The set of Integers is denoted by $\mathbb{Z}$ e.g. {⋯,−4,−3,−2,−1,0,1,2,3,4,⋯}
The set of Rational Numbers is denoted by $\mathbb{Q}$ e.g. $\{...1,\frac{3}{2},2,\frac{22}{7}...\}$
The set of Real Numbers is denoted by $\mathbb{R}$ e.g. $\{2,-4,613,\pi ,\sqrt{2},\dots \}$
Fields are denoted by $\mathbb{F}$, if they are a finite field or $\mathbb{K}$ for a field of real or complex numbers we also use ${\mathbb{Z}}_{p}^{\ast}$ 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.
When we write n mod k we mean simply the remainder when n is divided by k. Thus
25 mod 3 = 1,
15 mod 4 = 3,
−13 mod 5 = -3 , = 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)
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
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 $\mathbb{F}$.
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
An element can be represented as an integer greater or equal than 0 and less than the field’s order: {0, 1, …, p-1} in a simple field.
In a finite field of order $q$, the polynomial ${X}^{q}-X$ has all $q$ elements of the finite field as roots.
A homomorphism is a map between two algebraic structures of the same type, that preserves the operations of the structures.
This means a map $f:A\to B$ between two groups A, B equipped with the same structure such that,
if $\cdot $ is an operation of the structure (here a binary operation), then
$f(x\cdot y)=f(x)\cdot f(y)$
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. $3{x}^{2}+4x+3$
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. $5{x}^{2}+2x+1$) will go through them etc.
For n points, you can create a n-1 degree polynomial that will go through all of the points.
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
$(x-r)(Q(x))$
where
$Q(x)$ is a polynomial of degree $n-1$.
$Q(x)$ 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”.
The defining equation for an elliptic curve is for example ${y}^{2}={x}^{3}+ax+b$
For certain equations they will satisfy the group axioms
We often use 2 families of curves :
For example curve 22519 with equation ${y}^{2}={x}^{3}+486662{x}^{2}+x$
Generally this curve is considered over a finite field $\mathbb{K}$ (with order different from 2)
Curve 25519 gives 128 bits of security and is used in the Diffie–Hellman (ECDH) key agreement scheme
BN254 / BN_128 is the curve used in Ethereum for ZKSNARKS
BLS12-381 is the curve used by ZCash
The general equation is $a{x}^{2}+{y}^{2}=1+d{x}^{2}{y}^{2}$ with a = 1
If a <> 1 they are called Twisted Edwards Curves
Every twisted Edwards curve is birationally equivalent to a Montgomery curve
Bilinear pairings are functions that take two arguments and return one output, usually denoted by
e(G1, G2) --> GT.
with the following properties:
Order:All three groups must have order equal to a prime r.
Efficiency: the pairing function must be efficiently computable.
Bilinearity: For any elements P1, P2 of G1 and any elements Q1, Q2 of G2, the following holds true:
e(P1 + P2, Q1) = e(P1, Q1) e(P2, Q1)
e(P1, Q1 + Q2) = e(P1, Q1) e(P1, Q2)
This implies the following form, which is more often used (along with some other variants):
e(aP, bQ) = e(P,Q)ab = e(bP, aQ)
Non-degeneracy: the pairing of the generators of the first two groups is not the identity of the third group. If this were the case, every pairing would result in the same (the identity) element of GT:
$e(E1,E2)\ne 1T$
G2 is an elliptic curve, where points satisfy the same equation as G1, except where the coordinates are elements of ${F}_{{p}^{12}}$ (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 ${F}_{{p}^{12}}$
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 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.
(Taken from the ZCash explanation)
If $E(x)$ is a function with the following properties
The group ${\mathbb{Z}}_{p}^{\ast}$ with operations addition and multiplication allows this.
For a cyclic group ${G}_{q}$ of order $q$ with generator $g$.
In order to prove knowledge of $x={\mathrm{log}}_{g}y$, the prover interacts with the verifier as follows:
In the first round the prover commits himself to randomness $r$ therefore the first message $t={g}^{r}$ is also called commitment.
The verifier replies with a challenge $c$ chosen at random.
After receiving $c$, the prover sends the third and last message (the response) $s=r+cx$
The verifier accepts, if ${g}^{s}=t{y}^{c}$
Protocols which have the above three-move structure (commitment, challenge and response) are called sigma protocols
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 ${n}^{k}$ 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
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 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 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
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.