Reordering (#7)

* reorganization

* ported to md

* cleanup

* code blocks

* cleanup conversion

* dkg + vuf + vrf

* added render

* proper cred

* added build instr

notebooks/01_background/00_notation.md

@@ -0,0 +1,111 @@
+---
+title: Chaos Background
+author: Tristan Britt  (\texttt{tristan@warlock.xyz}), 0xAlcibiades (\texttt{alcibiades@warlock.xyz})
+date: July 2024
+---
+
+## Glossary of Notation
+
+| Notation | Meaning |
+|--------|---------|
+| $\{ \}$ | Set delimiters |
+| $\varnothing$ | Empty set |
+| $\in$ | Element of |
+| $\notin$ | Not an element of |
+| $\subset$ | Proper subset |
+| $\subseteq$ | Subset or equal to |
+| $\supset$ | Proper superset |
+| $\supseteq$ | Superset or equal to |
+| $\cup$ | Union |
+| $\cap$ | Intersection |
+| $\setminus$ | Set difference |
+| $\overline{A}$ | Complement of set $A$ |
+| $A^c$ | Complement of set $A$ (alternative notation) |
+| $\mathcal{P}(A)$ | Power set of $A$ |
+| $A \times B$ | Cartesian product of sets $A$ and $B$ |
+| $|A|$ | Cardinality (size) of set $A$ |
+| $\aleph_0$ | Cardinality of the natural numbers (countable infinity) |
+| $\mathfrak{c}$ | Cardinality of the real numbers (continuum) |
+| $\forall$ | For all |
+| $\exists$ | There exists |
+| $\exists!$ | There exists a unique |
+| $:$ or $\mid$ | Such that |
+| $\{ x \in A \mid P(x) \}$ | Set-builder notation: set of all $x$ in $A$ such that $P(x)$ is true |
+| $[a,b]$ | Closed interval from $a$ to $b$ |
+| $(a,b)$ | Open interval from $a$ to $b$ |
+| $[a,b)$ or $(a,b]$ | Half-open intervals |
+| $A \triangle B$ | Symmetric difference of sets $A$ and $B$ |
+| $\bigsqcup$ | Disjoint union |
+| $\bigcup_{i \in I} A_i$ | Union of a family of sets |
+| $\bigcap_{i \in I} A_i$ | Intersection of a family of sets |
+| $A^n$ | Cartesian product of $A$ with itself $n$ times |
+| $f: A \to B$ | Function $f$ from set $A$ to set $B$ |
+| $f(A)$ | Image of set $A$ under function $f$ |
+| $f^{-1}(B)$ | Preimage of set $B$ under function $f$ |
+| $\text{dom}(f)$ | Domain of function $f$ |
+| $\text{cod}(f)$ | Codomain of function $f$ |
+| $\text{range}(f)$ | Range of function $f$ |
+| $\text{id}_A$ | Identity function on set $A$ |
+| $f \circ g$ | Composition of functions $f$ and $g$ |
+| $f|_A$ | Restriction of function $f$ to set $A$ |
+| $f: A \twoheadrightarrow B$ | Surjective function from $A$ to $B$ |
+| $f: A \hookrightarrow B$ | Injective function from $A$ to $B$ |
+| $f: A \xrightarrow{\sim} B$ | Bijective function from $A$ to $B$ |
+| $\mathbb{Z}$ | set of all integers |
+| $\mathbb{Q}$ | set of all rational numbers |
+| $\mathbb{R}$ | set of all real numbers |
+| $\mathbb{C}$ | set of all complex numbers |
+| $\Leftrightarrow$, iff | if and only if |
