# Exercises - RSA and Public Key Codes

1. Find the decoding key $d$ for the code whose published values of $N$ and $e$ are
$$N = 233570063, e = 125$$

2. Decrypt, by hand, the following RSA-encoded message ($N=7387, \quad e=1357$):
$$2133 \quad 429 \quad 1126$$
You may assume the following was used to equate letter pairs with numbers:
$$\begin{array}{ccccccccccccc} A & B & C & D & E & F & G & H & I & J\\ 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \end{array}$$
$$\begin{array}{cccccccccc} K & L & M & N & O & P & Q & R & S & T\\ 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \end{array}$$
$$\begin{array}{ccccccccccccc} U & V & W & X & Y & Z\\ 31 & 32 & 33 & 34 & 35 & 36 \end{array}$$

3. Presume that a plaintext message is converted to a number by making the following substitutions:
blank = $99$, $A = 10$, $B = 11$, ..., $Z=35$. Decode, the following message. It is a quotation from Shakespeare's "Hamlet". The coded message consists of four integers
$$\begin{array}{r} 39 \; 25736 \; 57380 \; 83976\\ 8 \; 66571 \; 70599 \; 56870\\ 14569 \; 39934 \; 49451\\ 14 \; 57541 \; 36754 \; 04137 \end{array}$$
The public key used for encryption was
$$(N = 59 \; 11142 \; 11035 \; 79513, e = 123)$$

Hint: Don't do this one by hand. You may find it useful to use the Mathematica functions GCD, PowerMod, and FactorInteger in combination with www.wolframalpha.com. Look these functions up! Pay special attention to PowerMod, as it can be used not only to speed up successive squaring, but also to find multiplicative inverses in a given mod.

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