Maths Primer for Zero Knowledge Workshop

Terminology

Numbers

The set of Integers is denoted by Z e.g. {⋯,−4,−3,−2,−1,0,1,2,3,4,⋯}
The set of Rational Numbers is denoted by Q e.g. {...1,32,2,227...}
The set of Real Numbers is denoted by R e.g. {2,4,613,π,2,}

Fields are denoted by F, if they are a finite field or K for a field of real or complex numbers we also use Zp to represent a finite field of integers mod prime p with multiplicative inverses, these concepts will be explained further later.

We use finite fields for cryptography, because elements have “short”, exact representations.

Modular Arithmetic

When we write n mod k we mean simply the remainder when n is dividedby k. Thus
25 = 1 mod 3,
15 = 3 mod 4,
−13 =−3 mod 5 = 2 mod 5.
It is an important fact that modular arithmetic respects sums and products.
That is,
a+b mod n = a mod n+ b mod n
and
a·b mod n=(a mod n)·(b mod n)

Group Theory

Simply put a group is a set of elements {a,b,c,…} plus a binary operation, here we represent this as •
To be considered a group this combination needs to have certain properties

  1. Closure
    For all a, b in G, the result of the operation, a • b, is also in G
  2. Associativity
    For all a, b and c in G, (a • b) • c = a • (b • c)
  3. Identity element
    There exists an element e in G such that, for every element a in G, the equation e • a = a • e = a holds. Such an element is unique and thus one speaks of the identity element.
  4. Inverse element
    For each a in G, there exists an element b in G, commonly denoted a1 (or −a, if the operation is denoted “+”), such that a • b = b • a = e, where e is the identity element.

Fields

Formally, 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 set 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 elements 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: there exist two different elements 0 and 1 in F such that a + 0 = a and a · 1 = a.
  4. Additive inverses: for every a in F, there exists an element in F, denoted −a, called the additive inverse of a, such that a + (−a) = 0.
  5. Multiplicative inverses: for every a ≠ 0 in F, there exists an element in F, denoted by a1, called the multiplicative inverse of a, such that a ·a1 = 1.
  6. Distributivity of multiplication over addition: a · (b + c) = (a · b) + (a · c).

To try out operations on finite fields, see https://asecuritysite.com/encryption/finite

For a great introduction see http://coders-errand.com/zk-snarks-and-their-algebraic-structure/

The order of the field is the number of elements in the field’s set.
For a finite field the order must be either

In the simpler case, an element can be represented as an integer greater or equal than 0 and less than the field’s order: {0, 1, …, p-1}.

When a finite field has a non-prime (ie ‘composite’) order its elements can be represented as polynomials, and the field is called an extension field.

Group Homomorphisms

A homomorphism is a map between two algebraic structures of the same type, that preserves the operations of the structures.

This means a map f:AB between two groups A, B equipped with the same structure such that,

if is an operation of the structure (here a binary operation), then
f(xy)=f(x)f(y)

Polynomials

A polynomial is an expression that can be built from constants and variables by means of addition, multiplication and exponentiation to a non-negative integer power.

e.g. 3x2+4x+3

Lagrange Interpolation

If you have a set of points then doing a Lagrange interpolation on those points gives you a polynomial that passes through all of those points.

If you have two points on a plane, you can define a single straight line that passes through both, for 3 points, a single 2nd-degree curve (e.g. 5x2+2x+1) will go through them etc.
For n points, you can create a n-1 degree polynomial that will go through all of the points.

Adding, multiplying and dividing polynomials

We can add, multiply and divide polynomials, we don’t need to go onto the details here, for examples see https://en.wikipedia.org/wiki/Polynomial_arithmetic

For a polynomial P of a single variable x in a field K and with coefficients in that field, the root r of P is an element of K such that P(r)=0

B is said to divide another polynomial A when the latter can be written as

A=BC

with C also a polynomial,the fact that B divides A is denoted B|A

If one root r of a polynomial P(x) of degree n is known then polynomial long division can be used to factor P(x) into the form
(xr)(Q(x))
where
Q(x) is a polynomial of degree n1.
Q(x) is simply the quotient obtained from the division process; since r is known to be a root of P(x), it is known that the remainder must be zero.

Schwartz-Zippel Lemma stating that “different polynomials are different at most points”.

