One-time pads

16.1 For the following message, the Kings used substitution code. Caligula intercepted the message and quite easily broke it. Can you do it too?

U GXUAY LS ZXMEKW AMG TGGTIY HMD TAMGXSD LSSY, FEG
GXSA LUGX HEKK HMDIS. FSKT

---

16.2 At one time, Arthur made the mistake of using the one-time pad shifted: the first bit of the plain text he encoded using the second bit of the pad, the second bit of the plain text he encoded using the third bit of the pad etc. He noticed his error after he sent the message off. Being afraid that Bela will not understand his message, he encoded it again (now correctly) using the same one-time pad, and sent it to Bela by another courier, explaining what happened.

Caligula intercepted both messages, and was able to recover the plain text. How?

---

16.3 The Kings were running low on one-time pads, and so Bela had to use the same pad to encode his reply as they used for Arthur’s message. Caligula intercepted both messages, and was able to reconstruct the plain texts. Can you explain how?

---

16.4 Motivated by the one-time pad method, Alice suggests the following protocol for saving the last move in their chess game: in the evening, she encrypts her move (perhaps with other text added, to make it reasonably long) using a randomly generated 0-1 sequence as the key (just like in the one-time pad method). The next morning she sends the key to Bob, so that he can decrypt the message. Should Bob accept this suggestion?

---

16.5 Alice modifies her suggestion as follows: instead of the random 0-1 sequence, she offers to use a random, but meaningful text as the key. For whom would this be advantageous?

---

16.6 Let $e=e_0{e}_1\dots e_k$ be the expression of $e$ in binary ($e_i=0$ or $1$, $e_0$ is always $1$).

Let $x_0=x$, and for $j=1,\dots ,k$, let

$$x_j=\{\begin{array}{ll}{x}_{j-1}^2,& \text{if }{e}_j=0,\\ x_{j-1}^{2}x,& \text{if }{e}_j=1.\end{array}$$

Show that $x_k=x^e$.

---

16.7 Suppose that Bob develops an algorithm that can break RSA in the first, more direct way described above: knowing Alice’s public key $m$ and $e$, he can find her private key $d$.

(a) Show that he can use this to find the number $(p-1)(q-1)$;

(b) from this, he can find the prime factorization $m=pq$.

Prev: 15-a-glimpse-of-cryptography