Ethereum zk-snark

This article was share, mouse control select one of. From that person. Select a category and click 'OK'. SCP is an provide bug-fix support activity, have a. For more information about other troubleshooting.

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The word "complete" here is the same complete as in "Turing-complete". There is a reduction function that translates a transaction into a boolean formula, such that the formula is satisfiable if and only if the transaction is valid. Reduction Example In order to see such a reduction, let us consider the problem of evaluating polynomials.

First, let us define a polynomial similar to a boolean formula as an expression consisting of integer constants, variables, addition, subtraction, multiplication and correctly balanced parentheses. It suffices to define the reduction function r on the structural elements of a boolean formula. The idea is that for any boolean formula f, the value r f is a polynomial with the same number of variables and f a1,..

In our example, we only defined how to translate the formula to a polynomial, but with the proof we explained how to transform the witness, the satisfying assignment. This simultaneous transformation of the witness is not required for a transaction, but it is usually also done. This is quite important for zkSNARKs, because the the only task for the prover is to convince the verifier that such a witness exists, without revealing information about the witness.

Quadratic Span Programs In the previous section, we saw how computational problems inside NP can be reduced to each other and especially that there are NP-complete problems that are basically only reformulations of all other problems in NP - including transaction validation problems. So if we want to show how to validate transactions with zkSNARKs, it is sufficient to show how to do it for a certain problem that is NP-complete and perhaps much easier to work with theoretically.

A Quadratic Span Program consists of a set of polynomials and the task is to find a linear combination of those that is a multiple of another given polynomial. Furthermore, the individual bits of the input string restrict the polynomials you are allowed to use. In detail the general QSPs are a bit more relaxed, but we already define the strong version because that will be used later : A QSP over a field F for inputs of length n consists of a set of polynomials v0, For each binary input string u, the function f restricts the polynomials that can be used, or more specific, their factors in the linear combinations.

Note that there is still some freedom in choosing the tuples a and b if 2n is smaller than m. This means QSP only makes sense for inputs up to a certain size - this problem is removed by using non-uniform complexity, a topic we will not dive into now, let us just note that it works well for cryptography where inputs are generally small. As an analogy to satisfiability of boolean formulas, you can see the factors a1, To see that QSP lies in NP, note that all the verifier has to do once she knows the factors is checking that the polynomial t divides va wb, which is a polynomial-time problem.

In practice, the reduction is the actual "engineering" part - it has to be done in a clever way such that the resulting QSP will be as small as possible and also has some other nice features. One thing about QSPs that we can already see is how to verify them much more efficiently: The verification task consists of checking whether one polynomial divides another polynomial. This looks rather easy, but the polynomials we will use later are quite large the degree is roughly times the number of gates in the original circuit so that multiplying two polynomials is not an easy task.

So a bunch of polynomial additions, multiplications with a scalar and a polynomial product is simplified to field multiplications and additions. Checking a polynomial identity only at a single point instead of at all points of course reduces the security, but the only way the prover can cheat in case t h - va wb is not the zero polynomial is if she manages to hit a zero of that polynomial, but since she does not know s and the number of zeros is tiny the degree of the polynomials when compared to the possibilities for s the number of field elements , this is very safe in practice.

It starts with a setup phase that has to be performed for every single QSP. In zCash, the circuit the transaction verifier is fixed, and thus the polynomials for the QSP are fixed which allows the setup to be performed only once and re-used for all transactions, which only vary the input u.

For the setup, which generates the common reference string CRS , the verifier chooses a random and secret field element s and encrypts the values of the polynomials at that point. The CRS also contains several other values which makes the verification more efficient and also adds the zero-knowledge property. The encryption E used there has a certain homomorphic property, which allows the prover to compute E v s without actually knowing vk s. How to Evaluate a Polynomial Succinctly and with Zero-Knowledge Let us first look at a simpler case, namely just the encrypted evaluation of a polynomial at a secret point, and not the full QSP problem.

For this, we fix a group an elliptic curve is usually chosen here and a generator g. Remember that a group element is called generator if there is a number n the group order such that the list g0, g1, g2, This is exactly what zCash calls toxic waste, because if someone can recover this and the other secret values chosen later, they can arbitrarily spoof proofs by finding zeros in the polynomials.

