Zero-Knowledge Proofs: Beginners Guide 2026

Zero-Knowledge Proofs (ZKPs) can be extremely powerful tools for cryptography that have an array of useful uses. In this post well give brief outline of the basic principles of zero-knowledge proofs and not go too deeply into the mathematical concepts.

The discussion will focus on the fundamental theories of Zero-Knowledge Proofs with focus on fundamental concepts that were developed within computer science in the 80s. (So well not cover the latest protocols like ZK-SNARKs or Bulletproofs.)

If youve heard of the concept and wish to learn more about it or want to know more regarding how zero-knowledge proofs could have relevance to your project the following guide can be an excellent place to begin.

What are proofs of zero-knowledge?

Zero-knowledge proofs is cryptographic technique that enables one also known as provers in order to convince another person or verifier an assertion is factual and without divulging additional facts. The verifier is only informed of confirmation that the claim is factual.

A short background of Zero-Knowledge Proofs

The idea of Zero-Knowledge Proofs was first proposed in seminal research paper released in 1985 by the researchers Shafi Goldwasser Silvio Micali along with Charles Rackoff.

To understand the significance of their invention Lets look at the conventional concept of proof:

A typical proof does not just confirms the validity of claim but gives additional evidence to show how it was proved to be true.

Imagine that you have solved complicated mathematical equation. It is possible to show your partner the answer to show that your ability to solve it but youd also be revealing the solution. Do you have method to show that youve solved the problem without divulging the answer?

The result is more general issue:

Can you prove the authenticity of an assertion with no additional information other than the fact that it is accurate?

Goldwasser Micali and Rackoff offered positive response to the question. It resulted in the creation of the present-day Zero-knowledge evidences.

Since the time Zero-Knowledge Proofs have evolved from an abstract concept to being able to find various practical applications most notably those where security and privacy are crucial.

Proofs are not the same as. the evidence of understanding

Zero-knowledge proofs can be described as “zero-knowledge proofs of knowledge.”

These terms are used to describe similar yet technically different concepts theyre often employed interchangeably in the latest literature.

In essence proofs of zero-knowledge of knowledge constitute subset of proofs that are zero-knowledge. However for the most common purposes the term zero-knowledge proofs it refers to those proofs that demonstrate the proof-giver proves their understanding of the solution rather than merely proving the existence of solution. In this piece the term “ZKPs” is used to mean ZKPs which are zero-knowledge proofs (ZKPs) We specifically are referring to zero-knowledge proofs.

The Zero-Knowledge Proofs concept through the use of an example

Lets look at ZKPs through two scenarios:

1. Finding Waldo

Imagine looking at an “Wheres Waldo?” puzzle and discover Waldo hiding in the image. It is your goal to show another person that you are aware of exactly where Waldo is but do not want to disclose the exact spot of Waldo.

A way to accomplish this is to use huge piece of cardstock and make an indentation that is large enough to display Waldo but nothing else. Then you place the piece on top of the puzzle so where only Waldo is seen by the hole. In this way you can convince another individual that you can tell the exact location of Waldo is without divulging the position of his body.

The metaphor embodies the fundamentals of an evidence of zero knowledge: It is way to prove that you are aware of about something (Waldos place of residence) but without divulging the information in itself (the exact location of Waldo within the puzzle).

2. Coloring graphs

Coloring graphs is well-known mathematical problem.

Imagine graph made up from nodes (points) as well as edges (lines that connect the points) The aim is to shade each node so that two adjacent nodes have the exact color. This is done using just three colors. Imagine that you have found reliable color for the graph and would like to prove it an independent verifier without disclosing the exact hues.

For this it is possible to hide every nodes colors by putting it inside an envelope. Verifiers can then choose the edge they prefer. If they do they look through the envelopes to find the two nodes that are connected via that edge in order to show that theyre different colored. Although the verifier is able to see the adjacent nodes distinct colors they cant know the general color on the graph.

The verifier will probably remain skeptical after only couple of attempts. In order to convince the verifier that you are professional repeatedly repeat the procedure and repaint the graph in different way every time before showing the color again. The repeated display will convince the person you are aware of the solution without divulging the exact colours (even in part as the graph gets repainted each time).

