From ab4e1b35c39c124a320bb98969b235dbd187bade Mon Sep 17 00:00:00 2001
From: Aurore Guillevic <aurore.guillevic@inria.fr>
Date: Tue, 5 Sep 2023 22:24:58 +0200
Subject: [PATCH] another kss k=22 D=7 but with q=1 mod 11 possible
---
sage/tnfs/curve/kss.py | 145 ++++++++++++++++++++++++++---------------
1 file changed, 93 insertions(+), 52 deletions(-)
diff --git a/sage/tnfs/curve/kss.py b/sage/tnfs/curve/kss.py
index b724233..d682cf9 100644
--- a/sage/tnfs/curve/kss.py
+++ b/sage/tnfs/curve/kss.py
@@ -202,40 +202,58 @@ def polynomial_params(k, choice=None, D=None):
cofactor_r = [1, 1]
elif k == 22:
if D is None or D == 7:
- D = 7
- px = (x**24 - x**23 + 2*x**22 + 67*x**13 + 94*x**12 + 134*x**11 + 2048*x**2 + 5197*x + 4096)/7406
- rx = (x**20 - x**19 - x**18 + 3*x**17 - x**16 - 5*x**15 + 7*x**14 + 3*x**13 - 17*x**12 + 11*x**11 + 23*x**10 + 22*x**9 - 68*x**8 + 24*x**7 + 112*x**6 - 160*x**5 - 64*x**4 + 384*x**3 - 256*x**2 - 512*x + 1024)/23 # and sometimes / 23^2
- cx = (x**2 - x + 2) * (x**2 + x + 2) / (2 * 7 * 23)
- tx = (x**12 + 45*x + 46)/46
- yx = (x**12 - 4*x**11 - 47*x - 134)/322
- m = 161
- # D = 7 = 3 mod 4 -> Chi = x**2 + x + 2 has roots (-1+sqrt(-7))/2 and (-1 - sqrt(-7))/2
- betax = (11*x**23 - 20*x**22 + 53*x**21 - 58*x**20 + 106*x**19 - 116*x**18 + 212*x**17 - 232*x**16 + 424*x**15 - 464*x**14 + 848*x**13 - 191*x**12 + 1322*x**11 + 1051*x**10 - 172*x**9 + 2102*x**8 - 344*x**7 + 4204*x**6 - 688*x**5 + 8408*x**4 - 1376*x**3 + 16816*x**2 + 19776*x + 36142)/18515 # mod p
- lambx = (x**11 + 22)/23 # mod r
- if choice is None or choice == 0:
- u_mod_m = [32, 151] # and 7 | cxa = (x^2 - 2*x + 8) -> 2*u = 1 mod 7, 2*u mod 23 = [3, 18]
- else:
- u_mod_m = [4,18,25,39,81,95,116,123,144] # and 7 | cxa -> 2*u = 1 mod 7, 2*u mod 23 = [1, 8, 6, 13, 4, 2, 9, 16, 12]
- rx = rx/23
- cx = cx*23
- #px = (x**24 - 2*x**23 + 8*x**22 + 137216*x**13 + 385024*x**12 + 1097728*x**11 + 8589934592*x**2 + 43595595776*x + 68719476736)/124252061696
- #rx = (x**20 - 2*x**19 - 4*x**18 + 24*x**17 - 16*x**16 - 160*x**15 + 448*x**14 + 384*x**13 - 4352*x**12 + 5632*x**11 + 23552*x**10 + 45056*x**9 - 278528*x**8 + 196608*x**7 + 1835008*x**6 - 5242880*x**5 - 4194304*x**4 + 50331648*x**3 - 67108864*x**2 - 268435456*x + 1073741824)/24117248 # and sometimes 554696704
- #cx = (x**2 - 2*x + 8) * (x**2 + 2*x + 8)/5152
- #tx = (x**12 + 92160*x + 188416)/188416
- #yx = (x**12 - 8*x**11 - 96256*x - 548864)/1318912
- #m = 322
- # Diskrep([2, 7, 23], {2: [(1, 0)], 7: [(1, 1), (1, 6)], 23: [(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (1, 8), (1, 9), (1, 12), (1, 13), (1, 16), (1, 18)]})
- #u_mod_m = [6, 8, 36, 48, 50, 62, 64, 78, 104, 118, 146, 162, 174, 188, 190, 202, 216, 232, 246, 288, 300, 302]
- # if u_mod_m = [64, 302] (u=1 mod 7) then rx, px can take prime values.
