MAT331 homework problems

20.

(expires 8 May)     Write a procedure in Maple that counts the frequency of letters in a string of text. For example, here is what it looks like when I use mine:

freqs("time flies like an arrow, fruit flies like a bananna.");

[[" ",9], ["i",6], ["a",6], ["e",5], ["n",4], ["l",4], ["r",3], ["f",3], ["t",2], ["s",2], ["k",2], ["w",1], ["u",1], ["o",1], ["m",1], ["b",1], [",",1], [".",1]]

In the above phrase, there are 9 spaces, 6 each of the letters ``i'' and ``a'', ``e'' appears 5 times, and so on. [Hint: I found it useful to group identical letters in the text using Implode(sort(Explode(text)) , but you might not.]

21.

(expires 8 May)     The string below was encrypted using an affine cipher on the 27 letter alphabet `` abcdefghijklmnopqrstuvwxyz'' (there is a space in the $0^{th}$ position.) Decrypt it.

fmw segjaweoouanerj a ceyqrype aswaheoaqbrqabeafrua eeaojerf afmjeayperjpu

Hint: this phrase follows the the typical pattern in English where there are as many spaces as words (so spaces are very common), and the letter ``e'' is also very common. You can use the technique described in chapter 4 of the notes, section 7.3.

22.

(expires 8 May)     Recall that a Vignère cipher can be interpreted as a Caesar-like cipher on $n$-vectors, where $n$ is the length of the key phrase. Can every affine encipherment on digraphs (two-character codes) be interpreted as an affine matrix encipherment on 2-vectors? That is, suppose I encode a message by affine enciphering on digraphs. Can I always get the same crypttext from the same plaintext using an affine matrix enciphering (using a $2\times2$ matrix) on $2$-vectors? If your answer is yes, prove it. If no, give a counter-example that cannot be so interpreted.

EqOG3u0qaJTX.ccwlqa9UhjkFKRT8a)CWn1rs3u(PsfK8OPsNde2 9OW8OtURd(qdUx!nDo4R2

Hint: the above used an affine matrix cipher on 2-vectors, with the 67-letter alphabet

1234567890.! ()ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz

You should be able to guess what the first 20 letters of the message are.

24.

(expires 8 May)     Modify the AffineMatEncode routine we wrote in class so that you can use a text string as a key instead of a matrix and a vector. For example, if the phrase is $k$ characters long, the key should be an $n\times n$ matrix and an $n$-vector, where $n^2 + n \le k$. The elements of the key matrix and vector should be the numerical equivalents of the characters in the key phrase. Do something sensible with any extra letters (that is, if $k > n^2+n$). Be sure to check that the resulting matrix is nonsingular.