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Commit 107d4125 authored by Emmanuel Thomé's avatar Emmanuel Thomé
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reconcile normalisation choices between CocksPinchVariant.py and final_expo_k68.sage

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CocksPinchVariant.py
View file @ 107d4125
......@@ -535,6 +535,7 @@ class CocksPinchVariantResult(object):
self.t0 -= r
if abs(r-self.y0) < abs(self.y0):
self.y0 -= r
self.y0 = abs(self.y0)
# Determination of the lifted (t,y) from the solution mod r
......@@ -977,7 +978,10 @@ class CocksPinchVariantResult(object):
saved_max_B1 = self.max_B1
self.max_B1 = 600
dt0 = t0 - ((T**i+1) % r)
dy0 = y0 - ZZ((t0-2)/sqrt(Integers(r)(-fD)))
y0base = ZZ((t0-2)/sqrt(Integers(r)(-fD)))
if r - y0base < y0base:
y0base = r - y0base
dy0 = y0 - y0base
assert dt0 in [0,-r]
assert dy0 in [0,-r]
......@@ -1340,12 +1344,22 @@ class CocksPinchVariantSearch(object):
y0 = K(t0-2)/sqrt(K(-fD))
# Lift arbitrarily. Anyway we'll iterate over multiple
# possible representatives.
# The normalisation choice that we do in final_expo_k68
# (at least) is that we use the least positive integer
# representative of y0=\pm(t0-2)*inv_sqrt_D
#
# (as for t0, we have no sign indetermination, so we
# simply choose the representative of smallest absolute
# value, and that may mean a negative integer)
t0 = ZZ(t0)
y0 = ZZ(y0)
if abs(r-t0) < abs(t0):
t0 -= r
if abs(r-y0) < abs(y0):
y0 -= r
y0 = abs(y0)
# We want to constrain the bit length of t^2+fD*y^2{{{
# with t = t0 + ht * r and y = y0 + hy * r
......
cost_pairing.py
View file @ 107d4125
......@@ -167,11 +167,11 @@ def Hw(x) :
return len(bit_positions_2naf(x))
proof.arithmetic(False)
C5=CocksPinchVariantResult(5,10000000147,0xe000000000008000,1,ht=3,hy=-0x11e36418c7c8b454,max_B1=600)
C5=CocksPinchVariantResult(5,10000000147,0xe000000000008000,1,ht=3,hy=0x11e36418c7c8b454,max_B1=600)
C6=CocksPinchVariantResult(6,3,0xefffffffffffffe00000000000000000,1,ht=-1,hy=0xffbbffffffffffffc020,allowed_cofactor=420,allowed_size_cofactor=10,max_B1=600)
C7=CocksPinchVariantResult(7,20,0x5fffb820248,6,ht=-2,allowed_cofactor=1232,allowed_size_cofactor=10,max_B1=600)
#C8=CocksPinchVariantResult(8,4,0xffffffffeff7c200,5,ht=5,hy=-0xd700,allowed_cofactor=420,allowed_size_cofactor=10,max_B1=600)
C8=CocksPinchVariantResult(8,4,0xffc00020fffffffc,1,ht=1,hy=-0xdc04,allowed_cofactor=420,allowed_size_cofactor=10,max_B1=600)
C8=CocksPinchVariantResult(8,4,0xffc00020fffffffc,1,ht=1,hy=0xdc04,allowed_cofactor=420,allowed_size_cofactor=10,max_B1=600)
CMNT6=MNT6(u=873723667900031396506414143162332159382674816702805606206979732381600254701804231398281169537138620,a=209307816050232262803672282154940341360062431838092388077917610639183322072827259682607127795420474686833003315766797546568469776750651773087882545447646552119008299040167030969895802846139484415144,b=2319663192174958547181026340141410918530227127674793888869119262391240421488942353013995765010333162065568990954578077256489549792305772041454141172011940607053889955897003759289947924385489341215143,D=8317003,c=1)
......
