title: Understanding RSA broadcast attack
body: <p>I am trying to understand the RSA broadcast attack, and have become quite confused. Maybe someone on here can help me to understand and learn how to solve a problem.
  I have been given a message that was encrypted with three individual RSA public keys (N1,N2,N3), resulting in three cypher texts (C1,C2,C3). The public exponent e=3.
  I understand from Hastads's broadcast attack that:     </p>
  
  <p>c1≡m<sup>3</sup> mod N1,<br>
  c2≡m<sup>3</sup> mod N2,<br>
  c3≡m<sup>3</sup> mod N3.</p>
  
  <p>Using the Chinese remainder theorem, it should be possible to find a fourth cipher text value C4 that has the following properties:<br>
  c4≡c1 mod N1,<br>
  c4≡c2 mod N2,<br>
  c4≡c3 mod N3,   </p>
  
  <p>c4≡m<sup>3</sup> mod (N1 N2 N3)</p>
  
  <p>In theory, I should be able to solve for <code>m=∛c4</code>, or (<code>m=cube root of c4</code>).
  Here is where I'm stuck, and my woeful understanding of math is no help.</p>
  
  <p>Can I simply solve for c4 by letting <code>c4= C1 mod N1</code>? Using the values provided to me (actual hex numbers)?  And then use this c4 value in <code>m=∛c4</code> equation? This seems way too simple, and I don't think it will produce the correct value. 
  Any help is greatly appreciated; I'm trying to solve the problem, but also wish to gain a better understanding of Hastads attack and Chinese remainder theorem. Thank you!</p>