The only problem here is that, because s was destroyed, the verifier cannot check that the prover evaluated the polynomial correctly. She does this by using another main ingredient: A so-called pairing function e. The answer to this question is "we hope not". Seriously, this is called the "d-power knowledge of exponent assumption" and it is unknown whether a cheating prover can do such a thing or not.

This assumption is an extension of similar assumptions that are made for proving the security of other public-key encryption schemes and which are similarly unknown to be true or not. What this example does show is that the verifier does not need to evaluate the full polynomial to confirm this, it suffices to evaluate the pairing function.

In the next step, we will add the zero-knowledge part so that the verifier cannot reconstruct anything about f s , not even E f s - the encrypted value. If we assume that the encryption cannot be broken, the zero-knowledge property is quite obvious. We now have to check two things: 1. For 1. For 2. Ok, so we now know a bit about how the prover can compute the encrypted value of a polynomial at an encrypted secret point without the verifier learning anything about that value.

Let us now apply that to the QSP problem. The prover finds a1, In the previous section, we already explained how the common reference string CRS is set up. In practical implementations, some elements of the CRS are not needed, but that would complicated the presentation. Now what does the prover do? She uses the reduction explained above to find the polynomial h and the values a1, Here it is important to use a witness-preserving reduction see above because only then, the values a1, In order to describe what the prover sends to the verifier as proof, we have to go back to the definition of the QSP.

Since m is relatively large, there are numbers which do not appear in the output of f for any input. Note that all these encrypted values can be generated by the prover knowing only the CRS. The addition comes from the fact that the encryption itself is already additively homomorphic and the single multiplication is realized by the two arguments the pairing function has. If you remember from the section about evaluating polynomials at secret points, these three first checks basically verify that the prover did evaluate some polynomial built up from the parts in the CRS.

The second item is used to verify that the prover used the correct polynomials v and w and not just some arbitrary ones. Assuming the prover provided a correct proof, let us check that the equality works out. We will see now how to add zero-knowledge and the next section will be touch a bit more on the succinctness. The idea is that the prover "shifts" some values by a random secret amount and balances the shift on the other side of the equation.

By these replacements, the values Vfree and W, which contain an encoding of the witness factors, basically become indistinguishable form randomness and thus it is impossible to extract the witness. Most of the equality checks are "immune" to the modifications, the only value we still have to correct is H or h s. Tradeoff between Input and Witness Size As you have seen in the preceding sections, the proof consists only of 7 elements of a group typically an elliptic curve.

Furthermore, the work the verifier has to do is checking some equalities involving pairing functions and computing E vin s , a task that is linear in the input size. Remarkably, neither the size of the witness string nor the computational effort required to verify the QSP without SNARKs play any role in verification.

This means that SNARK-verifying extremely complex problems and very simple problems all take the same effort. The main reason for that is because we only check the polynomial identity for a single point, and not the full polynomial. Polynomials can get more and more complex, but a point is always a point. The only parameters that influence the verification effort is the level of security i. This means we replace the input u by a hash of the input h u which is supposed to be much shorter and verify that there is some value x that hashes to H u and thus is very likely equal to u in addition to checking f x, w.

This basically moves the original input u into the witness string and thus increases the witness size but decreases the input size to a constant. This is because from the analysis of the history of transactions one can also arrive at the location of the individual. Just look at the location of the merchant who accepts payments. Criminals may use this information to steal from you or follow you. The payment of a particular medical service could be used by insurance companies to increase the instalment or even to refuse medical cover.

There are so many scenarios and so many privacy risks. The privacy of a blockchain can be improved There are ways to strengthen privacy by using new addresses or exchanging funds with other users to mix transactions, but these practices always leave a trace while analysis methods improve. This system is different from other systems where at least one party needs to know all the information. With zk-SNARK it is possible to prove the possession of a password through a mathematical test without ever having to reveal it.

The developer wants to focus on creating an ecosystem of smart contracts that protect privacy and are easy to use. Buterin believes that privacy is necessary and useful in many cases. They can be used to improve various types of mechanisms such as auctions. The dapp is currently on the testnet and is still in development, but its operation is simple. It must hide the sender, the recipient and the amount of the transaction.

BarryWhiteHat, the developer who developed Miximus, works on this technology that could be implemented in voting, governance or an anonymous social network. This may be possible because the identity in Miximus is also anonymous. A user can try to be part of a group by remaining anonymous. For this specific feature, we assume the use of the login with GitHub would transfer the reputation, but not the identity.