final_expo_k68.sage
View file @ 107d4125
......@@ -59,20 +59,24 @@ def count_formula_k8(i, c):
c1 = c1 + y
d1 = 0
elif i==5:
c1 = c1 - y
c1 = c1 + y
d1 = 0
d1 = d1 + d0*V
assert c1 + d1/2 == (c0 + d0/4) * T + e1
c12 = c1*2
c2 = (c12 + d1) * 2*U + (c12 + d1) * V - y
c2 = (c12 + d1) * 2*U + (c12 + d1) * V
if i == 1:
c2 -= y
else:
c2 += y
d2 = 1
assert c2 + d2/4 == (c1 + d1/2) * T + e2
c22 = c2*2
c3 = (c22*2 + d2) * U + c22*V
if i == 7:
c3 = c3 + y
elif i == 3:
c3 = c3 - y
elif i == 3:
c3 = c3 + y
elif i == 1:
c3 = c3 + u - 1
elif i == 5:
......@@ -111,7 +115,7 @@ def count_formula_k8(i, c):
r = r * ay
s = s ** V
elif i == 5:
r = r * ayi
r = r * ay
s = s ** V
assert r.val == c1
assert s.val == d1
......@@ -120,7 +124,10 @@ def count_formula_k8(i, c):
r = r ** 2
r = r * s
r = (r ** 2) ** U * r ** V
r = r * ayi
if i == 1:
r = r * ayi
else:
r = r * ay
s = a
assert r.val == c2
assert s.val == d2
......@@ -129,9 +136,9 @@ def count_formula_k8(i, c):
r = r ** 2
r = (r ** 2 * s) ** U * r ** V
if i == 7:
r = r * ay
elif i == 3:
r = r * ayi
elif i == 3:
r = r * ay
elif i == 1:
r = r * au
r = r * ai
......@@ -172,17 +179,20 @@ def count_formula_k6(i, tr, c):
# For k=6, the expressions of t and y are:
# t = T + 1 +h_t*r
# y = 1/3*T^2 - 2/3*T +h_y*r
#
# t = T + 1 +h_t*r
# y = -1/3*T^2 - 2/3 +h_y*r
# y = 1/3*T^2 + 2/3 +h_y*r
#
# t = -T + 2 +h_t*r
# y = 1/3*T^2 - 2/3*T + 1 +h_y*r
#
# t = -T + 2 +h_t*r
# y = -1/3*T^2 + 1/3 +h_y*r
# y = 1/3*T^2 - 1/3 +h_y*r
# but if we reduce to the parity bit of t+y only:
# parity = T + 1 +h_t + T^2 +h_y = 1 + h_t + h_y
# parity = T + 1 +h_t - T^2 +h_y = 1 + h_t + h_y
# parity = -T +h_t + T^2 + 1 +h_y = 1 + h_t + h_y
# parity = -T +h_t - T^2 + 1 +h_y = 1 + h_t + h_y
# parity = T + 1 + h_t + T^2 + h_y = 1 + h_t + h_y
# parity = T + 1 + h_t + T^2 + h_y = 1 + h_t + h_y
# parity = T + h_t + T^2 + 1 + h_y = 1 + h_t + h_y
# parity = T + h_t + T^2 + 1 + h_y = 1 + h_t + h_y
# so that in all cases, we have either h_t odd and h_y
# even, or the converse.
......@@ -191,7 +201,7 @@ def count_formula_k6(i, tr, c):