- # if u_mod_m = [118,202] (u=6 mod 7) then rx can take prime values but 7 | px and 7 | cb = (x**2 + 2*x + 8)
- # if u_mod_m = [8,36,50,78,162,190,232,246,288] (u=1 mod 7) then rx/23 and px can take prime values
- # if u_mod_m = [6,48,62,104,146,174,188,216,300] (u=6 mod 7) then rx/23 can take prime values but 7 | px and and 7 | cxb
- #if choice is None or choice==0:
- # u_mod_m = [64, 302] # and 7 | cxa = (x**2 - 2*x + 8)
- #else:
- # u_mod_m = [8,36,50,78,162,190,232,246,288] # and 7 | cxa -> u = 0 mod 2, u = 1 mod 7, u mod 23 = [8, 13, 4, 9, 1, 6, 2, 16, 12]
- # rx = rx/23
- # cx = cx*23
+ if choice is None or choice in [0,1]:
+ D = 7
+ px = (x**24 - x**23 + 2*x**22 + 67*x**13 + 94*x**12 + 134*x**11 + 2048*x**2 + 5197*x + 4096)/7406
+ rx = (x**20 - x**19 - x**18 + 3*x**17 - x**16 - 5*x**15 + 7*x**14 + 3*x**13 - 17*x**12 + 11*x**11 + 23*x**10 + 22*x**9 - 68*x**8 + 24*x**7 + 112*x**6 - 160*x**5 - 64*x**4 + 384*x**3 - 256*x**2 - 512*x + 1024)/23 # and sometimes / 23^2
+ cx = (x**2 - x + 2) * (x**2 + x + 2) / (2 * 7 * 23)
+ tx = (x**12 + 45*x + 46)/46
+ yx = (x**12 - 4*x**11 - 47*x - 134)/322
+ m = 161
+ # D = 7 = 3 mod 4 -> Chi = x**2 + x + 2 has roots (-1+sqrt(-7))/2 and (-1 - sqrt(-7))/2
+ betax = (11*x**23 - 20*x**22 + 53*x**21 - 58*x**20 + 106*x**19 - 116*x**18 + 212*x**17 - 232*x**16 + 424*x**15 - 464*x**14 + 848*x**13 - 191*x**12 + 1322*x**11 + 1051*x**10 - 172*x**9 + 2102*x**8 - 344*x**7 + 4204*x**6 - 688*x**5 + 8408*x**4 - 1376*x**3 + 16816*x**2 + 19776*x + 36142)/18515 # mod p
+ lambx = (x**11 + 22)/23 # mod r
+ if choice is None or choice == 0:
+ u_mod_m = [32, 151] # and 7 | cxa = (x^2 - 2*x + 8) -> 2*u = 1 mod 7, 2*u mod 23 = [3, 18]
+ else:
+ u_mod_m = [4,18,25,39,81,95,116,123,144] # and 7 | cxa -> 2*u = 1 mod 7, 2*u mod 23 = [1, 8, 6, 13, 4, 2, 9, 16, 12]
+ rx = rx/23
+ cx = cx*23
+ #px = (x**24 - 2*x**23 + 8*x**22 + 137216*x**13 + 385024*x**12 + 1097728*x**11 + 8589934592*x**2 + 43595595776*x + 68719476736)/124252061696
+ #rx = (x**20 - 2*x**19 - 4*x**18 + 24*x**17 - 16*x**16 - 160*x**15 + 448*x**14 + 384*x**13 - 4352*x**12 + 5632*x**11 + 23552*x**10 + 45056*x**9 - 278528*x**8 + 196608*x**7 + 1835008*x**6 - 5242880*x**5 - 4194304*x**4 + 50331648*x**3 - 67108864*x**2 - 268435456*x + 1073741824)/24117248 # and sometimes 554696704
+ #cx = (x**2 - 2*x + 8) * (x**2 + 2*x + 8)/5152
+ #tx = (x**12 + 92160*x + 188416)/188416
+ #yx = (x**12 - 8*x**11 - 96256*x - 548864)/1318912
+ #m = 322
+ # Diskrep([2, 7, 23], {2: [(1, 0)], 7: [(1, 1), (1, 6)], 23: [(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (1, 8), (1, 9), (1, 12), (1, 13), (1, 16), (1, 18)]})
+ #u_mod_m = [6, 8, 36, 48, 50, 62, 64, 78, 104, 118, 146, 162, 174, 188, 190, 202, 216, 232, 246, 288, 300, 302]
+ # if u_mod_m = [64, 302] (u=1 mod 7) then rx, px can take prime values.