+| $\mathbb{Z}^+, \mathbb{Q}^+, \mathbb{R}^+$ | sets of all positive integers, rational numbers, and real numbers, respectively |
+| $a\mid b$ | $a$ divides $b$ |
+| $*$ | binary operation |
+| $\Delta$ | symmetric difference |
+| $e$ | identity element of a group |
+| $GL(2,\mathbb{R})$ | general linear group of degree 2 over $\mathbb{R}$ |
+| $P(X)$ | set of subsets $X$ |
+| $\mathbb{Z}_n$ | the set $\{0, 1, 2, \ldots, n-1\}$ |
+| $a \equiv b \pmod{n}$ | the integers $a$ and $b$ are congruent modulo $n$ |
+| $\oplus, \otimes$ | addition and multiplication modulo $n$ |
+| $o(x)$ | order of the element $x$ |
+| $\langle x \rangle$ | set of powers of the element $x$ |
+| $\|G\|$ | order of the group $G$ |
+| $V$ | Klein's 4-group |
+| $Z(G)$ | center of the group $G$ |
+| $GL(2,\mathbb{C})$ | general linear group of degree 2 over $\mathbb{C}$ |
+| $Q_8$ | group of unit quaternions |
+| $SL(2,\mathbb{R})$ | special linear group of degree 2 over $\mathbb{R}$ |
+| $Z(g)$ | centralizer of the element $g$ |
+| $G \times H$ | direct product of $G$ and $H$ |
+| $f:S \to T$ | $f$ is a function from $S$ to $T$ |
+| $f^{-1}$ | the inverse of the function $f$ |
+| $g \circ f$ | composite function |
+| $i_X$ | identity function on the set $X$ |
+| $S_X$ | symmetric group on $X$ |
+| $S_n$ | symmetric group of degree $n$ |
+| $A_n$ | alternating group of degree $n$ |
+| $D_4$ | group of symmetries of a square |
+| $x \equiv_H y$ | means $xy^{-1} \in H$ |
+| $x_H \equiv y$ | means $x^{-1}y \in H$ |
+| $[G:H]$ | the index of $H$ in $G$ |
+| $H \triangleleft G$ | $H$ is a normal subgroup of $G$ |
+| $G/H$ | quotient group of $G$ by $H$ |
+| $G \cong H$ | $G$ and $H$ are isomorphic |
+| $\varphi^{-1}(J)$ | inverse image of $J$ under $\varphi$ |
+| $\text{Aut}(G)$ | group of automorphisms of the group $G$ |
+| $\rho$ | canonical homomorphism |
+| $\ker(\varphi)$ | kernel of the homomorphism $\varphi$ |
+| $N(H)$ | normalizer of the subgroup $H$ |
+| $R \oplus S$ | direct sum of the rings $R$ and $S$ |
+| $M_2(\mathbb{R})$ | ring of all $2 \times 2$ real matrices |
+| $\mathbb{Z}[i]$ | ring of Gaussian integers |
+| $\mathbb{H}$ | ring of quaternions |
+| $R/I$ | quotient ring of $R$ by $I$ |
+| $R[X]$ | polynomial ring over $R$ |
+| $F(a)$ | field obtained by adjoining $a$ to the field $F$ |
+| $\text{irr}(a/F)$ | irreducible polynomial of $a$ over $F$ |
+| $\deg(a/F)$ | degree of $a$ over $F$ |
+| $[E : F]$ | degree of the field $E$ over the field $F$ |
+| $\mathbb{C}_c$ | field of constructible complex numbers |
+| $\Gamma(E/F)$ | Galois group of $E$ over $F$ |
+| $\Phi(H)$ | fixed field of the subgroup $H$ of $\Gamma(E/F)$ |
+| $\Gamma(f(X)/F)$ | Galois group of $f(X)$ over $F$ |