# This one expresses the result as a function of:
# u = h_t/2
# w = (h_y-z)/2
z = -1 if tr == 0 else 1
z = -1
# so we're assuming that h_t is even and h_y is odd.
new_c=horner_list(3*c(h_t=2*u,h_y=(2*w+z),T=tr+3*U),U)
new_c[1] /= 3
......@@ -252,14 +262,10 @@ def count_formula_k6(i, tr, c):
rplus3 = ar + a3
if parity_ht == 1:
c0 = c0 - a3u
if parity_ht == 0 and tr == 0:
if parity_ht == 0:
c0 = c0 - a6w
elif parity_ht == 1 and tr == 0:
elif parity_ht == 1:
c0 = c0 + a3w
elif parity_ht == 0 and tr == 1:
c0 = c0 + a6w
elif parity_ht == 1 and tr == 1:
c0 = c0 - a3w
assert c0 == e0
if i == 1:
if tr == 0:
......@@ -271,11 +277,11 @@ def count_formula_k6(i, tr, c):
c2 = ar
else:
if parity_ht == 0:
c1 = ar + a3u + a3w
c1 = ar + a3u - a3w
c2 = ar + a6u
else:
c1 = ar - a6w
c2 = ar + a3u - a9w
c1 = ar + a6w
c2 = ar + a3u + a9w
else:
if tr == 0:
if parity_ht == 0:
......@@ -286,10 +292,10 @@ def count_formula_k6(i, tr, c):
c2 = ar + a3u + a9w
else:
if parity_ht == 0:
c1 = ar - a3u + a3w
c2 = rplus3 + a3u + a9w
c1 = ar - a3u - a3w
c2 = rplus3 + a3u - a9w
else:
c1 = rplus3 - a6u - a6w
c1 = rplus3 - a6u + a6w
c2 = ar
assert c1 == e1
assert c2 == e2
......@@ -327,18 +333,18 @@ def count_formula_k6(i, tr, c):
acc = acc * ar
else:
if parity_ht == 0:
acc = acc * a6w
acc = acc * a6w^-1
acc = acc^U
ar3u = ar * a3u
acc = acc * ar3u * a3w
acc = acc * ar3u * a3w^-1
acc = acc^U
acc = acc * ar3u * a3u
else:
acc = acc * (a3u * a3w)^-1
acc = acc * a3u^-1 * a3w
acc = acc^U
acc = acc * ar * a6w^-1
acc = acc * ar * a6w
acc = acc^U
acc = acc * ar * a3u * a9w^-1
acc = acc * ar * a3u * a9w
else:
if tr == 0:
if parity_ht == 0:
......@@ -358,14 +364,14 @@ def count_formula_k6(i, tr, c):
else:
if parity_ht == 0:
ar3 = ar * a3
acc = acc * a6w
acc = acc * a6w^-1
acc = acc^U
acc = acc * ar * a3u^-1 * a3w
acc = acc * ar * a3u^-1 * a3w^-1
acc = acc^U
acc = acc * ar3 * a3u * a9w
acc = acc * ar3 * a3u * a9w^-1
else:
ar3 = ar * a3
a3u3w = a3u * a3w
a3u3w = a3u * a3w^-1
acc = acc * (a3u3w)^-1
acc = acc^U
acc = acc * ar3 * (a3u3w^2)^-1
......@@ -403,34 +409,45 @@ def formulas(k):
D=4
inv_sqrt_D = (1/sqrt(K(-D))).polynomial()
ld = inv_sqrt_D.list()[-1]
assert inv_sqrt_D.degree() < euler_phi(k)
# choose positive leading coefficient
if ld < 0:
inv_sqrt_D = -inv_sqrt_D
inv_sqrt_D = inv_sqrt_D(T)
# We do **NOT** normalize inv_sqrt_D. We could, if we wanted: the
# idea would be to do that depending on the congruence class of T,
# and choose the polynomial expression with the smallest leading
# coefficient in absolute value (for example).
#
# E.g. for k=6, inv_sqrt_D = \pm (2T-1)/3 ; if T is 1 mod 3, this is
# the same as if we add 0 = 2r/3 = 2*(T^2-T+1)/3, hence inv_sqrt_D =
# \pm (2T^2+1)/3, but then the reprensentatives \pm (T^2-3T+2)/3 are
# smaller.
#
# However, even though we know how to do this, we're better off doing
# this work on y0, which is the final data.
# Bottom line: below, we strive to write the formula for the
# (polynomial expression of the) least positive integer
# representative of y0=\pm(t0-2)*inv_sqrt_D, and this exact expression
# depends on the congruence class of D.
if k==6:
subfamilies=[
# Recall that T=2 mod 3 is forbidden since r=Phi_6(T)
# must be prime.
#
# For i==1, t0 == T+1
# Recall that T=2 mod 3 is forbidden since r=Phi_6(T).
# The representatives of (t0-2)*inv_sqrt_D in 1/3 * Z are:
# (2T^2-3T+1)/3 : outside range
# (T^2-2T)/3 = T*(T-2)/3 : good if T = 0,2 mod 3 (hence only 0)
# (-T-1)/3 : good if T is 2 mod 3, so *never good* !
# (-T^2-2)/3 : good if T is 1, 2 mod 3 (hence only 1)
(1, T+1, [(0,6,(T*(T-2))/3), (1,6,(1-T**2)/3-1), ]),
# what might correspond to CocksPinchVariant is:
#(1, T+1, [(0,6,(T*(T-2))/3), (1,6,(2+T**2)/3), ]),
# The representatives of \pm (t0-2)*inv_sqrt_D in 1/3 * Z are:
# \pm (T^2-2T)/3 = \pm T*(T-2)/3 : good if T = 0,2 mod 3 (hence only 0)
# \pm (T+1)/3 : good if T is 2 mod 3, so *never good* !
# \pm (T^2+2)/3 : good if T is 1, 2 mod 3 (hence only 1)
(1, T+1, [(0,3,(T*(T-2))/3), (1,3,(T**2+2)/3), ]),
# For i==5, t0 == 2-T
# The representatives of (t0-2)*inv_sqrt_D in 1/3 * Z are:
# (T-2*T^2)/3 = T*(1-2*T)/3 : outside range
# (1-T^2)/3 : good if T is 1 or 2 mod 3 (hence only 1)
# (2-T)/3 : good if T is 2 mod 3, so *never good* !
# (T^2-2*T+3)/3 : good if T is 0 or 2 mod 3 (hence only 0)
(5, 2-T, [ (0,3,1+T*(T-2)/3), (1,3,(1-T**2)/3), ]),
# what might correspond to CocksPinchVariant is:
#(5, 2-T, [ (0,3,1-T*(T-2)/3), (1,3,-(1-T**2)/3), ]),
# The representatives of \pm(t0-2)*inv_sqrt_D in 1/3 * Z are:
# \pm (T^2-1)/3 : good if T is 1 or 2 mod 3 (hence only 1)
# \pm (T-2)/3 : good if T is 2 mod 3, so *never good* !
# \pm (T^2-2*T+3)/3 : good if T is 0 or 2 mod 3 (hence only 0)
(5, 2-T, [ (0,3,1+T*(T-2)/3), (1,3,(T**2-1)/3), ]),
]
# Note that congruence classes on ht and hy will force p to be an
# integer, even though it seems to have a 1/4 in the denominator.
......@@ -444,11 +461,13 @@ def formulas(k):
# -> as a consequence, the formula should not be specific to one
# congruence class of T mod 4
elif k==8:
# Here T must be even, since T^4+1 must be prime. The minimal
# integer representatives of \pm (t0-2)*inv_sqrt_D are:
subfamilies=[
(1, T+1, [(0,2,(T-1)*T**2/2)]),
(3, T**3+1, [(0,2,-(T+1)*T/2)]),
(5, -T+1, [(0,2,(-T-1)*T**2/2)]),
(7, -T**3+1, [(0,2,(-T+1)*T/2)]),
(3, T**3+1, [(0,2,(T+1)*T/2)]),
(5, -T+1, [(0,2,(T+1)*T**2/2)]),
(7, -T**3+1, [(0,2,(T-1)*T/2)]),
]
else:
# just for completeness. This ignores the fact that the
......@@ -461,7 +480,9 @@ def formulas(k):
for i,t0,y0class in subfamilies:
for tr,tq,y0 in y0class:
assert (y0 - (t0-2)*inv_sqrt_D) % r == 0
# We don't check against inv_sqrt_D, since inv_sqrt_D is
# known only up to sign.
# assert (y0 - (t0-2)*inv_sqrt_D) % r == 0
assert (D * y0**2 + (t0-2)**2) % r == 0
t = t0 + h_t*r
y = y0 + h_y*r
......
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