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Ethereum creator Vitalik Buterin is back again with another creation he believes will take blockchain’s security to a whole new level. He calls it Circle STARKS, and I’m here to tell you everything you need to know about it. 
<h3 id="small-fields-changed-the-game">Small fields changed the game</h3> 
Circle STARKs is all about moving away from big, inefficient numbers to smaller, more manageable fields. Originally, STARKs used large 256-bit fields, but these were slow and wasted a lot of space.&nbsp; 
Now, with smaller fields like Goldilocks, Mersenne31, and BabyBear, everything runs faster and more efficiently.&nbsp; Starkware, for instance, can now handle 620,000 Poseidon2 hashes per second on an M3 laptop. 
<img loading="lazy" decoding="async" class="alignnone wp-image-559182 size-full" src="https://www.cryptopolitan.com/wp-content/uploads/2024/07/Screenshot-2024-07-23-at-15.47.46.png" alt="Everything to know about Vitalik Buterin's Circle STARKS" width="1191" height="921" srcset="https://www.cryptopolitan.com/wp-content/uploads/2024/07/Screenshot-2024-07-23-at-15.47.46.png 1191w, https://www.cryptopolitan.com/wp-content/uploads/2024/07/Screenshot-2024-07-23-at-15.47.46-300x232.png 300w, https://www.cryptopolitan.com/wp-content/uploads/2024/07/Screenshot-2024-07-23-at-15.47.46-768x594.png 768w" sizes="(max-width: 1191px) 100vw, 1191px"> 
Vitalik’s Circle STARKs, implemented in Starkware’s stwo and Polygon’s plonky3, offer unique solutions using the Mersenne31 field. 
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One of the main tricks in making hash-based proofs, or any proof, is to prove something about a polynomial by evaluating it at a random point. 
For example, if a proof system needs you to commit to a polynomial P(x), you might need to show P(z) = 0 for a random point z. 
This is simpler than proving things directly about P(x). If you know z in advance, you could cheat by making P(x) fit that point. To stop this, z is chosen after the polynomial is provided, often by hashing the polynomial. 
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It works fine with large fields, like in elliptic curve protocols, but small fields pose a problem. With small fields, an attacker could just try all possible values for z, making it much easier to cheat. 
To solve this, two main methods are used: multiple random checks and extension fields. The first is simple—check the polynomial at several points instead of just one. But this can get inefficient quickly.&nbsp; 
The second method, using extension fields, involves creating new, complex numbers that make it harder to guess z. 
<h3 id="the-magic-of-circle-starks">The magic of Circle STARKs</h3> 
Circle STARKs introduce a clever twist with Circle FRI. Given a prime p, there’s a group of size p-1 with properties that fit this method perfectly. For Mersenne31, this means using a set of points in a specific arrangement. 
<figure style="width: 100%" class="wp-caption aligncenter">
 <img loading="lazy" decoding="async" src="https://vitalik.eth.limo/images/circlestarks/S19YGbh8C.png" alt="Everything to know about Vitalik Buterin's Circle STARKS" width="451" height="451">
 <figcaption class="wp-caption-text">
 Credits: Vitalik Buterin
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</figure> 
The points follow a pattern similar to trigonometry or complex multiplication, making the math work out neatly. In Circle FRI, points are collapsed and combined in a way that keeps reducing their size, making the process efficient.&nbsp; 
Vitalik says this map doubles points on a circle, taking pairs of coordinates and converting them to new points. This method works well with existing 32-bit CPU/GPU operations, making it more efficient than BabyBear. 
Fast Fourier Transforms (FFTs) follow a similar path, converting evaluations of polynomials to coefficients and back. 
<img loading="lazy" decoding="async" class="aligncenter" src="https://vitalik.eth.limo/images/circlestarks/SJBxN-3UR.png" alt="Everything to know about Vitalik Buterin's Circle STARKS" width="451" height="451"> 
Circle FFTs work over what mathematicians call Riemann-Roch spaces, treating multiples of a base polynomial as zero, which simplifies the math. 
In STARK protocols, you often need to prove that a polynomial equals zero at certain points. Normally, you can use a simple line function to show this. In Circle STARKs, it’s a bit trickier because the equivalent line function needs to meet stricter conditions. 
<img loading="lazy" decoding="async" class="aligncenter" src="https://vitalik.eth.limo/images/circlestarks/HkGiSb28A.png" alt="Everything to know about Vitalik Buterin's Circle STARKS" width="451" height="451"> 
To handle this, Circle STARKs use interpolants—functions that equal zero at two points. By subtracting and dividing by these interpolants, you can show that the resulting quotient is a polynomial. 
Vanishing polynomials, which equal zero across an evaluation domain, also play a role. In regular STARKs, this is straightforward. In Circle STARKs, it involves repeating specific functions, which makes sure the math holds up. 
As Vitalik <a href="https://vitalik.eth.limo/general/2024/07/23/circlestarks.html" rel="nofollow noopener" target="_blank">puts</a> it, “The complexity of circle math is encapsulated, not systemic.”

Everything to know about Vitalik Buterin’s Circle STARKS

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