It is possible to experience the process by yourself as verifier. Check out this interactive demonstration.

Lets take take look at the way Zero-Knowledge Proofs operate on the ground by looking at the key characteristics of ZKPs.

Three-move protocol structure (three-move) protocol design

In this article we will discuss certain type of proof system that is zero-knowledge called S-protocols (Sigma-protocols).

They are particularly efficient and they follow specific three-move pattern: commitment the challenge and then the response. It is collaboration between two programs on computers that are the prover and verifyer.

Commitment (first message):

The proofer creates random number that is used to make the commitment. The decision is contingent on the random value as well as secret data (witness).
The proponent sends the commitment to the verifyer.

Challenge (second message):

The verifyer generates and issues an unintentional challenge to the proofer requesting the prover to reveal few parts of the agreement.

Response (third message):

The verifyer computes an answer using the witness as well as the random number in the commitment step as well as the challenge given by the verifier during the challenge phase.
The proofreader sends the answer to the verifier which checks the response against the original agreement and challenges it to verify that it is correct.

You might have noticed that this graph color illustration previously used fits exactly the three steps:

The proponent is bound to the correct coloration on the graph.
The verifier will send request to determine the color of nodes that are connected to particular edge.
The proofreader responds by providing the information required.

Whats an obligation?

Do you remember the envelopes that were used to conceal the color?

The term “commitment” (or commitment schemes) is the mathematical counterpart of these. commitment is binding agreement that binds the verifyer to certain amount but hides it from the verifyer until the committment is open.

Essential concepts in S-protocols

Interactivity

Zero-knowledge proofs usually rely on an inter-active communication between the prover and verifier. The interactivity of the protocol allows for convincing that the verification officer of the proofers expertise gradually. This protocol must be repeated several times until theres an extremely low chance that the proofer doesnt have the knowledge.

Randomness

A further important aspect is the significance of chance. The verifier will ask number of questions. They select the answers using an electronic version of flipping the coin. It adds randomness to each contest preventing the proofer from “faking” the proof and ensures that every time the process provides an chance to check the verifiers expertise.

The advantages of zero-knowledge network

To be officially recognized as non-knowledgeable it must satisfy three requirements:

Completeness statement is factual person who has the knowledge of proof could convince an independent verifier of its authenticity.
Soundness In the event that proposition cannot be proven then no proof could convince verifier the statement is factual (except at very low likelihood).
Zero-knowledge If fact is true the person verifying it is not able to know anything other than that the statement is accurate. The verification does not provide any further details about the assertion.

The first two characteristics are fairly easy to understand but finding method of formalize zero-knowledge is an actual problem. It is because its not entirely clear how to express the concept of “the verifier learns nothing” by proving.

The simulator algorithm

The answer was derived from Goldwasser Micali Goldwasser and Rackoff Its extremely intriguing.

The argument was that protocol is considered to be zero-knowledge if it is verified by every verifier the algorithm is in place which acts as simulation for zero-knowledge proof. The simulator could generate an account of the interaction between proofs that appears to be identical to genuine proof but not actually knowing the secrets.

Note the simulator is an imaginary construct employed in research exercise. Its not actually part of an actual run through the protocols. The existence of it is employed to show that the protocols transcripts reveal no information about the secrets.

What is the reason for this?

The simulator employs method that is known as “rewinding” that enables users to “revisit” earlier stages of interaction to the verifier. To simplify imagine it as similar process to returning to previous commit made in the context of version control system such as Git.

In order to demonstrate the technique of rewinding Lets take look at how an actual simulator will work with graph coloring:

The initial commit (to randomly selected value) If the simulator is aware of the edge that will be tested prior to the time it is scheduled the simulator can be committed to random values of different colors at the two edges of the edge and to random values in other places. It will not be able to distinguish this from the verifier due to the concealing characteristic of the commit scheme. Therefore the simulator will continually guess at the edge that needs to be checked prior to hopes that the verifier can inquire about this edge.
Challenge and Response The verifier issue challenge at random. If the verifier picks the edge that is expected then the simulator will open envelopes to the two nodes that are connected and the simulation for the entire process is completed. In the event that it is not so then rewinding will be used.
Rewind and try again In the event that the verifiers response isnt in line with the simulations predictions it reverses the verifier back towards the beginning of the process and then tries to do it again selecting different random edge. Repeat this process until the simulator is able to align its responses to the verifiers request in the number of times required.

