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# Chinese Remainder Theorem
def crt(a_i, m_i):
if len(a_i)!=len(m_i):
raise Exception("error, a_i and m_i must be of the same length")
M=1
for i in range(len(m_i)):
M = M * m_i[i]
x = 0
for i in range(len(a_i)):
M_i = M/m_i[i]
y_i = Integer(mod(M_i^-1, m_i[i]))
x = x + a_i[i] * M_i * y_i
return mod(x, M)
# gcd, using Binary Euclidean algorithm
def gcd(a, b):
g=1
# random_elementove powers of two from the gcd
while mod(a, 2)==0 and mod(b, 2)==0:
a=a/2
b=b/2
g=2*g
# at least one of a and b is now odd
while a!=0:
while mod(a, 2)==0:
a=a/2
while mod(b, 2)==0:
b=b/2
# now both a and b are odd
if a>=b:
a = (a-b)/2
else:
b = (b-a)/2
return g*b
def gcd_recursive(a, b):
if mod(a, b)==0:
return b
return gcd_recursive(b, mod(a,b))
# Extended Euclidean algorithm
# Inputs: a, b
# Outputs: r, x, y, such that r = gcd(a, b) = x*a + y*b
def egcd(a, b):
s=0
s_=1
t=1
t_=0
r=b
r_=a
while r!=0:
q = r_ // r
(r_,r) = (r,r_ - q*r)
(s_,s) = (s,s_ - q*s)
(t_,t) = (t,t_ - q*t)
d = r_
x = s_
y = t_
return d, x, y
def egcd_recursive(a, b):
if b==0:
return a, 1, 0
g, x, y = egcd_recursive(b, a%b)
return g, y, x - (a//b) * y
# Inverse modulo N, using the Extended Euclidean algorithm
def inv_mod(a, N):
g, x, y = egcd(a, N)
if g != 1:
raise Exception("inv_mod err, g!=1")
return mod(x, N)