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You can make a proof for the statement "I know a secret number such that if you take the word ‘cow', add the number to the end, and SHA256 hash it 100 million times, the output starts with 0x57d00485aa". The verifier can verify the proof far more quickly than it would take for them to run 100 million hashes themselves, and the proof would also not reveal what the secret number is.

In the context of blockchains, this has 2 very powerful applications: Perhaps the most powerful cryptographic technology to come out of the last decade is general-purpose succinct zero knowledge proofs, usually called zk-SNARKs ("zero knowledge succinct arguments of knowledge"). A zk-SNARK allows you to generate a proof that some computation has some particular output, in such a way that the proof can be verified extremely quickly even if the underlying computation takes a very long time to run. The "ZK" part adds an additional feature: the proof can keep some of the inputs to the computation hidden.

You can make a proof for the statement "I know a secret number such that if you take the word ‘cow', add the number to the end, and SHA256 hash it 100 million times, the output starts with 0x57d00485aa". The verifier can verify the proof far more quickly than it would take for them to run 100 million hashes themselves, and the proof would also not reveal what the secret number is.

In the context of blockchains, this has two very powerful applications:

1. Scalability: if a block takes a long time to verify, one person can verify it and generate a proof, and everyone else can just quickly verify the proof instead
2. Privacy: you can prove that you have the right to transfer some asset (you received it, and you didn't already transfer it) without revealing the link to which asset you received. This ensures security without unduly leaking information about who is transacting with whom to the public.

But zk-SNARKs are quite complex; indeed, as recently as in 2014-17 they were still frequently called "moon math". The good news is that since then, the protocols have become simpler and our understanding of them has become much better. This post will try to explain how ZK-SNARKs work, in a way that should be understandable to someone with a medium level of understanding of mathematics.

Why ZK-SNARKs "should" be hard

Let us take the example that we started with: we have a number (we can encode "cow" followed by the secret input as an integer), we take the SHA256 hash of that number, then we do that again another 99,999,999 times, we get the output, and we check what its starting digits are. This is a huge computation.

A "succinct" proof is one where both the size of the proof and the time required to verify it grow much more slowly than the computation to be verified. If we want a "succinct" proof, we cannot require the verifier to do some work per round of hashing (because then the verification time would be proportional to the computation). Instead, the verifier must somehow check the whole computation without peeking into each individual piece of the computation.

One natural technique is random sampling: how about we just have the verifier peek into the computation in 500 different places, check that those parts are correct, and if all 500 checks pass then assume that the rest of the computation must with high probability be fine, too?

Such a procedure could even be turned into a non-interactive proof using the Fiat-Shamir heuristic: the prover computes a Merkle root of the computation, uses the Merkle root to pseudorandomly choose 500 indices, and provides the 500 corresponding Merkle branches of the data. The key idea is that the prover does not know which branches they will need to reveal until they have already "committed to" the data. If a malicious prover tries to fudge the data after learning which indices are going to be checked, that would change the Merkle root, which would result in a new set of random indices, which would require fudging the data again... trapping the malicious prover in an endless cycle.

But unfortunately there is a fatal flaw in naively applying random sampling to spot-check a computation in this way: computation is inherently fragile. If a malicious prover flips one bit somewhere in the middle of a computation, they can make it give a completely different result, and a random sampling verifier would almost never find out.

It only takes one deliberately inserted error, that a random check would almost never catch, to make a computation give a completely incorrect result.

If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? There is a clever solution.

The international law firm Holland & Knight used an NFT built and airdropped by its asset recovery team to serve a defendant in a hacking case.

The law firms Holland & Knight and Bluestone used a nonfungible token to serve a defendant in a hacking case with a temporary restraining order, marking the first documented legal process assisted by an NFT.

The so-called "service token" or "service NFT" was served to an unknown defendant in a hacking case involving LCX, a cryptocurrency exchange based in Liechtenstein that was hacked for over $8 million in January. The attack compromised the platform's hot wallets, resulting in the loss of Ether (ETH), USD Coin (USDC), and other cryptocurrencies, according to Cointelegraph at the time.

