Quartz sync: Sep 18, 2024, 11:26 AM

content/BLS Threshold.md

@@ -9,15 +9,15 @@ tags: []
 Threshold cryptography allows a group of parties to jointly perform cryptographic operations such that only a subset (threshold) of the parties is required to collaborate, enhancing both security and availability. The Boneh-Lynn-Shacham (BLS) signature scheme is particularly well-suited for threshold implementations due to its simplicity and the properties of pairing-based cryptography.
 
 ## Overview
-A cryptographic algorithm that enables short signatures and efficient aggregation of signatures. Base on [[1726579238-bilinear-pairings|bilinear pairings]] over [[1726567251-elliptic-curves|elliptic curves]].
+A cryptographic algorithm that enables short signatures and efficient aggregation of signatures. Base on [[bilinear pairings]] over [[elliptic curves]].
 
 Key Features:
 - *Short Signatures* — signatures are elements of an elliptic curve group, resulting in compact representations
 - *Signature Aggregation* — multiple signatures can be combined into a single signature, reducing verification overhead
 - *Deterministic Signing* — signing process doesn't require randomness, simplifying implementation
 
 ### Mathematical Foundation
-Rely on properties of [[1726579238-bilinear-pairings|bilinear pairings]].
+Rely on properties of [[bilinear pairings]].
 
 ### Basic BLS Signature Scheme
 - Setup
@@ -53,7 +53,7 @@ Combining BLS with threshold cryptography results in *threshold BLS signatures*,
 - *Secret Sharing* — private key is shared among parties using a secret sharing scheme
 
 ### Secret Sharing Schemes
-See [[1725904360-shamirs-secret-sharing|Shamir's Secret Sharing]].
+See [[Shamir's Secret Sharing]].
 
 ### Threshold BLS Signature Protocol
 - Setup
@@ -66,7 +66,7 @@ See [[1725904360-shamirs-secret-sharing|Shamir's Secret Sharing]].
     2. Broadcast Partial Signatures
         - Parties share their $\sigma_i$ with the combiner
     3. Signature Reconstruction
-        - Using [[1725960857-lagrange-interpolation-formula|Lagrange interpolation]], combine $t$ partial signatures to form the full signature:
+        - Using [[Lagrange interpolation formula|Lagrange interpolation]], combine $t$ partial signatures to form the full signature:
         $$\sigma=\Pi_{i\in S,j\neq1}\sigma_i^{\lambda_i}$$
         where $S$ is the set of participating parties and $\lambda_i$ are Lagrange coefficients:
         $$\lambda_i=\Pi_{j\in S,j\neq i}\frac{j}{j-i}$$

content/Elliptic Curves.md

@@ -48,7 +48,7 @@ Uses the properties of elliptic curves to create cryptographic algorithms that a
 - *Elliptic Curve Digital Signature Algorithm (ECDSA)* — method for creating digital signatures, ensuring message integrity and authenticity
 - *Elliptic Curve Integrated Encryption Scheme (ECIES)* — hybrid encryption scheme combining ECC with symmetric encryption for data confidentiality
 
-## [[1726579238-bilinear-pairings|Pairings]] on Elliptic Curves
+## [[Bilinear pairings|Pairings]] on Elliptic Curves
 Bilinear maps that take two points on an elliptic curve and output an element in a finite field, enabling advanced cryptographic protocols.
 
 ### Definition
@@ -66,7 +66,7 @@ Each have respective applications for which they have better computational advan
 
 ### Applications in Cryptography
 - *Identity-Based Encryption (IBE)* — allows the use of arbitrary strings (e.g., email addresses) as public keys
-- *Short Signatures* — schemes such as [[1726567320-bls-threshold|BLS]] (Boneh-Lynn-Shacham) enable very short signatures with security based on hardness of certain problems in pairing-friendly groups
+- *Short Signatures* — schemes such as [[BLS Threshold|BLS]] (Boneh-Lynn-Shacham) enable very short signatures with security based on hardness of certain problems in pairing-friendly groups
 - *Attribute-Based Encryption (ABE)* — enables fine-grained access control over encrypted data
 
 ## Common Curves

content/KZG.md

@@ -16,12 +16,12 @@ KZG commitments allow a prover to:
 
 ## Mathematical Foundations
 Builds on:
-- [[1725898229-polynomial-arithmetic|Polynomial Arithmetic]]
-- [[1726567251-elliptic-curves#elliptic-curve-cryptography-ecc|Elliptic Curve Cryptography]]
-- [[1726567251-elliptic-curves#pairings-on-elliptic-curves|Pairings]]
+- [[Polynomial Arithmetic]]
+- [[Elliptic Curves#elliptic-curve-cryptography-ecc|Elliptic Curve Cryptography]]
+- [[Elliptic Curves#pairing-friendly-curves|Pairings]]
 
