random notes

README.md

@@ -18,7 +18,7 @@ ZK gives similar vibes to Machine Learning. More and more people just mention ZK
 
 Set of integers is denoted by Z. {...-4,-3,-2,-1,0,1,2,3,4..}
 
-Set of Rational numbers denoted by Q. {..1, 3/2, 2, 22/7,...} 
+Set of Rational numbers denoted by Q. {..1, 3/2, 2, 22/7,...}
 
 Set of Real Numbers denoted by R. {2, -4, 613, π, √2,..}
 
@@ -50,34 +50,94 @@ To be concidered a group the combination needs certain properties
 
 **Closure:** For all a, b in set G, the result of operation a•b, is also in G
 
-**Associativity:** for all a, b and c in set G, (a•b)•c 
+**Associativity:** for all a, b and c in set G, (a•b)•c
 
-**Identity element:** there exists an element e in G such that every element a in G, the equation e•a = a•e=a holds. 
+**Identity element:** there exists an element e in G such that every element a in G, the equation e•a = a•e=a holds.
 
 **Inverse element:** for each a in G, there exists an element b in G, denoted a<sup>-1</sup> (or -a if the operation is "+"), such that a•b = b•a = e, where e is the identity element
 
 ### Sub Groups
 
 if a subset of the elements in a group also satisfies the group properties, then that is a subgroup of the original group
 
-
 ### Cyclic groups and generators
 
 A finite group can be cyclic. That means it has a generator element. If you start at any point and apply a group operation with the generator as argument a certain number of times, you can go around the whole group and end in the same place.
 
-### Finding and inverse
+### Finding an inverse
+
+fermat's little theorem
 
 a<sup>-1</sup> ≡ a <sup>p-2</sup> (modp)
 
-Let p = 7 and a = 2
+Let p = 7 and a = 2. We can compute the inverse of a as:
 
+a<sup>p-2</sup> = 2<sup>5</sup> = 32 ≡ 4 mod 7
 
+Verify: 2 x 4 ≡ 1 mod 7
 
+### Equivalence classes
 
-- Homework 1
+since
 
-## Lesson 2: ZKP Theory / Zokrates
+6 mod 7 = 6
 
-- Homework 2
+13 mod 7 = 6
+
+20 mod 7 = 6
+
+6, 13, 20 form equivalence classes
+
+more formally
+
+i + kN | k EZ for some i between 0 and N - 1.
+
+Thus if we are trying to solve the equation
+
+x mod 7 = 6
+
+x could be 6, 13, 20 ....
+
+This gives us the basis for a one way function
+
+### Fields
+
+A field is a set of say integers together with two operations called addition and multiplication.
+
+One example of a field is the Real Numbers under addition and multiplication, another is a ser of integers mod a prime number with addition and multiplication.
 
+The field operations are required to satisfy the following field axioms. In these axioms, a, b and c are arbitrary element s of the field F.
 
+1. Associativity of addition and multiplication: a+(b+c) = (a+b)+c and a•(b•c)=(a•b)•c
+
+2. Commutativity of addition and multiplication: a+b=b+a and a•b=b•a
+
+3. Additive and multiplicative identity. Additve = 0, a + 0 = a, multiplicative = 1, a • 1 = a
+
+4. Additive inverse: For every a in F, there exists an element in F, denoted -a, called the additive invers of a, such that a + (-a) = 0
+
+5. Multicative inverse: For every a ≠ 0 in F, the exists an element in F, denoted by a<sup>-1</sup>, called the multiplicative inverse of a, such that a•a<sup>-1</sup>=1
+
+6. Distributivity of multiplication over addition: a•(b+c) = (a•b)+(a•c)
+
+### Finite fields and generators
+
+### Proving systems
+
+- Instance variables --> which are public
+- Witness variables ---> which are private
+
+
+- Interactive proofs --> Multiple rounds
+- Non-interactive proofs ---> no repeated communications between the prover and the verifier
+- Succint --->
+- Non Succint --->
+- Proof
+- Proof of Knowledge --->
+- Argument
+
+- [Homework 1](./homework/homework1.py)
+
+## Lesson 2: ZKP Theory / Zokrates
+
+- Homework 2

lesson2.md

@@ -0,0 +1,48 @@
+# Lesson 2
+
+This stuff is hard. Don't feel bad if you don't get it.
+
+- Fully Homomorphic encryption
+
+Plonomials in ZKPs
+
+If a prover claims to know some polynomial (no matter how large its degree is) that the verifier also knows, they can follow a simple protocol to verify the statement:
+
+- verifier chooses a random value for x and evaluates his polynomial locally
+- Verifier gives x to the prover and asks to evaluate the polynimial in question
+- prover evaluates her polynomial at x and the result to the verifier
+- Verifier checks if the local result is equal to the prover's result, and if so then the statement is proven with a high confidence
+
+Why is degree important
+
+in general, there is a rule that if a polynomial is zeor accross some set
+
+S = x1, x2 ... sn then it can be expressed as
+
+P(x) = Z(x) * H(s), where Z(x) = (x-x1) • (x-x2) •...•(x-xn) and H(x) is also a polynomial.
+
+In other words, any polynomial that equals zero accross set is a (polynomial) mulitiple of the (lowest-degree) polynomial that equals zero across that same set.
+
+## Homomorphic Hiding
+
+Taken from Zcash explanation
+
+if E(x) is a function with the following properties.
+
+- Given E(x) it is hard to find x
+- Different inputs lead to different outputs so if x≠yE(x) ≠ E(y)
+- We can compute E(x+y) given E(x) and E(y)
+
+The group Z<sub>p</sub> with operations addition and multiplication allows this.
+
+Here's a toy example of why Homomorphic Hiding is useful for Zero-Knowledge proofs.
+
+Suppoese Alice wants to prove to bob she knows numbers x,y such taht x+y = 7
+
+1. Alice sends E(x) and E(y) to Bob.
+2. Bob computes E(x+y) from these values (which he is to do since E is an HH).
+3. Bob also computes E(7), and now checks whether E(x+y) = E(7). He accepts Alice's proof pnly if equality holds.
+
+As different inputs are mapped by E to different hidings. Bob indedd accepts the proof oonly if Alice sent hidings of x,y such that x + y = 7. On the other hand, Bob does not learn x and y he just has acess to their hidings.
+
+## ZoKrates - xkSNARKs on Ethereum