notebooks/01_background/01_roadmap.md

@@ -0,0 +1,11 @@
+# Math Primer for BLS and BN254
+
+We'll begin with fundamental concepts in set theory, group theory, and ring theory. These will provide the basis for understanding more advanced structures like fields and vector spaces. Number theory and Euclidean domains will be introduced to provide essential tools for cryptographic applications.
+
+As we progress, we'll delve into algebraic varieties and elliptic curves, which are crucial for understanding the BN254 curve. We'll then explore bilinear pairings, which are fundamental to the BLS signature scheme.
+
+Finally, we'll apply these concepts to the specific case of the BN254 curve and the BLS signature scheme, culminating in an exploration of threshold signatures, R1CS, and distributed multi-party computation.
+
+By the end of this primer, readers will have a solid grasp of the mathematical concepts necessary to understand and implement BLS threshold signatures using the BN254 pairing. This knowledge is crucial for developing secure and efficient cryptographic protocols in various applications, including blockchain technology and distributed systems.
+
+The roadmap of our journey will be Sets → Groups → Rings → Domains → Fields → Vector Spaces → Algebraic Varieties → Elliptic Curves → Bilinear Pairings -> BN254 -> BLS -> BLS Thresholds -> R1CS -> Distributed MPC, rougly.

notebooks/01_background/02_sets_groups_and_rings.md

@@ -0,0 +1,210 @@
+## Set Theory
+
+Set theory forms the bedrock of modern mathematics. It provides us with a language to discuss collections of objects and the 
+relationships between them. For a more in-depth treatment of set theory, Halmos' "Naive Set Theory" and Suppes' "Axiomatic Set Theory" 
+are superb resources.
+
+### Basic Definitions
+
+1. A set is a collection of distinct objects, called elements or members of the set.
+2. If $a$ is an element of set $A$, we write $a \in A$.
+3. The empty set, denoted $\varnothing$, is the unique set with no elements.
+4. A set $A$ is a subset of set $B$, denoted $A \subseteq B$, if every element of $A$ is also an element of $B$.
+5. Every set $A \subseteq A$, or in other words, every set is contained by itself.
+
+### Set Operations
+
+Set theory defines several operations on sets:
+
+1. Union: $A \cup B = \{x : x \in A \text{ or } x \in B\}$
+   The union of two sets contains all elements that are in either set.
+
+2. Intersection: $A \cap B = \{x : x \in A \text{ and } x \in B\}$
+   The intersection contains all elements common to both sets.
+
+3. Difference: $A \setminus B = \{x : x \in A \text{ and } x \notin B\}$
+   The difference contains elements in $A$ but not in $B$.
+
+4. Symmetric Difference: $A \triangle B = (A \setminus B) \cup (B \setminus A)$
+   This operation results in elements that are in either set, but not in both.
+
+### Cartesian Product
+
+The Cartesian product of two sets $A$ and $B$, denoted $A \times B$, is the set of all ordered pairs where the first element comes 
+from $A$ and the second from $B$:
+
+$$A \times B = \{(a,b) : a \in A \text{ and } b \in B\}$$
+
+### Functions
+
+A function $f$ from set $A$ to set $B$, denoted $f: A \to B$, is a rule that assigns to each element of $A$ exactly one element of $B$. 
+We call $A$ the domain and $B$ the codomain of $f$. The set of all $f(a)$ for $a \in A$ is called the range of $f$.
+
+Functions can have special properties:
+
+1. Injective (one-to-one): $\forall a_1, a_2 \in A, f(a_1) = f(a_2) \implies a_1 = a_2$
+2. Surjective (onto): $\forall b \in B, \exists a \in A : f(a) = b$
+3. Bijective: Both injective and surjective
+
+### Cardinality
+
+The cardinality of a set $A$, denoted $|A|$, is the number of elements in $A$ if $A$ is finite. For infinite sets, cardinality 
