-
Notifications
You must be signed in to change notification settings - Fork 585
/
subquadratic_poly_utils.py
198 lines (174 loc) · 6.43 KB
/
subquadratic_poly_utils.py
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
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
modulus_poly = [1, 0, 0, 0, 0, 0, 0, 0,
0, 0, 1, 0, 1, 0, 0, 1,
1]
modulus_poly_as_int = sum([(v << i) for i, v in enumerate(modulus_poly)])
degree = len(modulus_poly) - 1
two_to_the_degree = 2**degree
two_to_the_degree_m1 = 2**degree - 1
def galoistpl(a):
# 2 is not a primitive root, so we have to use 3 as our logarithm base
if a * 2 < two_to_the_degree:
return (a * 2) ^ a
else:
return (a * 2) ^ a ^ modulus_poly_as_int
# Precomputing a log table for increased speed of addition and multiplication
glogtable = [0] * (two_to_the_degree)
gexptable = []
v = 1
for i in range(two_to_the_degree_m1):
glogtable[v] = i
gexptable.append(v)
v = galoistpl(v)
gexptable += gexptable + gexptable
# Add two values in the Galois field
def galois_add(x, y):
return x ^ y
# In binary fields, addition and subtraction are the same thing
galois_sub = galois_add
# Multiply two values in the Galois field
def galois_mul(x, y):
return 0 if x*y == 0 else gexptable[glogtable[x] + glogtable[y]]
# Divide two values in the Galois field
def galois_div(x, y):
return 0 if x == 0 else gexptable[(glogtable[x] - glogtable[y]) % two_to_the_degree_m1]
# Evaluate a polynomial at a point
def eval_poly_at(p, x):
if x == 0:
return p[0]
y = 0
logx = glogtable[x]
for i, p_coeff in enumerate(p):
if p_coeff:
# Add x**i * coeff
y ^= gexptable[(logx * i + glogtable[p_coeff]) % two_to_the_degree_m1]
return y
# Given p+1 y values and x values with no errors, recovers the original
# p+1 degree polynomial.
# Lagrange interpolation works roughly in the following way.
# 1. Suppose you have a set of points, eg. x = [1, 2, 3], y = [2, 5, 10]
# 2. For each x, generate a polynomial which equals its corresponding
# y coordinate at that point and 0 at all other points provided.
# 3. Add these polynomials together.
def lagrange_interp(pieces, xs):
# Generate master numerator polynomial, eg. (x - x1) * (x - x2) * ... * (x - xn)
root = mk_root_2(xs)
#print(root)
assert len(root) == len(pieces) + 1
# print(root)
# Generate the derivative
d = derivative(root)
# Generate denominators by evaluating numerator polys at each x
denoms = multi_eval_2(d, xs)
print(denoms)
# denoms = [eval_poly_at(d, xs[i]) for i in range(len(xs))]
# Generate output polynomial, which is the sum of the per-value numerator
# polynomials rescaled to have the right y values
factors = [galois_div(p, d) for p, d in zip(pieces, denoms)]
o = multi_root_derive(xs, factors)
# print(o)
return o
def multi_root_derive(xs, muls):
if len(xs) == 1:
return [muls[0]]
R1 = mk_root_2(xs[:len(xs) // 2])
R2 = mk_root_2(xs[len(xs) // 2:])
x1 = karatsuba_mul(R1, multi_root_derive(xs[len(xs) // 2:], muls[len(muls) // 2:]) + [0])
x2 = karatsuba_mul(R2, multi_root_derive(xs[:len(xs) // 2], muls[:len(muls) // 2]) + [0])
