Question

I wrote up the ruby script below to help my students understand public key encryption. I followed the “pencil and paper” method shown here: http://sergematovic.tripod.com/rsa1.html

This works fine as long as 29 isn’t chosen as either p or q. If 29 is picked, it hangs on calculating the secret key. Can anyone tell me why that is?

#!/usr/bin/env ruby -wKU

#initialize
primes, p, q, n, z, k, j, m,e,d = nil

def prime
  primes = [2,3,5,7,11,13,17,19,23,29,31]
  primes.sample
end

#pick p
p= prime
puts "p: " + p.to_s

#pick q
q=p
while p==q
  q = prime
end
puts "q: " + q.to_s

#find n
n=p*q
puts "n: " + n.to_s

#find z
z=(p-1)*(q-1)
puts "z: " + z.to_s

#pick a relative prime of the totient
k=7

puts "k: " + k.to_s

#calculate secret key
j=0
while j*k % z != 1
  j+=1
end

puts "j: " + j.to_s
#message
m=16
puts "Message: " + m.to_s

#encrypt
e = m**k % n
puts "Encrypted: " + e.to_s

#decrypt
d = e**j % n
puts "Decrypted: " + d.to_s

Answer 1

When 29 is chosen as p or q, z has 28 as a factor, and thus k = 7 is not a relative prime of the totient as your comment claims!

(This means j*k % z is always a multiple of 7, so your loop never terminates.)