+becomes more complex:
+
+- Countably infinite: A set with the same cardinality as the natural numbers, denoted $\aleph_0$.
+- Uncountable: An infinite set that is not countably infinite, such as the real numbers, with cardinality denoted $\mathfrak{c}$.
+
+---
+
+## Group Theory
+
+Group theory is the study of symmetries and algebraic structures. Professor Macauley's Visual Group Theory lectures on YouTube 
+and Nathan Carter's "Visual Group Theory" book provide a beautiful and approachable exposition. Saracino's "Abstract Algebra" 
+is approachable but in need of fresh typesetting. Lang's "Algebra" is also a good resource here and more generally on rings and 
+fields to come.
+
+### Groups
+
+A group is an ordered pair $(\mathbb{G}, *)$ where $\mathbb{G}$ is a set and $*$ is a binary operation on $\mathbb{G}$ satisfying 
+four axioms:
+
+1. Closure: $\forall a, b \in \mathbb{G}, a * b \in \mathbb{G}$
+2. Associativity: $\forall a, b, c \in \mathbb{G}, (a * b) * c = a * (b * c)$
+3. Identity: $\exists e \in \mathbb{G}, \forall a \in \mathbb{G}: a * e = e * a = a$
+4. Inverse: $\forall a \in \mathbb{G}, \exists a^{-1} \in \mathbb{G}: a * a^{-1} = a^{-1} * a = e$
+
+The identity element is often denoted as $e$, and the inverse of an element $a$ is written as $a^{-1}$. We also have "subtraction" defined through the binary operator of the inverse of an element.
+
+### Abelian Groups
+
+An Abelian group, named after Norwegian mathematician Niels Henrik Abel, is a group which is commutative under the binary operation $*$. A group $\mathbb{G}$ is abelian if $a * b = b * a, \forall a, b \in \mathbb{G}$.
+
+### Finite Groups
+
+A group $\mathbb{G}$ is finite if the number of elements in $\mathbb{G}$ is finite, which then has cardinality or order $|\mathbb{G}|$.
+
+### Lagrange's Theorem
+
+For a finite group $\mathbb{G}$ with $a \in \mathbb{G}$ and let there exist a positive integer $d$ such that $a^d$ is the smallest positive power of $a$ that is equal to $e$, the identity of the group. Let $n = |\mathbb{G}|$ be the order of $\mathbb{G}$, and let $d$ be the order of $a$, then $a^n = e$ and $d \mid n$.
+
+### Subgroups
+
+A subset $H$ of a group $\mathbb{G}$ is a subgroup if it forms a group under the same operation as $\mathbb{G}$. We denote this as $H \leq \mathbb{G}$. The order of a subgroup always divides the order of the group (Lagrange's Theorem). Similar to a set, every group $\mathbb{G} \subseteq \mathbb{G}$, and for every group there is a trivial subgroup containing only the identity.
+
+### Homomorphisms and Isomorphisms
+
+A function $f: G \to H$ between groups is a homomorphism if it preserves the group operation: $f(ab) = f(a)f(b) \forall a,b \in G$.
+
+An isomorphism is a bijective homomorphism. If there exists an isomorphism between groups $G$ and $H$, we say they are isomorphic and write $G \cong H$.
+
+### Cosets and Normal Subgroups
+
+For a subgroup $H$ of $G$ and an element $a \in G$, we define:
+
+- Left coset: $aH = \{ah : h \in H\}$
+- Right coset: $Ha = \{ha : h \in H\}$
+
+A subgroup $N$ of $G$ is called normal if $gN = Ng \forall g \in G$. We denote this as $$N \triangleleft G$$.
+
+### Cyclic Groups
+
+A group $\mathbb{G}$ is cyclic if there exists an element $g \in \mathbb{G}$ such that every element of $\mathbb{G}$ can be written as a power of $g$:
+
+$$\mathbb{G} = \langle g \rangle = \{g^n : n \in \mathbb{Z}\}$$
+
+Here, $g$ is called a generator of $\mathbb{G}$. Cyclic groups have several important properties:
+
+1. Every element $x \in \mathbb{G}$ can be written as $x = g^n$ for some integer $n$.
+2. If $\mathbb{G}$ is infinite, it is isomorphic to $(\mathbb{Z}, +)$.
+3. If $\mathbb{G}$ is finite with $|\mathbb{G}| = n$, it is isomorphic to $(\mathbb{Z}/n\mathbb{Z}, +)$.
