parameters_Q4_old.sage b/parameters_Q4_old.sage
@@ -1,123 +0,0 @@
-# Computes the l-divisibility parameters of G over Q4, given a good basis gb
-# over Q for G. Returns a list of pairs (di,hi).
-# If l is odd it just uses the good basis given to compute the parameters.
-def parameters_Q4( gb, l ):
- # Converts from "good basis format" to simple list
- b = []
- for x in gb:
- b += x
- ret = []
-
- if l != 2:
- for i in range( len( gb ) ):
- for j in gb[i]:
- ret.append( (i,0) )
- return ret
- else:
- R.<y> = PolynomialRing( QQ )
- pol = R(y^2+1)
- Q4.<eye> = NumberField( pol ) # I already use i for other things
-
- # Factorize basis elements over Q4 and so on.
- d = []
- B = []
- h = []
- ideals_list = set()
- M = [] # Exponent matrix of the Bi's
-
- # Pre-process to find all ideals appearing in the factorization and fix
- # a chosen generator for each of them. This is important in order to
- # compute the "sign" (h-parameter) of an element with respect to it Bi.
- for g in b:
- factorization_list = list( Q4.ideal(g).factor() )
- ideals_list |= set( [ x[0] for x in factorization_list ] )
- ideals_list = list( ideals_list )
- # Chooses a generator of each principal ideal in the list
- irreducibles_list = [ J.gens_reduced()[0] for J in ideals_list ]
-
- # Compute the Q4-parameters of the given basis b. Also computes the
- # exponent matrix of the Bi's
- for g in b:
- factorization_list = list( Q4.ideal(g).factor() )
- exps = [ x[1] for x in factorization_list ]
- d.append( divisibility( exps, l ) )
- Bg = 1
- for j in range(len(factorization_list)):
- a = 0
- for i in range( len( ideals_list ) ):
- if ideals_list[i] == factorization_list[j][0]:
- a = irreducibles_list[i]
- break
- Bg *= a ^ (exps[j]/(l^d[-1]))
- B.append(Bg)
- u = g / (Bg^(l^d[-1]))
- if not u.is_unit():
- print "Error: g is not the right power of the computed Bg."
- print "g:", g, ", Bg:", Bg, ", exponent:", l^d[-1]
- if u == 1:
- h.append( 0 )
- elif u == -1:
- h.append( 1 )
- else:
- h.append( 2 )
-
- # Make the exponent matrix M (for now as a list of rows)
- for g in B:
- row = [0] * len(ideals_list)
- for i in range(len(ideals_list)):
- I = ideals_list[i]
- ee = 1
- while (I^ee).divides(g):
- ee += 1
- row[i] = ee-1
- M.append(row)
-
- # If the Bi's are not strongly independent, apply the algorithm (only
- # once) to produce a new basis. The new basis has maximal parameters.
- coeffs = find_combination( matrix(M), l )
- if coeffs != []:
-
- maxi = -1
- maxd = -1
- for i in range(len(d)):
- if d[i] > maxd and coeffs[i] != 0:
- maxd = d[i]
- maxi = i
-
- x = [(a/coeffs[maxi]).lift() for a in coeffs] # Now a vector of int
-
- new_element = 1
- for i in range(len(d)):
- new_element *= b[i]^( x[i] * l^(d[maxi]-d[i]) )
-
- b[maxi] = new_element
-
- # Compute new B, d and so on.
- factorization_list = list( Q4.ideal(b[maxi]).factor() )
- exps = [ x[1] for x in factorization_list ]
- d[maxi] = divisibility( exps, l )
- Bg = 1
-
- for j in range(len(factorization_list)):
- a = 0
- for i in range( len( ideals_list ) ):
- if ideals_list[i] == factorization_list[j][0]:
- a = irreducibles_list[i]
- break
- Bg *= a ^ (exps[j]/(l^d[maxi]))
- B[maxi] = Bg
- M[maxi] = [ x[1] for x in list( Q4.ideal(Bg).factor() ) ]
- u = b[maxi] / (Bg^(l^d[maxi]))
- if not u.is_unit():
- print "Error: new element is not the right power of B."
- print "New el.:", b[maxi], ", B:", Bg, ", exponent:", l^d[maxi]
- if u == 1:
- h[maxi] = 0
- elif u == -1:
- h[maxi] = 1
- else:
- h[maxi] = 2
-
- return [(d[i],h[i]) for i in range(len(d))]
-
-