+ # if u_mod_m = [118,202] (u=6 mod 7) then rx can take prime values but 7 | px and 7 | cb = (x**2 + 2*x + 8)
+ # if u_mod_m = [8,36,50,78,162,190,232,246,288] (u=1 mod 7) then rx/23 and px can take prime values
+ # if u_mod_m = [6,48,62,104,146,174,188,216,300] (u=6 mod 7) then rx/23 can take prime values but 7 | px and and 7 | cxb
+ #if choice is None or choice==0:
+ # u_mod_m = [64, 302] # and 7 | cxa = (x**2 - 2*x + 8)
+ #else:
+ # u_mod_m = [8,36,50,78,162,190,232,246,288] # and 7 | cxa -> u = 0 mod 2, u = 1 mod 7, u mod 23 = [8, 13, 4, 9, 1, 6, 2, 16, 12]
+ # rx = rx/23
+ # cx = cx*23
+ else: #elif choice in [2,3]:
+ D = 7
+ px = (x**24 - 7*x**23 + 14*x**22 - 7**6*23*x**13 + 7**6*94*x**12 - 7**7*46*x**11 + 14**11*x**2 + 7**12*967*x + 14**12)/(2*7**12*67**2)
+ rx = (x**20 - 7*x**19 + 35*x**18 - 147*x**17 + 539*x**16 - 1715*x**15 + 4459*x**14 - 7203*x**13 - 12005*x**12 + 184877*x**11 - 1126069*x**10 + 2588278*x**9 - 2352980*x**8 - 19765032*x**7 + 171296944*x**6 - 922368160*x**5 + 4058419904*x**4 - 15495785088*x**3 + 51652616960*x**2 - 144627327488*x + 289254654976)/(7**10*67)
+ cx = (x**4 - 21*x**2 + 196)/6566
+ tx = (-x**12 + 45*7**6*x + 134*7**6)/(2*7**6*67)
+ yx = (-x**12 + 4*x**11 + 47*7**5*x - 46*7**6)/(2*7**6*67)
+ betax = (x**23 - 16*x**22 + 91*x**21 - 350*x**20 + 1274*x**19 - 4900*x**18 + 17836*x**17 - 68600*x**16 + 249704*x**15 - 960400*x**14 + 3495856*x**13 - 16151527*x**12 + 44941918*x**11 - 87648505*x**10 + 418359844*x**9 - 1227079070*x**8 + 5857037816*x**7 - 17179106980*x**6 + 81998529424*x**5 - 240507497720*x**4 + 1147979411936*x**3 - 3367104968080*x**2 + 20121276936768*x - 39068021788194)/44381098746635# root of x^2+x+2 mod px
+ lambx = (x**11 - 114*7**5)/(7**5*67)# root of x^2+x+2 mod rx
+ m = 5159
+ u_mod_m = [161, 1008, 1316, 1393, 2009, 2856, 2933, 3087, 3395, 3626, 4011]
+ if choice == 2:
+ u_mod_m = [2933, 3626]
+ else:
+ u_mod_m = [161, 1008, 1316, 1393, 2009, 2856, 3087, 3395, 4011]
+ rx = rx/67
+ cx = cx*67
elif D == 3:
D = 3
px = (x**24 - 6*x**23 + 21*x**22 - 37431234*x**13 + 223805916*x**12 - 786055914*x**11 + 350277500542221*x**2 - 2087000802936846*x + 7355827511386641)/267191528688
@@ -574,19 +592,35 @@ def poly_cofactor_twist_g1_g2(k, choice=None, D=None):
assert (E2_order % rx) == 0
g2cx = E2_order // rx # reducible with 3 factors
if D is None or D == 7:
- m = 161
- cxa = (x**2 - x + 2)/7
- cxb = (x**2 + x + 2)
- g2cxa = (x**2 + x + 2)