+4. All cyclic groups are Abelian.
+5. Subgroups of cyclic groups are cyclic.
+6. The order of $\mathbb{G}$ is the smallest positive integer $m$ such that $g^m = e$.
+
+### Quotient Groups
+
+If $N \triangleleft G$, we can form the quotient group $G/N$, whose elements are the cosets of $N$ in $G$.
+
+---
+
+## Ring Theory
+
+### Rings
+
+A ring $(R, +, \cdot)$ is an algebraic structure consisting of a set $R$ with two binary operations, addition $(+)$ and multiplication $(\cdot)$, satisfying the following axioms:
+
+1. $(R, +)$ is an abelian group:
+   - Closure: $\forall a, b \in R, a + b \in R$
+   - Associativity: $\forall a, b, c \in R, (a + b) + c = a + (b + c)$
+   - Commutativity: $\forall a, b \in R, a + b = b + a$
+   - Identity: $\exists 0 \in R, \forall a \in R, a + 0 = 0 + a = a$
+   - Inverse: $\forall a \in R, \exists (-a) \in R, a + (-a) = (-a) + a = 0$
+
+2. $(R, \cdot)$ is a monoid:
+   - Closure: $\forall a, b \in R, a \cdot b \in R$
+   - Associativity: $\forall a, b, c \in R, (a \cdot b) \cdot c = a \cdot (b \cdot c)$
+
+3. Distributivity:
+   - Left distributivity: $\forall a, b, c \in R, a \cdot (b + c) = (a \cdot b) + (a \cdot c)$
+   - Right distributivity: $\forall a, b, c \in R, (a + b) \cdot c = (a \cdot c) + (b \cdot c)$
+
+A ring is called commutative if multiplication is commutative, i.e., $\forall a, b \in R, a \cdot b = b \cdot a$. If a ring has a multiplicative identity element $1 \neq 0$ such that $\forall a \in R, 1 \cdot a = a \cdot 1 = a$, it is called a ring with unity.
+
+### Ideals
+
+An ideal of a ring $R$ is a subset $I \subseteq R$ where:
+
+1. $(I,+)$ is a subgroup of $(R,+)$, meaning:
+   a. $I$ is non-empty
+   b. For all $a,b \in I$, $a - b \in I$
+2. For all $r \in R$ and $i \in I$, both $r \cdot i \in I$ and $i \cdot r \in I$ (absorption property)
+
+The absorption property of ideals interacts with both ring operations, as it involves multiplication by any ring element and the result remains in the ideal.
+
+### Quotient Rings
+
+For a ring $R$ and ideal $I$, the quotient ring $R/I$ is defined as:
+
+$R/I = \{r + I : r \in R\}$
+
+where $r + I = \{r + i : i \in I\}$ is the coset of $r$ modulo $I$.
+
+Operations in $R/I$ are defined as:
+
+1. Addition: $(a + I) + (b + I) = (a + b) + I$
+2. Multiplication: $(a + I) \cdot (b + I) = (a \cdot b) + I$
+
+These operations are well-defined because of the ideal properties, particularly the absorption property.
+
+### Polynomial Rings
+
+Given a ring $R$, the polynomial ring $R[x]$ is defined as the set of all formal sums of the form:
+
+$$f(x) = \sum_{i=0}^n a_i x^i = a_0 + a_1x + a_2x^2 + ... + a_nx^n$$
+
+where:
+
+1. $n \in \mathbb{Z^+}$
+2. $a_i \in R$ (called coefficients)
+3. $x$ is an indeterminate (or variable)
+4. Only finitely many $a_i$ are non-zero
+
+The ring structure of $R[x]$ is defined by the following operations:
+
+1. Addition: For $f(x) = \sum a_ix^i$ and $g(x) = \sum b_ix^i$,
+
+   $(f + g)(x) = \sum_{i=0}^{\max(\deg(f),\deg(g))} (a_i + b_i)x^i$
+
+2. Multiplication: For $f(x) = \sum a_ix^i$$ and $$g(x) = \sum b_ix^i$,
+
+   $(f \cdot g)(x) = \sum_{k=0}^{\deg(f)+\deg(g)} (\sum_{i+j=k} a_ib_j)x^k$
+
+Key properties:
+
+1. The zero polynomial, denoted $0$, has all coefficients equal to $0$.
+2. If $R$ has a unity $1 \neq 0$, then $R[x]$ has a unity, which is the constant polynomial $1$.
+3. $R$ is embedded in $R[x]$ as the set of constant polynomials.
+4. If $R$ is commutative, then $R[x]$ is commutative.
+5. The degree of a non-zero polynomial $f(x)$, denoted $\deg(f)$, is the highest power of $x$ with a non-zero coefficient.
+
+This definition treats polynomials as formal algebraic objects, not as functions. The construction can be extended to multiple variables, e.g., $R[x,y] = (R[x])[y]$.