# 9. Public Key Encryption

In symmetric encryption, we assumed that the two parties had a shared key in advance. If the two parties do not have a shared key, public-key encryption can be used to encrypt messages. Public Key...

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In symmetric encryption, we assumed that the two parties had a shared key in advance. If the two parties do not have a shared key, public-key encryption can be used to encrypt messages. Public Key...

This is a brief comparison of HTTP and HTTPS HTTP: HyperText Transfer Protocol HTTPS: HyperText Transfer Protocol Secure Uses certificates, encryption, TLS. Used for privacy....

Suppose that we’re using RSA, Alice has public key $(N, e)$ and private key $d$. Anyone can send messages to Alice using $(N, e)$. But because anyone can generate $(N, e)$, we are not sure whether ...

In symmetric key cryptography, we have a problem with key sharing and management. More info in the first few paragraphs of Key Exchange (Modern Cryptography). Public Key Cryptography We use two k...

Background Number Theory Let $n$ be a positive integer and let $p$ be prime. Notation. Let $\mathbb{Z}$ denote the set of integers. We will write $\mathbb{Z}_n = \left\lbrace 0, 1, \dots, n -...

Exponential Inverses Suppose we are given integers $a$ and $N$. For any integer $x$ that is relatively prime to $N$, we choose $b$ so that [\tag{$*$} ab \equiv 1 \pmod{\phi(N)}.] Then we have [...

Exponentiation by Squaring Suppose we want to calculate $a^n$ where $n$ is very large, like $n \approx 2^{1000}$. A naive multiplication would take $\mathcal{O}(n)$ multiplications. We will ignore...

In symmetric key encryption, we assumed that the two parties already share the same key. We will see how this can be done. In symmetric key settings, a user has to agree and store every key for ev...

Hash functions are functions that take some input an compress them to produce an output of fixed size, usually just called hash or digest. A desired property of hash function is collision resistanc...

Previously, we focused on semantic security against passive adversaries, that only eavesdrop on the ciphertext. But in the real world, there are active adversaries that interfere with the communica...