On June 7, LCX claimed that around 60% of the stolen cash had been frozen, with investigations ongoing in Liechtenstein, Ireland, Spain, and the United States. Based on a court judgment from the New York Supreme Court, Centre Consortium, a company created by USDC issuer Circle and crypto exchange Coinbase, has frozen around $1.3 million in USDC.

The monies were laundered through Tornado Cash, according to LCX, but were later tracked using "algorithmic forensic analysis." The organization was also able to identify wallets linked to the hacker as a result of the investigation.

In light of these findings, the law firms representing LCX, Holland & Knight and Bluestone, served the unnamed defendant with a temporary restraining order issued on-chain using an NFT. According to LCX, this system "was allowed by the New York Supreme Court and is an example of how innovation can bring legitimacy and transparency to a market that some say is ungovernable."

Six Thinking Hats was written by Dr. Edward de Bono. "Six Thinking Hats" and parallel thinking allow groups to plan thinking processes in a detailed and cohesive way, improving collaboration.

Fundamental ideas

In order to develop strategies for thinking about specific issues, the method assumes that the human brain thinks in a variety of ways that can be intentionally challenged. De Bono identifies six brain-challenging directions. In each direction, the brain brings certain issues into conscious thought (e.g. gut instinct, pessimistic judgement, neutral facts). Some may find wearing hats unnatural, uncomfortable, or counterproductive.

The example of "mismatch" sensitivity is compelling. In the natural world, something out of the ordinary may be dangerous. This mode causes negative judgment and critical thinking.

Colored hats represent each direction. Putting on a colored hat symbolizes changing direction, either literally or metaphorically. De Bono first used this metaphor in his 1971 book "Lateral Thinking for Management" to describe a brainstorming framework. These metaphors allow more complete and elaborate thought separation. Six thinking hats indicate ideas' problems and solutions.

Similarly, his CoRT Thinking Programme introduced "The Five Stages of Thinking" method in 1973.

Strategies and programs

After identifying the six thinking modes, programs can be created. These are groups of hats that encompass and structure the thinking process. Several of these are included in the materials for franchised six hats training, but they must often be adapted. Programs are often "emergent," meaning the group plans the first few hats and the facilitator decides what to do next.

The group agrees on how to think, then thinks, then evaluates the results and decides what to do next. Individuals or groups can use sequences (and indeed hats). Each hat is typically used for 2 minutes at a time, although an extended white hat session is common at the start of a process to get everyone on the same page. The red hat is recommended to be used for a very short period to get a visceral gut reaction – about 30 seconds, and in practice often takes the form of dot-voting.

Use

Speedo's swimsuit designers reportedly used the six thinking hats. "They used the "Six Thinking Hats" method to brainstorm, with a green hat for creative ideas and a black one for feasibility.

Typically, a project begins with extensive white hat research. Each hat is used for a few minutes at a time, except the red hat, which is limited to 30 seconds to ensure an instinctive gut reaction, not judgement. According to Malcolm Gladwell's "blink" theory, this pace improves thinking.

De Bono believed that the key to a successful Six Thinking Hats session was focusing the discussion on a particular approach. A meeting may be called to review and solve a problem. The Six Thinking Hats method can be used in sequence to explore the problem, develop a set of solutions, and choose a solution through critical examination.

Everyone may don the Blue hat to discuss the meeting's goals and objectives. The discussion may then shift to Red hat thinking to gather opinions and reactions. This phase may also be used to determine who will be affected by the problem and/or solutions. The discussion may then shift to the (Yellow then) Green hat to generate solutions and ideas. The discussion may move from White hat thinking to Black hat thinking to develop solution set criticisms.

Because everyone is focused on one approach at a time, the group is more collaborative than if one person is reacting emotionally (Red hat), another is trying to be objective (White hat), and another is critical of the points which emerge from the discussion (Black hat). The hats help people approach problems from different angles and highlight problem-solving flaws.