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* PUSH NOTE : 수학 공부에 대한 고찰.md

* PUSH NOTE : 09. Lp Functions.md

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@@ -15,7 +15,7 @@ date: 2023-10-09
github_title: 2023-10-09-public-key-cryptography
---
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)](../modern-cryptography/2023-10-03-key-exchange.md).
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)](../../modern-cryptography/2023-10-03-key-exchange/).
## Public Key Cryptography
@@ -32,7 +32,7 @@ These keys are created to be used in **trapdoor one-way functions**.
A **one-way function** is a function that is easy to compute, but hard to compute the pre-image of any output. Here are some common examples.
- *Cryptographic hash functions*: [Hash Functions (Modern Cryptography)](../modern-cryptography/2023-09-28-hash-functions.md#collision-resistance).
- *Cryptographic hash functions*: [Hash Functions (Modern Cryptography)](../../modern-cryptography/2023-09-28-hash-functions/#collision-resistance).
- *Factoring a large integer*: It is easy to multiply to integers even if they're large, but factoring is very hard.
- *Discrete logarithm problem*: It is easy to exponentiate a number, but it is hard to find the discrete logarithm.
@@ -80,14 +80,14 @@ But a problem still remains. How does one verify that this key is indeed from th
## Diffie-Hellman Key Exchange
Choose a large prime $p$ and a generator $g$ of $\mathbb{Z}_p^\ast$. The description of $g$ and $p$ will be known to the public.
Choose a large prime $p$ and a generator $g$ of $\mathbb{Z} _ p^\ast$. The description of $g$ and $p$ will be known to the public.
> 1. Alice chooses some $x \in \mathbb{Z}_p^\ast$ and sends $g^x \bmod p$ to Bob.
> 2. Bob chooses some $y \in \mathbb{Z}_p^\ast$ and sends $g^y \bmod p$ to Alice.
> 1. Alice chooses some $x \in \mathbb{Z} _ p^\ast$ and sends $g^x \bmod p$ to Bob.
> 2. Bob chooses some $y \in \mathbb{Z} _ p^\ast$ and sends $g^y \bmod p$ to Alice.
> 3. Alice and Bob calculate $g^{xy} \bmod p$ separately.
> 4. Eve can see $g^x \bmod p$, $g^y \bmod p$ but cannot calculate $g^{xy} \bmod p$.
Refer to [Diffie-Hellman Key Exchange (Modern Cryptography)](../modern-cryptography/2023-10-03-key-exchange.md#diffie-hellman-key-exchange-(dhke)).
Refer to [Diffie-Hellman Key Exchange (Modern Cryptography)](../../modern-cryptography/2023-10-03-key-exchange/#diffie-hellman-key-exchange-(dhke)).
## Message Integrity