1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
|
from sage.all import *
import coin
import const_precomp
class UV_params:
"""
Parameter set for running the algorithm and finding solutions to the coin
equation. By default, takes as input the securty level
(500|1000|1500|2000|4000) and proceed to setup the predefined parameters
for such level. Optionally, different parameters can be provided as a
dictionary.
Parameters:
- p: the underlying prime
- O: them maximal order Z[(pi + 1)/2]
- uv_e: the 2-torsion available for solving the norm equation; by default
is v_2(p+1) - 2;
- spr: precomputed rounded square root of p;
- allowed_primes: small elkies primes that can be removed from the norm
equation and computed separately; a different (heuristically) optimal set
is available for each level; to update the list, and all the associated
parameters, use the functions `set_allowed_primes` and
`add_allowed_prime` described below;
- allowed_primes_prod: the product of all allowed_primes;
- sos_allowed_primes: primes that can be removed from the sum of squares
step; consists of the primes 3 mod 4 in allowed_primes;
- norm_bound: scaling factor used to bound the norms of the short vectors
sampled from the ideal lattice; only vectors with norm smaller than
norm_bound*spr are considered;
- norm_cff_bound: short vectors are computed as linear combinations of an
LLL basis; this value bound the coefficients of the combinations; by
default is the (rounded) square root of norm_bound;
- comb_bound: the number of short vectors that are combined to try to find
a solution to the coin equation; by default is 10, meaning 100
combinations are tried;
- n_squares: how many between u and v must be sums of squares; default 2;
- sol_bound: how many solutions to try before returning the best one found;
default 1;
Methods:
- set_allowed_primes(ls): takes in input a list `ls` and set
`self.allowed_primes` to the Elkies primes contained in `ls`; updates the
related parameters accordingly;
- add_allowed_prime(ell): if the given prime `ell` is Elkies, it is added
to `self.allowed_primes` and the function returns True; otherwise, it
returns False.
"""
def __init__(self, level, params=None):
level = int(level)
if level == 500:
self.init_500()
elif level == 1000:
self.init_1000()
elif level == 1500:
self.init_1500()
elif level == 2000:
self.init_2000()
elif level == 4000:
self.init_4000()
else:
raise ValueError('unknown level')
# Arithmetic
self.Fp = GF(self.p)
self.Fp2 = GF((self.p, 2), name='i', modulus=var('x')**2 + 1)
self.K = NumberField(name="pi", polynomial = var('x')**2 + self.p)
self.w = self.K.gens()[0]
self.max_order = self.K.maximal_order()
self.order = self.K.order_of_conductor(2)
# Starting curve
E_start = EllipticCurve(self.Fp, [0, self.A, 0, 1, 0])
self.E_start = E_start
# UV parameters
self.uv_e = self.e - 3
self.spr = isqrt(self.p)
p_3mod4 = list(filter(lambda x: x%4 == 3, self.allowed_primes))
self.sos_allowed_primes = p_3mod4
self.allowed_primes_prod = prod(self.allowed_primes)
self.norm_bound = ZZ(self.p).nbits()
self.norm_cff_bound = isqrt(self.norm_bound) + 1
self.comb_bound = 10
self.n_squares = 2
self.sol_bound = 1
# Optional parameter update
if params:
for p_key, p_val in params.items():
setattr(self,p_key,p_value)
def init_500(self):
self.f = 33
self.e = 503
self.p = self.f * 2**self.e - 1
self.A = 846923981206860774667188923127927404866138431504857441785556155002892474407686465068998563030997400041497236660113832719116298437902864009587446028542241
allowed_primes = [3, 7, 11, 13]
self.allowed_primes = [ZZ(li) for li in allowed_primes]
def init_1000(self):
self.f = 15
self.e = 1004
self.p = self.f * 2**self.e - 1
self.A = 2447932165031884901753526747235802754200581132492013990029309589916334680445261935406178846882142279733036010435356728900482083533977052489550706097041090570437350041707592382243329641233414772511433265807801169694848478362818691704488235495552842938554477902999119670514198118427921959251521459174283757
allowed_primes = [3, 5, 7, 11]
