Suppose a message is RSA-encrypted with modulus $N=323$ and encrypting key $e=49$. Find the decrypting key $d$.
Find an integer $x$ that satisfies the congruence below.
$$x^{169} \equiv 119 \pmod{1452}$$
Consider the function $f: \mathbb{Z} \rightarrow 2\mathbb{Z}$ defined by $f(x) = 2x^2 - 4$. Is $f(x)$ injective? Is it surjective? Justify your claims.
Find or describe a bijection between the points in the open interval $(0,1)$ and the points in $(-\infty,\infty)$.
Prove that the cardinality of the power set of a given set is never equal to the cardinality of the set itself.
Suppose $\text{ord}_n b = 3$. How many positive integer solutions does $b^x \equiv 1 \pmod{n}$ have for $x$ that are less than $26$?
Suppose that $r$ is a primitive root for $n$, prove the following:
Alice and Bob agree to create a secret encryption key for future correspondence according to the Diffie-Hellman Key Exchange Protocol. Over a public communications channel, they agree upon a prime base of $5$ and a modulus of $23$. Alice then sends Bob the value $8$, and Bob sends Alice the value $19$. What is their agreed upon secret encryption key? What could they have done to better protect the secret key they generated? (Hint: You may wish to refer to the table of powers and indexes $\pmod{23}$ provided in the next question.)
Use the table of powers and indexes provided below and index arithmetic to solve the following congruences:
In a plane, given two points $F_1$ and $F_2$, a hyperbola is defined as the set of all points $P$ where the positive difference between the distance from $P$ to $F_1$ and the distance from $P$ to $F_2$ is constant. Suppose $F_1 = (-2,0)$, $F_2 = (2,0)$, the aforementioned constant positive difference is 3, and distances are found with the taxicab metric. Draw the corresponding "hyperbola".
The inhabitants of planet Quatro wish to beam a message to Earth. The alphabet used on Quatro has four letters: A, B, C, and D, which the people of Quatro have encoded as $00000$, $10110$, $01011$, and $11101$, respectively. What is the maximum number of single digit errors that this encoding scheme can correct in the transmission of any one letter?
Show that the powers of $\alpha = x$ generates the finite field with elements constructed using the irreducible polynomial $f(x) = x^3 + x^2 + 1$ from $\mathbb{Z}_2[x]$.
Construct a Zech's Log Table for the finite field generated by $\alpha = x$ and the irreducible polynomial $f(x) = x^4 + x + 1$ with coefficients taken from $\mathbb{Z}_2$. Now let $y$ be given by the following: $$ y = x^{10} + \frac{x^7 + x^{23}}{x^4 + x^9} + 1 $$ Use the table constructed to write $y$ as a power of $x$.