- g2cxb = (x**22 - 2*x**21 + 2*x**20 + 2*x**19 - 6*x**18 + 2*x**17 + 10*x**16 - 14*x**15 - 6*x**14 + 34*x**13 - 22*x**12 + 21*x**11 + 278*x**10 - 186*x**9 - 370*x**8 + 742*x**7 - 2*x**6 - 1482*x**5 + 1486*x**4 + 1478*x**3 - 4450*x**2 + 1494*x + 9454)/(7 * 23)
- if choice is None or choice==0:
- u_m = [32, 151]
- cxb = cxb/23
- g2cxa = g2cxa/23
- else:
- u_m = [4,18,25,39,81,95,116,123,144]
- g2cxb = g2cxb/23
- g2cxc = g2cx//(g2cxa*g2cxb)
+ if choice is None or choice in [0, 1]:
+ m = 161
+ cxa = (x**2 - x + 2)/7
+ cxb = (x**2 + x + 2)
+ g2cxa = (x**2 + x + 2)
+ g2cxb = (x**22 - 2*x**21 + 2*x**20 + 2*x**19 - 6*x**18 + 2*x**17 + 10*x**16 - 14*x**15 - 6*x**14 + 34*x**13 - 22*x**12 + 21*x**11 + 278*x**10 - 186*x**9 - 370*x**8 + 742*x**7 - 2*x**6 - 1482*x**5 + 1486*x**4 + 1478*x**3 - 4450*x**2 + 1494*x + 9454)/(7 * 23)
+ if choice is None or choice == 0:
+ u_m = [32, 151]
+ cxb = cxb/23
+ g2cxa = g2cxa/23
+ else:
+ u_m = [4,18,25,39,81,95,116,123,144]
+ g2cxb = g2cxb/23
+ g2cxc = g2cx//(g2cxa*g2cxb)
+ else: # if choice in [2, 3]:
+ m = 5159
+ # cx divided by 2 * 7^2 * 67
+ cxa = (x**2 - 7*x + 14)/14
+ cxb = (x**2 + 7*x + 14)/7
+ g2cxa = (x**2 + 7*x + 14)/14
+ g2cxb = (x**22 - 14*x**21 + 98*x**20 - 490*x**19 + 2058*x**18 - 7546*x**17 + 24010*x**16 - 62426*x**15 + 100842*x**14 + 168070*x**13 - 2588278*x**12 + 13059039*x**11 - 52000858*x**10 + 143296482*x**9 - 275063362*x**8 - 80707214*x**7 + 4415837566*x**6 - 29780961966*x**5 + 146645007838*x**4 - 609581587342*x**3 + 2214041001662*x**2 - 6964144788846*x + 21802004668318)/(7**11*67)
+ if choice == 2:
+ u_m = [2933, 3626]
+ cxb = cxb/67
+ g2cxa = g2cxa/67
+ else:
+ u_m = [161, 1008, 1316, 1393, 2009, 2856, 3087, 3395, 4011]
+ g2cxb = g2cxb/67
+ g2cxc = g2cx//(g2cxa*g2cxb)
elif D == 3:
cxa = (x**2 - 6*x + 21)/12
cxb = (x**2 + 6*x + 21)/84
@@ -795,11 +829,18 @@ def congruence_constraints(k, choice=None, D=None):
cofactor_r = [1, 1]
elif k == 22:
if D is None or D == 7:
- m = 161
- if choice is None or choice == 0:
- u_m = [32, 151]
- else:
- u_m = [4,18,25,39,81,95,116,123,144]
+ if choice is None or choice in [0,1]:
+ m = 161
+ if choice is None or choice == 0:
+ u_m = [32, 151]
+ else:
+ u_m = [4,18,25,39,81,95,116,123,144]
+ else: # choice in [2,3]
+ m = 5159
+ if choice == 2:
+ u_m = [2933, 3626]
+ else:
+ u_m = [161, 1008, 1316, 1393, 2009, 2856, 3087, 3395, 4011]
elif D == 3:
m = 2814
if choice is None or choice == 0:
--
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