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######################################################################
#
# Functions for Minkowski-reduction of generators, and naive reduction
# of torsion generators and of generators mod torsion.
#
######################################################################
pt_wt = lambda P: len(str(P))
def reduce_tgens(tgens, verbose=False):
"""
tgens: list of torsion generators (if two, sorted by order)
Return a new list of generators which is minimal with respect to string length.
"""
r = len(tgens)
if r == 0:
return tgens
if r == 1: # cyclic
P1 = tgens[0]
n1 = P1.order()
if n1 == 2: # no choice
return tgens
Plist = [i * P1 for i in range(1, n1) if n1.gcd(i) == 1]
Plist.sort(key=pt_wt)
Q = Plist[0]
if Q != P1 and verbose:
print("Replacing torsion [{}] generator {} with {}".format(n1, P1, Q))
return [Q]
# now r=2 and P1 has order n1=2 while n2 = 2, 4, 6, 8.
# -- we use brute force
assert r == 2
if tgens[0].order() > tgens[1].order():
tgens.reverse()
P1, P2 = tgens
n1 = P1.order() # = 2
n2 = P2.order() # = 2, 4, 6 or 8
assert n1 == 2 and n2 in [2, 4, 6, 8]
m = n2 // 2
P1a = m * P2 # other 2-torsion
P1b = P1 + P1a # points
gen_pairs = []
for j in range(n2):
jP2 = j * P2
for i in range(n1):
Q = i * P1 + jP2
if Q.order() != n2:
continue
Q2 = m * Q
for P in [P1, P1a, P1b]:
if P != Q2:
gen_pairs.append((P, Q))
pt_wt2 = lambda PQ: sum(pt_wt(P) for P in PQ)
gen_pairs.sort(key=pt_wt2)
rtgens = list(gen_pairs[0])
# for structure [2,2] we swap the two gens over so that the first has smallest x-coordinate
if n2 == 2:
rtgens.sort(key=lambda P: list(P)[0])
if rtgens != tgens and verbose:
print("Replacing torsion [{},{}] generators {} with {}".format(n1, n2, tgens, rtgens))
return rtgens
def check_minkowski(gens):
"""
Check the points are Minkowski-reduced (rank up to 3 only)
"""
r = len(gens)
if r < 2 or r > 3:
return True
if r == 2:
P1, P2 = gens
h1 = P1.height()
h2 = P2.height()
h3 = (P1 + P2).height()
return h1 < h2 and h2 < h3 and h3 < 2 * h1 + h2
# r=3
if not check_minkowski(gens[:2]):
return False
if not check_minkowski(gens[1:]):
return False
if not check_minkowski(gens[::2]):
return False
P1, P2, P3 = gens
h3 = P3.height()
P4 = P1 + P2
if h3 > (P3 + P4).height() or h3 > (P3 - P4).height():
return False
P4 = P1 - P2
return h3 < (P3 + P4).height() and h3 < (P3 - P4).height()
def reduce_mod_2d(P3, P1, P2, debug=False):
"""
Assuming [P1,P2] reduced, return P3+n1*P1+n2*P2 of minimal height
"""
if debug:
print("Reducing {} mod [{},{}]".format(P3, P1, P2))
h1 = P1.height()
h2 = P2.height()
assert h1 <= h2
P12 = P1 + P2
h12 = (P12.height() - h1 - h2) / 2
assert 2 * h12.abs() <= h1
# now the height of x*P1+y*P2 is ax^2+2bxy+cy^2
h3 = P3.height()
h13 = ((P1 + P3).height() - h1 - h3) / 2
h23 = ((P2 + P3).height() - h2 - h3) / 2
d = h1 * h2 - h12 * h12
y1 = (h2 * h13 - h12 * h23) / d
y2 = (h1 * h23 - h12 * h13) / d
# now y1*P1+y2*p2 is the orthogonal projection of P3 onto the P1,P2-plane
n1 = y1.round()
n2 = y2.round()
if debug:
print("orthog proj has coords ({},{}), rounded to ({},{})".format(y1, y2, n1, n2))
Q3 = P3 - (n1 * P1 + n2 * P2) # approximate answer
if debug:
print("base reduction is {}, height {}".format(Q3, Q3.height()))
P21 = P1 - P2
Q3list = [Q3, Q3 - P1, Q3 + P1, Q3 - P2, Q3 + P2, Q3 - P12, Q3 + P12, Q3 - P21, Q3 + P21]
Q3list.sort(key=lambda P: P.height())
if debug:
print("candidates for reduction: {}".format(Q3list))
print(" with heights: {}".format([Q.height() for Q in Q3list]))
R = P3 - P1 - P2
print(" P3-P1-P2 ={} has height {}".format(R, R.height()))
return Q3list[0]
def mreduce_gens(gens, debug=False):
r = len(gens)
if r < 2 or r > 3:
return gens
if r == 2:
P1, P2 = gens
h1 = P1.height()
h2 = P2.height()
h12 = ((P1 + P2).height() - h1 - h2) / 2
while True:
x = (h12/h1).round()
y = h12 - x * h1
P1, P2 = P2 - x * P1, P1
h1, h2 = h2, h1
h1 = h1 - x * (y + h12)
h12 = y
if h1 > h2:
return [P2, P1]
# now r=3
if check_minkowski(gens):
return gens
P1, P2, P3 = gens
if debug:
print("--------------------------------------------------------")
while True:
if debug:
print("At top of loop: {}".format([P1, P2, P3]))
P1, P2 = mreduce_gens([P1, P2]) # recursive
if debug:
print("After one 2D step: {}".format([P1, P2, P3]))
P3 = reduce_mod_2d(P3, P1, P2)
if debug:
print("After reducing P3 mod [P1,P2]: {}".format([P1, P2, P3]))
h3 = P3.height()
if h3 >= P2.height():
newgens = [P1, P2, P3]
if not check_minkowski(newgens):
label = gens[0].curve().label()
print("{}: gens = {}, newgens = {} are not Minkowski-reduced!".format(label, gens, newgens))
print("heights are {}".format([P.height() for P in newgens]))
raise RuntimeError
return newgens
P1, P2, P3 = (P1, P3, P2) if P1.height() < h3 else (P3, P1, P2)
if debug:
print("After resorting: {}".format([P1, P2, P3]))
def reduce_gens(gens, tgens, verbose=False, label=None):
"""
gens: list of generators mod torsion
tgens: list of torsion generators
(1) Reduce the torsion generators w.r.t. string length
(2) Minkowki reduce the mod-torsion gens (or just LLL-reduce if rank>3)
(3) Reduce the mod-torsion gens mod torsion w.r.t. string length
"""
rtgens = reduce_tgens(tgens)
if not gens:
return [], rtgens
E = gens[0].curve()
newgens = mreduce_gens(E.lll_reduce(gens)[0]) # discard transformation matrix
Tlist = E.torsion_points() if tgens else [E(0)]
def reduce_one(P):
mP = -P
Plist = [P + T for T in Tlist] + [mP + T for T in Tlist]
Plist.sort(key=pt_wt)
return Plist[0]
newgens = [reduce_one(P) for P in newgens]
if verbose and len(gens) > 1 and newgens != gens:
print("replacing {} generators ({}) {} with {}".format(len(gens), label, gens, newgens))
return newgens, rtgens
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