 ### Bilinear Pairings
-A [[1726579238-bilinear-pairings|bilinear pairing]] is a map:
+A [[bilinear pairings|bilinear pairing]] is a map:
 $$e:G_1\times G_2 \rightarrow G_T$$
 
 where:
@@ -58,7 +58,7 @@ Alternatively, using polynomial notation:
 - Represent $f(\tau)$ as an element in $G_1$:
     - $C=G^{f(\tau)}$
 
-This uses the [[1726580434-homomorphism|homomorphism]] between polynomials evaluated at $\tau$ and group elements.
+This uses the [[homomorphism]] between polynomials evaluated at $\tau$ and group elements.
 
 ### Open
 To prove that $y=f(s)$ for some $s\in \Bbb{F}_\tau$, the prover computes a proof $\pi$ as follows:
@@ -95,7 +95,7 @@ $$e(\pi, H^{\tau-s})=e(G^{\frac{f(\tau)-y}{\tau-s}},H^{\tau-s})=e(G^{f(\tau)-y},
 Thus, both sides are equal.
 
 ### Binding
-The binding property relies on the [[1726567251-elliptic-curves#discrete-logarithm-problem-dlp|discrete logarithm problem]] and the assumption that the prover cannot find two different polynomials $f(x)$ and $f'(x)$ such that $f(\tau)=f'(\tau)$, unless $f(x)=f'(x)$.
+The binding property relies on the [[elliptic curves#discrete-logarithm-problem-dlp|discrete logarithm problem]] and the assumption that the prover cannot find two different polynomials $f(x)$ and $f'(x)$ such that $f(\tau)=f'(\tau)$, unless $f(x)=f'(x)$.
 
 #### Security Assumption
 - *Computational Diffie-Hellman (CDH) Problem* — hardness of computing $G^{ab}$ given $G^a$ and $G^b$
@@ -124,7 +124,7 @@ Both the commitment $C$ and the proof $\pi$ are single elements in $G_1$, regard
 
 ### Prover Efficiency
 - The prover's work involves computing $w(x)$ and exponentiations
-- Efficient algorithms like [[1725903044-fast-fourier-transform|Fast Fourier Transforms (FFT)]] can optimise polynomial operations when dealing with large degrees.
+- Efficient algorithms like [[fast fourier transform|Fast Fourier Transforms (FFT)]] can optimise polynomial operations when dealing with large degrees.
 
 ## Applications of KZG Commitments
 ### zk-SNARKs

content/Lagrange interpolation formula.md

@@ -17,7 +17,7 @@ Where:
 - `yi` are the y-coordinates of the known points
 - `Li(x)` are the Lagrange basis polynomials
 
-In the [[1725904360-shamirs-secret-sharing|Shamir's Secret Sharing]] code, this formula is implemented as follows:
+In the [[Shamir's Secret Sharing]] code, this formula is implemented as follows:
 
 1. The outer loop `for i in 0..threshold` corresponds to the summation in the formula.
 

content/Polynomial Commitments.md

@@ -35,7 +35,7 @@ Essential for constructing efficient cryptographic protocols where the size of t
     - *Fast Verification* — verifier's work should be minimal, enabling practical development
 
 ## Types of Commitment Schemes
-### Kate Commitments ([[1726567313-kzg|KZG]] Commitments)
+### Kate Commitments ([[KZG]] Commitments)
 - *Setup* — trusted setup generates public parameters, including sequence $\{\tau^i\}$ for $i=0$ to $d$, where $\tau$ is a secret
 - *Commit* — given polynomial $f(x)=\sum^d_{i=0}{f_ix^i}$, the commitment is:
 $$C=\sum_{i=0}^d{f_iG^{\tau^i}}$$
@@ -44,7 +44,7 @@ where $G$ is a generator of an elliptic curve group.
 - *Verify* — verifier checks pairing equations to confirm proof's validity
 
 #### Use of Pairings
-KZG commitments rely on [[bilinear pairings]] on [[1726567251-elliptic-curves|elliptic curves]], which enable efficient verification through pairing-based equations.
+KZG commitments rely on [[bilinear pairings]] on [[elliptic curves]], which enable efficient verification through pairing-based equations.
 - *Bilinear Pairing* — a map $e:G_1\times G_2\rightarrow G_T$ satisfying bilinearity, non-degeneracy, and computability
 - *Verification Equation* — $e(C-y\cdot G, H)=e(\pi, G-s\cdot H)$, here, $H$ is another generator, and $s$ is the evaluation point
 
