def gcd(e, z):
if z == 0:
return e
else:
return gcd(z, e % z)
e = int(input("Please enter the first number:"))
z = int(input("Please enter the second number:"))
print ("The GCD of ",e," and ",z," is ",gcd(e,z))
d = 1
while d < e:
if d * e == 1 % z:
print (d," * ",e," = 1 (mod ",z,")")
d = d + 1
else:
d = d + 1
I am trying to use this code to find candidates for rsa by brute force it seems like it should work but it doesn't can anyone help me out?
z = (p − 1)(q − 1)
for the calculation of z is used prior to this
with p = 47 and q = 59, e = 17 and d = 157 but after the program runs it finds no matches but it should.
Where you have
if d * e == 1 % z:
you seem to want to check "Is d*e equal (mod z) to 1"
But what you're doing is performing 1 % z (which gives 1) and checking if d * e is equal to 1.
Change it to:
if (d*e) % z == 1:
and it should do the calculation you intend.
some problems from your code:
1) if z = (p-1)(q-1), d*e = 1 (mod z), then z and e are coprime, gcd(e, z) will always equal 1, see
2) if you say d*e = 1 (mod z), the code should be if d * e % z == 1
Related
For context, I am trying to find all viable solutions to this question:
here's the code I have so far, but I am having trouble with the part that is supposed to iterate through every possible combination.
x = 1
y = 1
z = 10
a = 10
while x < 10 and y < 10 and z < 100 and a < 100: #iterates through every possible combination
x = x + 1
y = y + 1
z = z + 1
a = a + 1
if x != y: #checks if x and are the same
if a/x == z/y or z/x == a/y: #checks if x and y are proportional to a and z
a = str(a) #converting each int to string
z = str(z)
x = str(x)
y = str(y)
if a.count(x) < 1 and a.count(y) < 1 and z.count(y) <1 and z.count(x) < 1: #checks if any number reapeats
print(x, y, z, a) #prints viable solution```
You have six boxes to fill. Just run through all permutations of 6 members and check the conditions:
import itertools
for a,b,c,d,e,f in itertools.permutations([0,1,2,3,4,5,6,7,8,9],6):
if a == 0 or b == 0 or c == 0 or e == 0:
continue
if (10*c + d) / a == (10*e + f) / b:
print( a, b, c*10+d, e*10+f )
It looks like there are 57 solutions.
So I am very unexperienced with Python, I know basically nothing, and our teacher gave us the task to write a code that makes a partial fraction decomposition with this function:
I don't really know how to start or even how to define that function. I tried this at first: `
def function(x):
a = (x^4)-(3*x^2)+x+5
b = (x^11)-(3*x^10)-(x^9)+(7*x^8)-(9*x^7)+(23*x^6)-(11*x^5)-(3*x^4)-(4*x^3)-(32*x^2)-16
return a/b
But our maths script says that we need to split up the denominator and then make a system of equations out of it and solve it.
So I was thinking about defining each part of the function itself and then make a function somehow like a = 7*x and use it like f(x) = b/a^7 if this works but I don't really know. We are unfortunately not allowed to use "apart" which I think is a sympy-function?
Thank you so much in advance!
Sincerely, Phie
Addition: So after a few hours of trying I figured this. But I am very sure that this is not the way to do it. Also it tells me that variable l is not defined in z and I am sure that all the others aren't as well. I don't know what to do.
def function(x):
global a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v
a = (x^4)-(3*x^2)+x+5
b = 11
c = 10
d = 9
e = 8
f = 7
g = 6
h = 5
i = 4
j = 3
k = 2
l = x**b
m = 3*x**c
n = x**d
o = 7*x**e
p = 9*x**f
q = 23*x**g
r = 11*x**h
s = 3*x**i
t = 4*x**j
u = 32*x**k
v = 16
return a/(l-m-n+o-p+q-r-s-t-u-v)
print("We are starting the partial fraction decomposition with this
function: (x^4)-(3*x^2)+x+5 / (x^11)-(3*x^10)-(x^9)+(7*x^8)-(9*x^7)+
(23*x^6)-(11*x^5)-(3*x^4)-(4*x^3)-(32*x^2)-16")
z = l-m-n+o-p+q-r-s-t-u-v
while c >= 0:
c = c-1
z = z-l
while d >= 0:
d = d-1
z = z-m
while e >= 0:
e = e-1
z = z-n
while f >= 0:
f = f-1
z = z+o
while g >= 0:
g = g-1
z = z-p
while h >= 0:
h = h-1
z = z+q
while i >= 0:
i = i-1
z = z-r
while j >= 0:
j = j-1
z = z-s
while k >= 0:
k = k-1
z = z-t
print(z)
Since I just solved this myself, here's some input:
Let poly = function() for your function, although be careful to replace ^ with **. Include both from sympy import * and from sympy.abc import a, b, c, d, e, f, g, h, i, j, k, x.