o = [v1 ^ v2 for v1, v2 in zip(x1, x2)][:len(xs)]
# print(len(R1), len(x1), len(xs), len(o))
return o
def multi_root_derive_1(xs, muls):
o = [0] * len(xs)
for i in range(len(xs)):
_xs = xs[:i] + xs[(i+1):]
root = mk_root_2(_xs)
for j in range(len(root)):
o[j] ^= galois_mul(root[j], muls[i])
return o
a = 124
b = 8932
c = 12415
assert galois_mul(galois_add(a, b), c) == galois_add(galois_mul(a, c), galois_mul(b, c))
def karatsuba_mul(p1, p2):
L = len(p1)
# assert L == len(p2)
if L <= 16:
o = [0] * (L * 2)
for i, v1 in enumerate(p1):
for j, v2 in enumerate(p2):
if v1 and v2:
o[i + j] ^= gexptable[glogtable[v1] + glogtable[v2]]
return o
if L % 2:
p1 = p1 + [0]
p2 = p2 + [0]
L += 1
halflen = L // 2
low1 = p1[:halflen]
high1 = p1[halflen:]
sum1 = [l ^ h for l, h in zip(low1, high1)]
low2 = p2[:halflen]
high2 = p2[halflen:]
sum2 = [l ^ h for l, h in zip(low2, high2)]
z2 = karatsuba_mul(high1, high2)
z0 = karatsuba_mul(low1, low2)
z1 = [m ^ _z0 ^ _z2 for m, _z0, _z2 in zip(karatsuba_mul(sum1, sum2), z0, z2)]
o = z0[:halflen] + \
[a ^ b for a, b in zip(z0[halflen:], z1[:halflen])] + \
[a ^ b for a, b in zip(z2[:halflen], z1[halflen:])] + \
z2[halflen:]
return o
def mk_root_1(xs):
root = [1]
for x in xs:
logx = glogtable[x]
root.insert(0, 0)
for j in range(len(root)-1):
if root[j+1] and x:
root[j] ^= gexptable[glogtable[root[j+1]] + logx]
return root
def mk_root_2(xs):
if len(xs) >= 128:
return karatsuba_mul(mk_root_2(xs[:len(xs) // 2]), mk_root_2(xs[len(xs) // 2:]))[:len(xs) + 1]
root = [1]
for x in xs:
logx = glogtable[x]
root.insert(0, 0)
for j in range(len(root)-1):
if root[j+1] and x:
root[j] ^= gexptable[glogtable[root[j+1]] + logx]
return root
def derivative(root):
return [0 if i % 2 else r for i, r in enumerate(root[1:])]
# Credit to http://people.csail.mit.edu/madhu/ST12/scribe/lect06.pdf for the algorithm
def xn_mod_poly(p):
if len(p) == 1:
return [galois_div(1, p[0])]
halflen = len(p) // 2
lowinv = xn_mod_poly(p[:halflen])
submod_high = karatsuba_mul(lowinv, p[:halflen])[halflen:]
med = karatsuba_mul(p[halflen:], lowinv)[:halflen]
med_plus_high = [x ^ y for x, y in zip(med, submod_high)]
highinv = karatsuba_mul(med_plus_high, lowinv)
o = (lowinv + highinv)[:len(p)]
print(halflen, lowinv, submod_high, med, highinv)
# assert karatsuba_mul(o, p)[:len(p)] == [1] + [0] * (len(p) - 1)
return o
def mod(a, b):
assert len(a) == 2 * (len(b) - 1)
L = len(b)
inv_rev_b = xn_mod_poly(b[::-1] + [0] * (len(a) - L))[:L]
quot = karatsuba_mul(inv_rev_b, a[::-1][:L])[:L-1][::-1]
subt = karatsuba_mul(b, quot + [0])[:-1]
o = [x ^ y for x, y in zip(a[:L-1], subt[:L-1])]
# assert [x^y for x, y in zip(karatsuba_mul(quot + [0], b), o)] == a
return o
def multi_eval_1(poly, xs):
return [eval_poly_at(poly, x) for x in xs]
def multi_eval_2(poly, xs):
if len(xs) <= 1024:
return [eval_poly_at(poly, x) for x in xs]
halflen = len(xs) // 2
return multi_eval_2(mod(poly, mk_root_2(xs[:halflen])), xs[:halflen]) + \
multi_eval_2(mod(poly, mk_root_2(xs[halflen:])), xs[halflen:])
# [eval_poly_at(poly, xs[-2]), eval_poly_at(poly, xs[-1])]