Problems

Upon the installation of a keypad lock, each of the 26 letters located on the lock’s keypad is assigned an arbitrary natural number known only to the owner of the lock. Different letters do not necessarily have different numbers assigned to them. After a combination of different letters, where each letter is typed once at most, is entered into the lock a summation is carried out of the corresponding numbers to the letters typed in. The lock opens only if the result of the summation is divisible by 26. Prove that for any set of numbers assigned to the 26 letters, there exists a combination that will open the lock.

To transmit messages by telegraph, each letter of the Russian alphabet () ( and are counted as identical) is represented as a five-digit combination of zeros and ones corresponding to the binary number of the given letter in the alphabet (letter numbering starts from zero). For example, the letter is represented in the form 00000, letter -00001, letter -10111, letter -11111. Transmission of the five-digit combination is made via a cable containing five wires. Each bit is transmitted on a separate wire. When you receive a message, Cryptos has confused the wires, so instead of the transmitted word, a set of letters is received. Find the word you sent.

The meeting of the secret agents took place in the green house.

Considering the numbers in the windows of the green house, what should be drawn in the empty frame?

Find one way to encrypt letters of Latin alphabet as sequences of \(0\)s and \(1\)s, each letter corresponds to a sequence of five symbols.

Pinoccio keeps his Golden Key in the safe that is locked with a numerical password. For secure storage of the Key he replaced some digits in the password by letters (in such a way that different letters substitute different digits). After replacement Pinoccio got the password \(QUANTISED17\). Honest John found out that:
• the number \(QUANTISED\) is divisible by all integers less than 17, and
• the difference \(QUA-NTI\) is divisible by \(7\).
Could he find the password?

Using the representation of Latin alphabet as sequences of \(0\)s and \(1\)s five symbols long, encrypt your first and last name.

Decipher the quote from Philip Pullmans "His Dark Materials":
Erh csy wlepp orsa xli xvyxl, erh xli xvyxl wlepp qeoi csy jvii.
The same letters correspond to the same in the phrase, different letters correspond to different. We know that no original letters stayed in place, meaning that in places of e,r,h there was surely something else.

Decipher the following quote from Alice in Wonderland:
Lw zrxog eh vr qlfh li vrphwklqj pdgh vhqvh iru d fkdqjh.
The same letters correspond to the same in the phrase, different letters correspond to different. We know that no original letters stayed in place, meaning that in places of e,r,h there was surely something else.

Elon is studying the Twitter server. Inside the software he found two integer variables \(a\) and \(b\) which change their values when special search queries “RED”, “GREEN”, and “BLUE” are processed. More precisely the pair \((a, b)\) changes into \((a + 18b, 18a - b)\) when processing the query “RED”, to \((17a + 6b, -6a + 17b)\) when processing “GREEN”, and to \((-10a - 15b, 15a - 10b)\) when processing “BLUE”. When any of \(a\) or \(b\) reaches a multiple of \(324\), it resets to \(0\). If \((a, b) = (0, 0)\) the server crashes. On the server startup, the variables \((a, b)\) are set to \((20, 20)\). Prove that the server will never crash with these initial values, regardless of the search queries processed.

After mastering the Caesar shift cypher one may wonder how to generalize it. One possible way is to use Affine cypher. The difference between these two methods can be described as follows:

• In case of Caesar cypher we took a letter with position \(n\) from \(1\) to \(26\) and added to its position a number \(d\) obtaining the number \(n+d\), then we compute its residue modulo \(26\).

• In case of affine cypher we take a letter with position \(n\) and consider a number \(nx + d\) modulo \(26\).

To decipher such code we need to know values \(x\) and \(d\), then if we have a letter in the code with position \(m\), we can find \(n\) as \(n= (m-d)x^{-1}\) modulo \(26\). Here we have to explain what is \(x^{-1}\): for a number \(x < 26\) we are looking for such a number \(y\), that \(26\) divides \(xy-1\).

• Does there always exist a number \(x^{-1}\) modulo \(26\) for any \(x\)?

• Using data \(x=3\), \(d=8\) encrypt the word "SOLUTION".