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* PUSH NOTE : 09. Transport Layer Security.md

* PUSH ATTACHMENT : is-09-tls-handshake.png

* PUSH NOTE : 08. Public Key Infrastructure.md

* PUSH ATTACHMENT : is-08-certificate-validation.png

* PUSH NOTE : 07. Public Key Cryptography.md

* PUSH NOTE : 06. RSA and ElGamal Encryption.md

* PUSH NOTE : 05. Modular Arithmetic (2).md

* PUSH NOTE : 04. Modular Arithmetic (1).md

* PUSH NOTE : 03. Symmetric Key Cryptography (2).md

* PUSH ATTACHMENT : is-03-feistel-function.png

* PUSH ATTACHMENT : is-03-ecb-encryption.png

* PUSH ATTACHMENT : is-03-cbc-encryption.png

* PUSH ATTACHMENT : is-03-cfb-encryption.png

* PUSH ATTACHMENT : is-03-ofb-encryption.png

* PUSH ATTACHMENT : is-03-ctr-encryption.png

* PUSH NOTE : 02. Symmetric Key Cryptography (1).md

* PUSH NOTE : 01. Security Introduction.md

* PUSH ATTACHMENT : is-01-cryptosystem.png

* PUSH NOTE : 9. Public Key Encryption.md

* PUSH ATTACHMENT : mc-09-ss-pke.png

* PUSH NOTE : 7. Key Exchange.md

* PUSH ATTACHMENT : mc-07-dhke.png

* PUSH ATTACHMENT : mc-07-dhke-mitm.png

* PUSH ATTACHMENT : mc-07-merkle-puzzles.png

* PUSH NOTE : 6. Hash Functions.md

* PUSH ATTACHMENT : mc-06-merkle-damgard.png

* PUSH ATTACHMENT : mc-06-davies-meyer.png

* PUSH ATTACHMENT : mc-06-hmac.png

* PUSH NOTE : 5. CCA-Security and Authenticated Encryption.md

* PUSH ATTACHMENT : mc-05-ci.png

* PUSH ATTACHMENT : mc-05-etm-mte.png

* PUSH NOTE : 4. Message Authentication Codes.md

* PUSH ATTACHMENT : mc-04-mac.png

* PUSH ATTACHMENT : mc-04-mac-security.png

* PUSH ATTACHMENT : mc-04-cbc-mac.png

* PUSH ATTACHMENT : mc-04-ecbc-mac.png

* PUSH NOTE : 2. PRFs, PRPs and Block Ciphers.md

* PUSH ATTACHMENT : mc-02-block-cipher.png

* PUSH ATTACHMENT : mc-02-feistel-network.png

* PUSH ATTACHMENT : mc-02-des-round.png

* PUSH ATTACHMENT : mc-02-DES.png

* PUSH ATTACHMENT : mc-02-aes-128.png

* PUSH ATTACHMENT : mc-02-2des-mitm.png

* PUSH NOTE : 16. The GMW Protocol.md

* PUSH ATTACHMENT : mc-16-beaver-triple.png

* PUSH NOTE : 13. Sigma Protocols.md

* PUSH ATTACHMENT : mc-13-sigma-protocol.png

* PUSH ATTACHMENT : mc-10-schnorr-identification.png

* PUSH ATTACHMENT : mc-13-okamoto.png

* PUSH ATTACHMENT : mc-13-chaum-pedersen.png

* PUSH ATTACHMENT : mc-13-gq-protocol.png

* PUSH NOTE : 12. Zero-Knowledge Proofs (Introduction).md

* PUSH ATTACHMENT : mc-12-id-protocol.png

* PUSH NOTE : 10. Digital Signatures.md

* PUSH ATTACHMENT : mc-10-dsig-security.png

* PUSH NOTE : 1. OTP, Stream Ciphers and PRGs.md

* PUSH ATTACHMENT : mc-01-prg-game.png

* PUSH ATTACHMENT : mc-01-ss.png

* DELETE FILE : _posts/Lecture Notes/Internet Security/2023-09-10-security-intro.md

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Sungchan Yi
2024-11-12 20:54:38 +09:00
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_posts/lecture-notes/modern-cryptography/2023-09-28-hash-functions.md
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@@ -14,9 +14,9 @@ title: 6. Hash Functions
date: 2023-09-28
github_title: 2023-09-28-hash-functions
image:
path: assets/img/posts/Lecture Notes/Modern Cryptography/mc-06-merkle-damgard.png
path: assets/img/posts/lecture-notes/modern-cryptography/mc-06-merkle-damgard.png
attachment:
folder: assets/img/posts/Lecture Notes/Modern Cryptography
folder: assets/img/posts/lecture-notes/modern-cryptography
---
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 resistance**.
@@ -107,7 +107,7 @@ Now we want to construct collision resistant hash functions that work for arbitr
The Merkle-Damgård transform gives as a way to extend our input domain of the hash function by iterating the function.
![mc-06-merkle-damgard.png](../../../assets/img/posts/Lecture%20Notes/Modern%20Cryptography/mc-06-merkle-damgard.png)
![mc-06-merkle-damgard.png](../../../assets/img/posts/lecture-notes/modern-cryptography/mc-06-merkle-damgard.png)
> **Definition.** Let $h : \left\lbrace 0, 1 \right\rbrace^n \times \left\lbrace 0, 1 \right\rbrace^l \rightarrow \left\lbrace 0, 1 \right\rbrace^n$ be a hash function. The **Merkle-Damgård function derived from $h$** is a function $H$ that works as follows.
>
@@ -152,7 +152,7 @@ Now we only have to build a collision resistant compression function. We can bui
Number theoretic primitives will be shown after we learn some number theory.[^3] An example is shown in [collision resistance using DL problem (Modern Cryptography)](./2023-10-03-key-exchange.md#collision-resistance-based-on-dl-problem).
![mc-06-davies-meyer.png](../../../assets/img/posts/Lecture%20Notes/Modern%20Cryptography/mc-06-davies-meyer.png)
![mc-06-davies-meyer.png](../../../assets/img/posts/lecture-notes/modern-cryptography/mc-06-davies-meyer.png)
> **Definition.** Let $\mathcal{E} = (E, D)$ be a block cipher over $(\mathcal{K}, X, X)$ where $X = \left\lbrace 0, 1 \right\rbrace^n$. The **Davies-Meyer compression function derived from $E$** maps inputs in $X \times \mathcal{K}$ to outputs in $X$, defined as follows.
>
@@ -217,7 +217,7 @@ This can be thought of as blocking the length extension attack from prepending t
### HMAC Definition
![mc-06-hmac.png](../../../assets/img/posts/Lecture%20Notes/Modern%20Cryptography/mc-06-hmac.png)
![mc-06-hmac.png](../../../assets/img/posts/lecture-notes/modern-cryptography/mc-06-hmac.png)
This is a variant of the two-key nest, but the difference is that the keys $k_1', k_2'$ are not independent. Choose a key $k \leftarrow \mathcal{K}$, and set