@@ -54,7 +54,7 @@ KZG commitments rely on [[bilinear pairings]] on [[1726567251-elliptic-curves|el
 
 #### Drawbacks
 - *Trusted setup* — requires a secure generation of secret $\tau$
-- *Security assumptions* — relies on hardness of [[1726567251-elliptic-curves#discrete-logarithm-problem-dlp|Discrete Logarithm Problem]] and *Computational Diffie-Hellman Problem* in pairing groups.
+- *Security assumptions* — relies on hardness of [[Elliptic Curves#discrete-logarithm-problem-dlp|Discrete Logarithm Problem]] and *Computational Diffie-Hellman Problem* in pairing groups.
 
 ### Other Schemes
 #### Pedersen Commitments
@@ -90,7 +90,7 @@ Drawbacks
 - *Ethereum 2.0* uses KZG commitments in *danksharding* proposal for scalable data availability proofs
 - *Plonk Protocol* is a universal SNARK protocol that uses polynomial commitments for efficient proof generation and verification
 
-## Connection to [[1725960857-lagrange-interpolation-formula|Lagrange Interpolation]]
+## Connection to [[Lagrange interpolation formula|Lagrange Interpolation]]
 Any polynomial $f(x)$ of degree $d$ can be uniquely distributed by $d+1$ evaluations at distinct points, which is fundamental in constructing and verifying polynomial commitments.
 
 ## Implementation Considerations

content/Week 3.md

@@ -6,15 +6,15 @@ tags: []
 ---
 
 # Key Ideas
-- [[1726567251-elliptic-curves|Elliptic Curves]]
-    - [[1726567251-elliptic-curves#pairings-on-elliptic-curves|Pairings]]
-    - [[1726567251-elliptic-curves#common-curves|Common Curves]]
-- [[1726567296-polynomial-commitments|Polynomial Commitments]]
-- [[1726567313-kzg|KZG]]
-- [[1726567320-bls-threshold|BLS Threshold]]
+- [[Elliptic Curves]]
+    - [[Elliptic Curves#pairing-friendly-curves|Pairings]]
+    - [[Elliptic Curves#common-curves|Common Curves]]
+- [[Polynomial Commitments]]
+- [[KZG]]
+- [[BLS Threshold]]
 
 # Exercises
-- Implement [[1726567313-kzg|KZG]] commitment, proof
+- Implement [[KZG]] commitment, proof
 
 # References
 - https://github.com/pluto/ronkathon/blob/main/src/curve/README.md

content/bilinear pairings.md

@@ -23,5 +23,5 @@ Each have respective applications for which they have better computational advan
 
 ### Applications in Cryptography
 - *Identity-Based Encryption (IBE)* — allows the use of arbitrary strings (e.g., email addresses) as public keys
-- *Short Signatures* — schemes such as [[1726567320-bls-threshold|BLS]] (Boneh-Lynn-Shacham) enable very short signatures with security based on hardness of certain problems in pairing-friendly groups
+- *Short Signatures* — schemes such as [[BLS Threshold|BLS]] (Boneh-Lynn-Shacham) enable very short signatures with security based on hardness of certain problems in pairing-friendly groups
 - *Attribute-Based Encryption (ABE)* — enables fine-grained access control over encrypted data

content/index.md

@@ -55,15 +55,15 @@ If you'd like to follow along, you may navigate to [this Notion page](https://ww
 - Moonmath Manual Polynomial Arithmetic
 
 ### [[Week 3]]
-- [[1726567251-elliptic-curves|Elliptic Curves]]
-    - [[1726567251-elliptic-curves#pairings-on-elliptic-curves|Pairings]]
-    - [[1726567251-elliptic-curves#common-curves|Common Curves]]
-- [[1726567296-polynomial-commitments|Polynomial Commitments]]
-- [[1726567313-kzg|KZG]]
-- [[1726567320-bls-threshold|BLS Threshold]]
+- [[Elliptic Curves]]
+    - [[Elliptic Curves#pairing-friendly-curves|Pairings]]
+    - [[Elliptic Curves#common-curves|Common Curves]]
+- [[Polynomial Commitments]]
+- [[KZG]]
+- [[BLS Threshold]]
 
 #### EXERCISES
-- Implement [[1726567313-kzg|KZG]] commitment, proof
+- Implement [[KZG]] commitment, proof
 
 #### REFERENCES
 - https://github.com/pluto/ronkathon/blob/main/src/curve/README.md