In the case of the verifier the simulation is similar to the real-time protocols execution. Both times the verifier is able to see an array of “envelopes” and two different randomly colored envelopes being open. One difference is that there is no rewinding during true proof contrary to what is observed in the model. But this isnt obvious to verifier as the transcripts of protocol appear exactly the same.

For more detailed and technical explanation–including how it is possible to technically “rewind” the verifier the relationship between soundness and zero-knowledgeness and what is required to achieve negligible probability–we recommend the Tutorial on the Simulation Proof Technique.

What is the significance of this?

The concept is the creation of the appearance of fake protocol transcript that appears identical to the original one without being aware of the secret behind it–demonstrates that the authentic protocol transcript isnt divulging anything about the secrets.

The examiner is unable to distinguish between genuine proof as opposed to fake this means that they have no knowledge about the actual secret and therefore the procedure is “zero-knowledge.”

*Non-interactive zero-knowledge proofs

Initially zero-knowledge proof systems require prover and verifier to communicate with multiple parties. In some situations it is not necessary or unpractical.

This resulted in the development of zero-knowledge proofs that are non-interactive (NIZKs) Zero-Knowledge Succinct Arguments that are Non-Interactive of Knowledge (zk-SNARKs) as well as Zero-Knowledge Successive transparent Arguments of Knowledge (zk-STARKs).

They eliminate the need to have the prover and verifyer to be connected at the same time which makes them more effective for certain situations. (But this effectiveness comes with some trade-offs).

Practical application

Proofs that are zero knowledge extend to go beyond coloring graphs.

They are able to be applied to every combinatorial issue in which the goal is to demonstrate knowledge while not revealing the truth.

In addition ZKPs have been effectively utilized in complex situations in the real world across variety of areas:

Blockchains and cryptocurrency Zero-Knowledge Proofs are extensively employed in crypto currencies to provide the privacy of transactions. In particular cryptocurrencies with focus on privacy such as Zcash use ZK-SNARKs in order to ensure an anonymity for transactions.
Authentication systems: Consider server authentication. For secure communications server sharing its public key while keeping its secret key secure. Prior to connecting to the server user might need to ensure that its connecting to legitimate server and not fake. In order to do this the client needs to ask the server for non-knowledge test to prove that it is aware of the secret code.
Signatures that are digital are the Schnorr protocol is an essential zero-knowledge proofing technique that is the basis of numerous modern signature systems. In particular the EdDSA signatures are founded on Schnorr and serves as crucial element in various technologies like OpenSSH GnuPG and several other secure protocols for communication.
Secure identity in digital identification systems Zero-Knowledge Proofs may be utilized to guard the privacy of users. The traditional methods of digital identification typically need users to provide greater amounts of information than is necessary which exposes unnecessary identity attributes as well as cryptographic information. By using Zero-Knowledge Proofs people are able to prove just the data required by the verification process and limit data disclosure. BBS/BBS+ signatures and Camenisch-Lysyanskaya signatures are examples of ZKP-based solutions that enable privacy-preserving identification.

ZKPs as well as privacy

Because of their inherent nature zero-knowledge proofs can be the ideal method to prove reliability of computational computations without divulging the inputs. Also they are able to prove that computation was conducted in fair and honest manner. This is what makes Zero-Knowledge Proofs especially valuable in cryptocurrency which are widely utilized.

In their entirety Zero-Knowledge Proofs do not grant total privacy to online interactions. The most determined and resourceful actors may have the ability to trace and monitor the users. When in conjunction with other privacy-preserving tools Zero-Knowledge Proofs make it much more difficult for criminals to gain access to sensitive or personal data. Therefore ZKPs should be considered in conjunction with wider security program that considers the entire range of possible security threats.