The Theorem that Won the War: Activities for Part 3.3 (Breaking the Full Enigma)

Jeff Suzuki (Brooklyn College)


Note:  If you want to produce your own Enigma encryptions, there are several online Enigma emulators, such as this one.

  1. Suppose the following message keys are intercepted:\[\begin{array}{lllllll}rsw \,\,\,\,     sia               & \,\,\,\,\,\,\,\,\,\,&pos \,\,\,\, eyc               &\,\,\,\,\,\,\,\,\,\,&zlo \,\,\,\,qks               &\,\,\,\,\,\,\,\,\,\,&qju \,\,\,\,fdv   \\   wyn \,\,\,bci &&ylo \,\,\,\,dks        &&vll \,\,\,\,okm    &&mwn \,mxi\\ktz \,\,\,\,ygl    \,&&llx \,\,\,\,pkz      &&iaf \,\,\,vqp         &&hln \,\,\,\,hki  \\  ohb \,\,\,zwt &&nwl\, gxm     &&blw \,\,\,jka       &&dll\,\,\,\,ckm\\all \,\,\,\,\,wkm   &&xln \,\,aki       &&uwl \,\,uxm     &&gll \,\,\,\,\,kkm\\eln \,\,\,rki       &&slk \,\,\,lkr             &&flx \,\,\,ikz               &&cle \,\,\,\,\,nkk\\pln \,\,\,eki    &&jlb \,\,\,xkt       &&spq \,\,\,lax        &&rry \,\,\,\,slb\end{array}\]
    1. Find as much of the cycle decomposition of \(E_{4}E_{1}\) as possible.
    2. Find as much of the cycle decomposition of \(E_{5}E_{2}\) as possible.
    3. Find as much of the cycle decomposition of \(E_{6}E_{3}\) as possible.
  2. Suppose \(E_{4}\) and \(E_{1}\) are proper involutions. Prove the following.
    1. If both contain a transposition \((\alpha\beta)\), then \(\alpha, \beta\) will not appear in any cycle in the product \(E_{4}E_{1}\).
    2. If \(E_{1}\) contains the transposition \((\alpha\beta)\) and \(E_{4}\) contains the transposition \((\alpha\gamma)\), then \(E_{4}E_{1}\) contains a cycle that includes \((\ldots \beta\gamma \ldots)\).
  3. Most histories of cryptography claim that the allied decryption efforts were easier because Enigma operators didn't use randomly chosen letters, but rather used sequences that were easy to type.
    1. Suppose all operators used the message code \(aaa\, aaa\). If this was the message code used by all operators, would it be harder or easier to recover \(E_{4}E_{1}\), and subsequently the Enigma encryptions \(E_{4}\) and \(E_{1}\)?  Why/why not?
    2. What if instead of random letter sequences, operators chose three letter words?  Use the Enigma emulator to encrypt repeated three letter words as message codes. Would this practice make it easier or harder to recover the Enigma encryptions?  Why/why not?
  4. Some histories of cryptography claim that the Germans ended their messages with “Heil Hitler!” This isn’t true, but suppose it was. Would the use of such a standard closing message make it easier to break Enigma? Why/why not?  Suggestion: The first “H” would be encrypted using \(E_{k}\) for some value \(k\) that depended on the message length. Could this be used to recover \(E_{k}\)?

Return to the overview of Part 3.3 (Breaking the Full Enigma).

Continue to the Conclusion.