#
**Xli wigvix gsrjiwwmsr sj xli epqmklxc vypiv**

*To all those who persist in the face of adversity, **as** **insurmountable as it may seem, until they find the truth*

AEW XLEX LMQ? M xlsyklx M geyklx e kpmqtwi sj lmq mr xli qmhhpi sj xli fpsshxlmvwxc lsvhi gsqmrk yt xli wxemvw. M waiev li pssoih e psx pmoi lmq, ex piewx jvsq e hmwxergi. M asyph izir wec xlex M wea e kpmqtwi sj vigskrmxmsr mr lmw iciw ew aipp. Qc lsti, fsvr sj hiwtivexmsr erh qehriww, xsso vijyki mr xli viqsxi tswwmfmpmxc xlex M aew avsrk erh xlex mx aew rsx lmq ejxiv epp, sv xlex li aew epws qmwxeoir, erh xlex M aew rsx qcwipj imxliv.

Xlic gsrxmryih xs ehzergi viwspyxipc xsaevhw qi erh, ew li geqi gpswiv, mx figeqi temrjyppc sfzmsyw xlex mx aew lmq, erh xlex li aew mr rs hsyfx efsyx als M aew. Alir M wyhhirpc jsyrh qcwipj wyvvsyrhih fc xli mrxmqmhexmrk, herkivsyw kvsyt, li mrhiih xyvrih syx xs fi als M xlsyklx li aew, erh li oria zivc aipp als M aew.

Xli Fexxpi sj Dipe erh qc jeqsyw "Zirm, zmhm, zmgm" wiiqih wygl e psrk aec sjj rsa. Xshec M aew ksmrk xs pswi. Alex gsyph M wec xs hmkrmjc wygl e qiqsvefpi sggewmsr? "Epie megxe iwx" wiiqih xss sfzmsyw erh vitixmxmzi. Xli srpc xlmrk xlex sggyvvih xs qi aew xli xihmsyw "Ix xy, Fvyxi?" almgl wyvipc rs sri ampp iziv viqiqfiv.

Izir xli qmrh sj e fvmppmerx, ewxyxi erh pygmh wtieoiv wygl ew M wleqijyppc erh yrgsrjiwwefpc gpsyhih sziv alir mx wirwih xlex MX AEW GSQMRK XS ER IRH.

## Mathematical note

We humans are naturally curious beings. Our interest in the world around us has been fundamental to our species and our evolution to what we are now. However, this thirst for knowledge has its dark side. All you need is two people to decide to engage in a personal conversation for a third to succumb to the temptation to eavesdrop on their private communication.

To defend the rights of the two people having the conversation, coders develop mechanisms and methods to encrypt communication so that it can be kept private. To help the third, the busybody, decoders strive to find mechanisms and methods to decipher the private talk between the first two without having permission to do so.

The story of encoding (and decoding) is very interesting and full of anecdotes. A classical encoding method was already in use in the time of Julius Caesar. It consisted of changing each letter of the original (and therefore uncoded) text for another letter, thus producing an encoded text. What this encoding method does is to change each letter of the original alphabet for another letter four positions down the alphabet. The coding table used by Julius Caesar was this one:

To encode texts, you need to find the letter you want to encode in the first row and change it for the corresponding letter in the second row. To decode texts, all you have to do is find the letter in the second row and change it for the corresponding letter in the first row. For example, to decipher a text, the encoded letter 'E' corresponds to the uncoded letter 'A'; the encoded letter 'F' corresponds to the uncoded 'B'; etc. Julius Caesar must have felt confident that the task of decoding his text without knowing the table was sufficiently difficult to discourage nosy decoders.

This method of encrypting texts can be defined from a mathematical point of view. There are two sets (the set of uncoded letters and the set of encoded letters) and a relation between the two sets (each letter is moved four positions down the alphabet). A decoder who did not know the relation between the two sets would seek to find it.

