Question 1.

Ciphertext: OAHWP

Deciphering steps:

1. Guvf vf n Pnrfne Pvcure (nccyvrq gb qrpvcurevat fgrcf, abg gur bevtvany pvcuregrkg). Rnpu fuvsg pbagevohgrf gb gur xrl sbe gur pvcuregrkg naq gurer ner ab qhcyvpngr fuvsgf, bayl fuvsgf sbe zber frpgvbaf.
2. Bnki pda lwnwcnwld ceraj xahks pda heop, bejz pda iwteiqi zeopnexqpekj kb whh pda happano ej pda bkni haoo pdwj 1
3. Kyv vetipgkzfe giftvjj zj rj wfccfnj:
   
   Vxktc 2 ztnh \(z_u\) pcs \(z_x\) (uadpi pcs xci gthetrixktan). Detgpixdc xh sdct xc iwt hipcspgs Tcvaxhw paewpqti lxiw 26 tcjbtgpits rwpgpritgh.
   
   Tigmz 1 (Innqvm Tigmz): Bpqa tigmz acjabqbcbma mikp kpizikbmz dqi \(K=(iX+j)\) eqbp \(X\) nzwu xtiqvbmfb, \(K\) ia wcbxcb ivl \(i, j\) lmzqdml nzwu \(s_n\). Tmb \(l_1, l_2\) jm bpm nqzab vwv-hmzw lqoqba. \(i=l_1+1\pmod{26}\). Aqvkm \(j\) qa i apqnb, \(j=l_2\pmod{26}\)
   
   Ujhna 2 (Yxuhjuyqjkncrl Bqroc): \(t_r\) mncnavrwnb ynarxm \(U\) jwm bqroc ejudnb \(b_0, b_1, ..., b_{U-1}\). \(U\) rb bdv xo mrprcb xo \(w\). \(b_r\) rw bnzdnwln \(B\) rb mncnavrwnm jb cqn \(r\)-cq mrprc xo \(t_r\) oaxv cqn arpqc (cjtnw \(\bmod{26}\)) jwm anynjcb dwcru \(b_{U-1}\) xena wdvkna xo mrprcb xo \(t_r\)
   Yt fuuqd, ljy \(J_o\) ktw jajwd qjyyjw tk Qfdjw 1 tzyuzy (rfuuji yt szrgjw) \(H_o\) anf \(J_o = H_o + x_{o \bmod{Q}} \pmod{26}\). Ymjs rfu \(J_o\) yt qjyyjw fx ymj knsfq hnumjwyjcy \(J\).

Each shift in the caesar cipher(s) is concatenated to form the large key.

Question 2.

Let a labelled set \(A={a_1, a_2, a_3, a_4, a_5, a_6}\)
Let also corresponding weights \(w=(1,3,4,6,9,10)\)
A triad is a 3-element subset of \(A\). For each triad \(T = \{a_i, a_j, a_k\}\) with indices \(i < j < k\), define its value as \[V(T) = (w_i + w_j + w_k) + (j - i) + (k - j)\] And let \[\mathcal{T} = \{T_1, T_2, \dots, T_{20}\}\] be the family of all possible triads.
Define an ordering function \(\Phi()\) for \(\mathcal{T}\):

1. Sort in ascending order of \(V(T)\)
2. Sort triads with the same \(V(T)\) lexicographically by their indices

\[\Phi(\mathcal{T}) = [T_1', T_2', \dots, T_{20}']\]

Let sequence \(S = (s_1, s_2, s_3, s_4)\)

1. \(s_1\) is \(i + j + k\) of the 3rd triad in \(\Phi(\mathcal{T})\)
2. \(s_2\) is \(V(T)\) of the 7th triad
3. \(s_3\) is the number of distinct parity patterns among all 20 triads when classifying each triad by the parity (even/odd) of its weight sum \(w_i + w_j + w_k\)
4. \(s_4\) is the position (1-indexed) in \(\Phi(\mathcal{T})\) of the triad \(\{a_2, a_4, a_6\}\)

Lastly, map \(s_i\) to \(c_i \in C\) and concatenate \(C\) for the final sequence

Question 3.

Let \(K_{p,q}\) be a (p, q) torus knot.

The length of \(K_{p,q}\) is given by \[L(p,q)=2\pi R \sqrt{p^2 + (\frac{r}{R})^2 q^2}\] Suppose \(\frac{r}{R}=\frac{\sqrt{3}}{2}\) and \(p,q\) are coprime positive integers satisfying the Diophantine equation \[p^2 - 3q^2 = 1\] Let \[N=\left\lfloor{\frac{L(p,q)}{R}}\right\rfloor\] Encode the triple \(S=(p,q,N)\) as a three character sequence \[S_i = \begin{cases} S_i, & \text{if } S_i < 10,\\ \text{the uppercase letter corresponding to } S_i - 10, & \text{if } S_i \ge 10, \end{cases}\]

Question 4.

Let \(G\) be the group of all \(4 \times 4\) permutation metrices with entries in \(\mathbb{F}_{13}\), under matrix multiplication. For each \(M\in G\), let \(j_i\) denote the column position of the 1 in the \(i\)-th row of \(M\). Define \[\phi(M)=(\sum_{i=1}^{4}ij_i\sum_{i=1}^{4}i^2j_i\sum_{i=1}^{4}i^3j_i\sum_{i=1}^{4}i^4j_i)\bmod{13}\] Let \[S=\{M\in G\ \vert\ det(M+M^{-1})\equiv 5\bmod{13}\}\] Define \[\Phi=\bigoplus_{M\in S}\phi(M)\] where \(\bigoplus\) denotes direct addition in \(\mathbb{F}_{13}^4\) Then, map each coordinate \(x\in\mathbb{F}_{13}\) to a character: \[\chi(x) = \begin{cases} A+x, & 0\le x\le 9,\\ \text{digit }(x-10), & 10\le x\le 12, \end{cases}\] yielding \(\chi(\Phi_1)\chi(\Phi_2)\chi(\Phi_3)\chi(\Phi_4)\)

Clue 1.