Using factor(exp) you can find all the roots of your function, use these to define the 11 terms term_1 = a/(x-2), term_2 = b/(x2-)**2, ... , term_6 = (f*x + g)/(x**2 +1), ..., term_8 = (j*x + k)/(x**2 + 1) (you get the gist). Define your_sum = term_1 + ... + term_8, eq = Eq(your_sum, poly)
Define the variable your_sum = sum(term_1, ..., term_8), and use solve_undetermined_coeffs(eq, [a,b, ..., k], x))) to get the result.
I'm a bit stuck on a python problem.
I'm suppose to write a function that takes a positive integer n and returns the number of different operations that can sum to n (2<n<201) with decreasing and unique elements.
To give an example:
If n = 3 then f(n) = 1 (Because the only possible solution is 2+1).
If n = 5 then f(n) = 2 (because the possible solutions are 4+1 & 3+2).
If n = 10 then f(n) = 9 (Because the possible solutions are (9+1) & (8+2) & (7+3) & (7+2+1) & (6+4) & (6+3+1) & (5+4+1) & (5+3+2) & (4+3+2+1)).
For the code I started like that:
def solution(n):
nb = list(range(1,n))
l = 2
summ = 0
itt = 0
for index in range(len(nb)):
x = nb[-(index+1)]
if x > 3:
for index2 in range(x-1):
y = nb[index2]
#print(str(x) + ' + ' + str(y))
if (x + y) == n:
itt = itt + 1
for index3 in range(y-1):
z = nb[index3]
if (x + y + z) == n:
itt = itt + 1
for index4 in range(z-1):
w = nb[index4]
if (x + y + z + w) == n:
itt = itt + 1
return itt
It works when n is small but when you start to be around n=100, it's super slow and I will need to add more for loop which will worsen the situation...
Do you have an idea on how I could solve this issue? Is there an obvious solution I missed?
This problem is called integer partition into distinct parts. OEIS sequence (values are off by 1 because you don't need n=>n case )
I already have code for partition into k distinct parts, so modified it a bit to calculate number of partitions into any number of parts:
import functools
#functools.lru_cache(20000)
def diffparts(n, k, last):
result = 0
if n == 0 and k == 0:
result = 1
if n == 0 or k == 0:
return result
for i in range(last + 1, n // k + 1):
result += diffparts(n - i, k - 1, i)
return result
def dparts(n):
res = 0
k = 2
while k * (k + 1) <= 2 * n:
res += diffparts(n, k, 0)
k += 1
return res
print(dparts(201))
Is there a function in Sage which returns the fundamental discriminant associated to a given discriminant?
See https://en.wikipedia.org/wiki/Fundamental_discriminant
This is the function I wrote, not finding an existing one:
def getFund(D):
if D % 4 == 2 or D % 4 == 3:
raise ValueError("Not a discriminant.")
if D == 0:
raise ValueError("There is no fundamental associated to 0.")
P = sign(D)
for p in factor(D):
if p[1] % 2 == 1:
P *= p[0]
if P % 4 == 2 or P % 4 == 3:
P *= 4
return P
What's the problem with fundamental_discriminant?
sage: fundamental_discriminant(1)
1
sage: fundamental_discriminant(3)
12
sage: fundamental_discriminant?
Signature: fundamental_discriminant(D)
Docstring:
Return the discriminant of the quadratic extension K=Q(sqrt{D}),
i.e. an integer d congruent to either 0 or 1, mod 4, and such that,
at most, the only square dividing it is 4.
INPUT:
* "D" - an integer
OUTPUT:
* an integer, the fundamental discriminant
I don't know if the function is implemented in SageMath.
But if I understand the definition correctly, you can define the function as follows:
def fundamental_discriminant(d):
if d % 4 == 1:
return d.squarefree_part()
if d % 4 == 0:
k = d.valuation(4)
dd = d // 4^k
if dd % 4 == 1:
return dd.squarefree_part()
if dd % 4 == 2:
return 4 * dd.squarefree_part()
if dd % 4 == 3:
return 4 * dd.squarefree_part()
raise ValueError("Not a discriminant.")
I have Miller-Rabin implemntation
def MillerRabin(n,a):
e = 0
q = n-1
while q % 2 == 0:
e += 1
q = q/2
if a**q % n == 1:
return 1
for i in range(e):
if a ** (q * 2 ** i) % n == n-1:
return 1
return 0
(n, minA, maxA) = map(int, sys.argv[1:4])
print [MillerRabin(n, a) for a in range(minA,maxA)]
There are three inputs: number, min-base, max-base. Function works fine when number is low. But when number is way too big, I get an error (test case is number = 12530759607784496010584573923, min-base = 16, max-base = 32)
exponent must be at most 9223372036854775807
Use the builtin pow function. It can take an optional mod parameter
>>> help(pow)
Help on built-in function pow in module __builtin__:
pow(...)
pow(x, y[, z]) -> number
With two arguments, equivalent to x**y. With three arguments,
equivalent to (x**y) % z, but may be more efficient (e.g. for longs).
def MillerRabin(n, a):
e = 0
q = n-1
while q % 2 == 0:
e += 1
q = q // 2
if pow(a, q, n) == 1:
return 1
for i in range(e):
if pow(a , (q * 2 ** i) , n) == n - 1:
return 1
return 0