self.allowed_primes = [ZZ(li) for li in allowed_primes]
def init_1500(self):
self.f = 9
self.e = 1551
self.p = self.f * 2**self.e - 1
self.A = 470690138325573302381284028925922066259197385545434613961546293117200415888454393210985745811672527707750649809464293986828934656433695254827553343115710988038803620829829607213986283428022502352610750230825225753910990410861379903133396479857538580518638524776922388484859923418301448669856130467309798601801912903218678382572894516230515348525582714674329329665444893417613957336453258663721257841338989369604936267692408989319582217755257612152597484632197929506422
allowed_primes = [3, 5, 11]
self.allowed_primes = [ZZ(li) for li in allowed_primes]
coin.good_prime_prod = const_precomp.good_prime_prod_100k
coin.bad_prime_prod = const_precomp.bad_prime_prod_100k
def init_2000(self):
self.f = 51
self.e = 2026
self.p = self.f * 2**self.e - 1
self.A = 81522175295034447595829362752089781064926402323585792843903881571299469475765731013589255502541646344652265632612080053850583985808199619939412577427961666539089410884159333672468730652939947531656027845338622561223839271607859219023997914401035458486761172816051586319123853202351034903432110329760972693423592535540529210197485000054794934251253577983683962395651835481200021277459090885409493264948676810331609354889543129654381859017748250839312739412603528665722325984598111481835819726605526313101275796618189346971789271155935744998546622047982841689962568590619138079006213242588187631607235805859652542
allowed_primes = [3, 7, 11, 17]
self.allowed_primes = [ZZ(li) for li in allowed_primes]
coin.good_prime_prod = const_precomp.good_prime_prod_100k
coin.bad_prime_prod = const_precomp.bad_prime_prod_100k
def init_4000(self):
self.f = 63
self.e = 4084
self.p = self.f * 2**self.e - 1
self.A = 3861182440791393427115834319919792363082397953235366672989642232032664601358637530587592475724492228157994152478762370779212684014665350275282740708992301403517860204937961301178353993394123923639955895951016030909047562464435151058330155507538738419078593938731652956043256633693827720453080631845604091259290706126622597714437085574776283248730437225629704782127771641525437187101408298218133833634726212421345805005641803578982727095077266614175977073363364911048284637511496334424899430835066269145012363608931646817082193451123867313226197275905378376442493746564842029440001132446848573232704980649005361798250530086069725335482701101306335776380418104681645935311890062938192551774656426376713685296742945987418112888455314815190689399358568871101254106213839110567602195750401800211431397684678743127540890669195457658925252532743828619079040054116684464335116945294570022631103926522225020600966809347462518428416543333615647560326220108209234883366833121125898408957117320377370772129934485893908787438550276617471673661822552984223791369029779538709395655155777458500406072451055588404055831790775259870437154343571081246432780811947116682159194253450328736841478964648801067542070339026439293541698676665959781451048203
allowed_primes = [3, 7, 11, 17, 19]
self.allowed_primes = [ZZ(li) for li in allowed_primes]
coin.good_prime_prod = const_precomp.good_prime_prod_100k
coin.bad_prime_prod = const_precomp.bad_prime_prod_100k
def set_allowed_primes(self, allowed_primes):
"""
Set the list of allowed primes to a given list by filtering non-Elkies
primes, and update parameters accordingly.
"""
self.allowed_primes = list(filter(
lambda ell: kronecker(-self.p, ell) == 1, allowed_primes))
self.allowed_primes_prod = prod(self.allowed_primes)
p_3mod4 = list(filter(lambda x: x%4 == 3, self.allowed_primes))
self.sos_allowed_primes = p_3mod4
def add_allowed_prime(self, ell):
"""
Check if ell is an elkies prime; if so, add it to the list of allowed
primes, update the related parameters accordingly, and return True;
otherwise, return False.
"""
if ell in self.allowed_primes or kronecker(-self.p, ell) != 1:
return False
self.allowed_primes.append(ell)
self.allowed_primes_prod *= ell
if ell%4 == 3:
self.sos_allowed_primes.append(ell)
return True
|