Elliptic Curves

The defining equation for an elliptic curve is for example y2=x3+ax+b
For certain equations they will satisfy the group axioms

We often use 2 families of curves :

Montgomery Curves

Example Curve

For example curve 22519 with equation y2=x3+486662x2+x
Generally this curve is considered over a finite field K (with order different from 2)

BN254 / BN_128 is the curve used in Ethereum for ZKSNARKS
BLS12-381 is the curve used by ZCash

Edwards Curves

The general equation is ax2+y2=1+dx2y2 with a = 1

If a <> 1 they are called Twisted Edwards Curves
Every twisted Edwards curve is birationally equivalent to a Montgomery curve

Pairings

Bilinear pairings are functions that take two arguments and return one output, usually denoted by

    e(G1, G2) --> GT.

A bilinear pairing is a function e: G1 x G2 --> GT with the following properties:

G2 is an elliptic curve, where points satisfy the same equation as G1, except where the coordinates are elements of Fp12 (this is an extension field, where the elements of the field are polynomials of degree 12)
Gt is the type of object that the result of the elliptic curve goes into. In the curves that we look at, Gt is also Fp12

Complexity Theory

Complexity theory looks at the time or space requirements to solve a problem, particularly in terms of the size of the input.
We can classify problems according to the time required to find a solution, for some problems there may exist an algorithm to find a solution in a reasonable time, whereas for other problems we may not know of such an algorithm, and may have to ‘brute force’ a solution, trying out all potential solutions until one is found.

For example the travelling salesman problem tries to find the shortest route for a salesman required to travel between a number of cities, visiting every city exactly once. For a small number of cities, say 3, we can quickly try all alternatives to find the shortest route, however as the number of cities grows, this quickly becomes unfeasible.

Based on the size of the input n , we classify problems according to how the time required to find a solution grows with n.
If the time taken in the worst case grows as a polynomial of n, that is roughly proportional to nk for some value k, we put these problems in class P. These problems are seen as tractable.

We are also interested in knowing how long it takes to verify a potential solution once it has been found.

A computational problem can be viewed as an infinite collection of instances together with a solution for every instance. The input string for a computational problem is referred to as a problem instance, and should not be confused with the problem itself.

https://en.wikipedia.org/wiki/Computational_complexity_theory

Decision Problem: A problem with a yes or no answer

Complexity Classes

Complexity Classes

P

P is a complexity class that represents the set of all decision problems that can be solved in polynomial time. That is, given an instance of the problem, the answer yes or no can be decided in polynomial time.

NP

NP is a complexity class that represents the set of all decision problems for which the instances where the answer is “yes” have proofs that can be verified in polynomial time.
This means that if someone gives us an instance of the problem and a witness to the answer being yes, we can check that it is correct in polynomial time.

NP-Complete

NP-Complete is a complexity class which represents the set of all problems X in NP for which it is possible to reduce any other NP problem Y to X in polynomial time.
Intuitively this means that we can solve Y quickly if we know how to solve X quickly. Precisely, Y is reducible to X, if there is a polynomial time algorithm f to transform instances y of Y to instances
x = f(y) of X in polynomial time, with the property that the answer to y is yes, if and only if the answer to f(y) is yes

NP-hard

Intuitively, these are the problems that are at least as hard as the NP-complete problems. Note that NP-hard problems do not have to be in NP, and they do not have to be decision problems.
The precise definition here is that a problem X is NP-hard, if there is an NP-complete problem Y, such that Y is reducible to X in polynomial time.
But since any NP-complete problem can be reduced to any other NP-complete problem in polynomial time, all NP-complete problems can be reduced to any NP-hard problem in polynomial time. Then, if there is a solution to one NP-hard problem in polynomial time, there is a solution to all NP problems in polynomial time.

Homomorphic Encryption

Homomorphic encryption is a form of encryption with an additional evaluation capability for computing over encrypted data without access to the secret key. The result of such a computation remains encrypted. Homomorphic encryption can be viewed as an extension of either symmetric-key or public-key cryptography. Homomorphic refers to homomorphism in algebra: the encryption and decryption functions can be thought as homomorphisms between plaintext and ciphertext spaces.

Bitcoin split-key vanity mining

Bitcoin addresses are hashes of public keys from ECDSA key pairs. A vanity address is an address generated from parameters such that the resultant hash contains a human-readable string (e.g., 1BoatSLRHtKNngkdXEeobR76b53LETtpyT). Given that ECDSA key pairs have homomorphic properties for addition and multiplication, one can outsource the generation of a vanity address without having the generator know the full private key for this address.

For example,
Alice generates a private key (a) and public key (A) pair, and publicly posts A.
Bob generates a key pair (b, B) such that hash(A + B) results in a desired vanity address. He sells b and B to Alice.
A, B, and b are publicly known, so one can verify that the address = hash(A + B) as desired.
Alice computes the combined private key (a + b) and uses it as the private key for the public key (A + B).
Similarly, multiplication could be used instead of addition.