One feasible technique with Julius Caesar's encryption method is to analyze the frequencies of the letters that appear in the encoded text. For example, the nosy decoder would notice that the letter 'I' appears most often in the encoded story you have tried to read, and that the most commonly used letter in English is the letter 'E'. Therefore, you could safely replace all the encoded 'Is' with 'Es'. Likewise, the letter 'X' is the second most common letter in the encoded text, and in English the second most commonly used letter is 'T'.

Once you have replaced a few letters (better to start with vowels), you could try to find out the missing letters by looking for the words that are most used in English. For example, the most common word is the article 'THE' and in the encoded text the word 'XLI' appears many times: so the coded letter 'L' could correspond to the letter 'H'. Working in this way, nosy decoders will have the text decrypted in a few hours if done by hand, and much faster if they have the help of a computer.

Coders aspire to finding mathematical rules that are extremely difficult to work out. Luckily for decoders, calculation machines have been developed that speed up computing processes. The faster the computer, the more likely it is that the decoder will end up finding the coding rule no matter how complex it is.

For example, during World War II, the Germans designed a machine that used very complex coding rules, which would take thousands of hours of calculation before they were worked out. And if they eventually were, their machine, Enigma, would have allowed them to easily change the old rules for new rules. Fortunately for the Allies, they had Alan Turing in their ranks, who designed calculating machines that tested a great number of rules very quickly. Although his machines used mechanical components, they are regarded as primitive computers.

Nowadays, all the information circulating on the Internet is encrypted using mathematics and the computing potential of computers. The encryption algorithm is known as public key encryption (or RSA) and was the result of work done by Ron Rivest, Adi Shamir and Leonard Adleman in 1972. The encoding rule is based on modular arithmetic. When this arithmetic was first reported, it had no practical use and was not much more than a game to keep mathematicians entertained.

To encode a text with RSA, two prime numbers are needed which are subject to operations based on modular algebra. Although the coding rule is known to everyone, one of the prime numbers used in the rule is not. The private prime number is information that only the coder has and is not shared with anyone. The other prime number is public and known to everyone. When someone wants to encode a text, they use their private prime number and the public prime number of the person who is to decode the text. When they receive the message, which has been encrypted with the public prime number, recipients will be able to decode it without difficulty.

To prevent (uninvited) computers from testing prime numbers to see if they can illegally decipher the content of an encoded text, the prime numbers used are huge. They are so large that no one could have imagined that knowing them would be of any practical use. Can you imagine your favorite football player having the number 77119407959408992776820155300527263864372200742365678630027678105714399623154045939140731123684692285690942961135490292162986944327353101381252303737932482823451114141958194053467498712484469550569500487994952905559075632709934056435491415286132329302472787689019397900722796720491316092481635212364527611217 printed on her shirt? No one would be able to remember it. Not to mention the fact that the shirt would have to be huge. The numbers used to encrypt data on the Internet are usually of this magnitude, or even higher. It would take a lifetime for a nosy computer to find out which prime numbers had been used to encode the information you just searched for. And even if it eventually succeeded, it would take so long that it would no longer matter in the slightest to anyone what the encoded content was.

We must not forget, however, that the constant struggle between coders and decoders tends to be resolved in favor of the latter. No matter how sophisticated and ingenious the coder's procedures and rules are, decoders always manage to figure them out. All in all, it's just a question of having a better and faster computer.

With the technology available today, the mathematical rules of the RSA and the magnitude of the prime numbers used, the security of our communications is ensured for the time being. However, a new generation of computers based on quantum technology is about to see the light of day. With these new machines, the decoders will again have the tools to win the game.

So the team of coders will need people with ingenious minds who know how to use the potential of mathematics to ensure our privacy once again. They will have to be young, be able to persist in the face of adversity and, above all, be very creative. Does that sound like you?

*Urbano Lorenzo Seva, Reus 2020*

*Translation by John F. Bates and Urbano Lorenzo Seva*