Starks explained to my grandma
Kindly note that she had a MS in mathematics though
Let's get cooking!
Low degree polynomial extension with its FRIed constraints
Ingredients
- a problem to solve
- an arithmetization of the problem
- some notions about polynomials
- having heard about finite fields
~ integers bounded by a prime number - knowing that if x0 is a root of a polynomial f, then f has the form
f:x↦(x−x0)⋅g(x)
Step 1
Have a problem to solve
Given an incomplete sudoku grid, I know a valid solution to this grid
Step 2
Arithmetize it
x0x9⋮x72x1x10x73⋯⋯⋱⋯x8x17⋮x80
\begin{bmatrix}
x_0 & x_1 & \cdots & x_8 \\
x_9 & x_{10} & \cdots & x_{17} \\
\vdots & & \ddots & \vdots \\
x_{72} & x_{73} & \cdots & x_{80}
\end{bmatrix}
(xi−1)⋅(xi−2)⋯(xi−9)=0∀i∈[0,80]
(x_i - 1) \cdot (x_i - 2) \cdots (x_i - 9) =0 \quad \forall i \in [0, 80]
(xi−1)⋅(xi+1−1)⋯(xi+8−1)+(xi−2)⋅(xi+1−2)⋯(xi+8−2)+⋯+(xi−9)⋅(xi+1−9)⋯(xi+8−9)=0∀i∈0,8,⋯,72
(x_i - 1) \cdot (x_{i+1} - 1) \cdots (x_{i+8} - 1) \\
+ (x_i - 2) \cdot (x_{i+1} - 2) \cdots (x_{i+8} - 2) \\
+ \cdots \\
+ (x_i - 9) \cdot (x_{i+1} - 9) \cdots (x_{i+8} - 9) \\
=0 \quad \forall i \in {0, 8, \cdots, 72}
All values are in [1,9]
Lines are valid
The Low degree extension sauce
Where the actual Stark journey starts
Choose a special g value and associate all the Sudoku's xi to its exponents
g0g1⋯g80x0x1x80
\begin{vmatrix}
g^0 & x_0 \\
g^1 & x_1 \\
\cdots \\
g^{80} & x_{80}
\end{vmatrix}
Step 1
Interpolate a polynomial going through all those points, let's call it f
Step 2
Then add a shitload of points on the polynomial, say 10 times the original points
Step 3
Commit these points
Step 4
Prepare the boiled constraints
Transform your problem into polynomial constraints
(xi−1)⋅(xi−2)⋯(xi−9)=0∀i∈[0,80]
(x_i - 1) \cdot (x_i - 2) \cdots (x_i - 9) =0 \quad \forall i \in [0, 80]
(f(gi)−1)⋅(f(gi)−2)⋯(f(gi)−9)=0∀i∈[0,80]
(f(g^i) - 1) \cdot (f(g^i) - 2) \cdots (f(g^i) - 9) =0 \quad \forall i \in [0, 80]
giare roots of(f(x)−1)⋅(f(x)−2)⋯(f(x)−9)
g^i \: \text{are roots of} \: (f(x) - 1) \cdot (f(x) - 2) \cdots (f(x) - 9)
∃p0:{p0is a polynomial(f(x)−1)⋅(f(x)−2)⋯(f(x)−9)=(x−g0)⋅(x−g1)⋯(x−g80)∗p0(x)
\exists p_0: \begin{cases}
p_0 \: \text{is a polynomial}\\
(f(x) - 1) \cdot (f(x) - 2) \cdots (f(x) - 9) = (x - g^0) \cdot (x - g^1) \cdots (x - g^{80}) * p_0(x)
\end{cases}
p0:x↦(x−g0)⋅(x−g1)⋯(x−g80)(f(x)−1)⋅(f(x)−2)⋯(f(x)−9)is a polynomial
p_0: x \mapsto \frac {(f(x) - 1) \cdot (f(x) - 2) \cdots (f(x) - 9)} {(x - g^0) \cdot (x - g^1) \cdots (x - g^{80})} \; \text{is a polynomial}
Step 1
Group the constraints
cp:x↦α⋅p0+β⋅p1+γ⋅p2is a polynomial
cp: x \mapsto \alpha \cdot p_0 + \beta \cdot p_1 + \gamma \cdot p_2 \; \text{is a polynomial}
Let's say you have p0, p1 and p2 contraints
Choose 3 random numbers α,β ,γ
Commit evaluations of cp
Step 2
FRI the constraints
or proving that cp is a polynomial
Proving that cp is a polynomial is hard, let's prove that it's close to a polynomial of low degree d instead
What's FRI anyways?
Proving f is close to a polynomial of low degree d
Proving g is close to another polynomial of low degree d/2
Receive a random number α from the verifier
Separate f between odd and even exponents
f(x)=g(x2)+x⋅h(x2)
f′(y)=g(y)+α⋅h(y)
f'(y) = g(y) + \alpha \cdot h(y)
Commit evaluations of f′
Step 1
Rince and repeat
dividing the degree d until d =0
...and commit the constant value of f
Step 2
Let it simmer for 2 hours
and serve it warm to the verifier
What are we serving to the verifier?
- The trace commitment: Merkle(xi∈[0,80])
- The cp commitment: Merkle(cp(x))
- The commitment at all FRI steps
- The constant value after applying FRI until d=0
What to serve with this?
- Starkware's Stark101: https://starkware.co/stark-101/
- Vitalik's Starks article series: https://vitalik.ca/general/2017/11/09/starks_part_1.html
- Proving Brainf*ck: https://aszepieniec.github.io/stark-brainfuck/
Bon appétit !
No time left for questions but do reach out to me for easy questions or Eli for hard ones
Starks explained to my grandma Kindly note that she had a MS in mathematics though
Starks explained to my grandma
By bernardstanislas
Starks explained to my grandma
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