Cryptographic Commitments (The Math): Locking Your Cryptographic Answer in a Box

Cryptography has a recurring problem.

Sometimes, I want to tell you that I know something.

But, I don’t want to tell you what I know.

Sometimes, I want to choose something now, but I don’t want you to know what I chose until later.

And, sometimes I want to prove that I made a particular choice before I knew what everyone else was going to choose.

This is where cryptographic Commitments come in.

The basic idea is commit to a value now. Reveal it later. Make it computationally infeasible to change your mind.

It is the cryptographic equivalent of putting your answer in an envelope, sealing the envelope, and then loudly announcing that you have written down your answer.

The difference is that cryptography lets everyone verify that you didn’t swap the contents of the envelope while nobody was looking.

The Two Properties

A Commitment scheme has two fundamental security properties:

Hiding

Given the Commitment, an observer should not be able to determine the committed value.

Binding

Once the Commitment has been created, the committer should not be able to produce a valid opening to a different value.

In notation, a Commitment scheme generally has two algorithms:

C = Commit(m, r)

m = Open(C, r)

where:

• m is the message or value being committed to
• r is random secret data
• C is the Commitment

The random value is important.

Without it, a Commitment to a small value can often be brute-forced.

If I publish:

C = SHA-256(“42”)

and the possible answers are numbers from 1 through 100, congratulations: I have invented a guessing game.

An attacker simply hashes all 100 possibilities.

Instead, we can calculate:

C = SHA-256(m || r)

where r is a sufficiently random secret value and the operation || means concatenation.

Now, the attacker can’t simply hash every possible m without also knowing r.

Commitments Aren’t Encryption

This distinction is worth making early.

Encryption provides confidentiality: “You can’t read this without the key.”

A Commitment provides a different property: “I have locked myself into this value, but I’m not telling you what it is yet.”

The Commitment doesn’t necessarily need a secret key.

And, unlike encryption, the goal isn’t primarily to recover the original value later.f

The goal is to establish that the value was fixed at an earlier point in time.

That temporal aspect turns out to be extremely useful.

Hash Commitments

The simplest commitment construction is based on a cryptographic hash.

Conceptually:

For example:

C = SHA-256(m || r)

Later, the committer publishes:

m r

The Verifier calculates:

SHA-256(m || r)

and checks that it equals C.

That’s it.

There is no trusted database.

There is no online authority.

There is no special “Commitment server.”

Just mathematics and an agreed-upon hash function.

Why Does This Work?

The binding property comes from the difficulty of finding two different openings that produce the same Commitment.

An attacker would need something like:

m1 || r1 != m2 || r2

but

H(m1 || r1) = H(m2 || r2)

That’s essentially a collision problem.

The hiding property comes from the randomness of r.

If r has sufficient entropy, the Commitment doesn’t reveal useful information about m.

There are more formal ways of defining all of this, but the basic intuition is:

Binding → You can’t change the answer.

Hiding → Nobody can see the answer.

Two extremely useful properties from one tiny primitive.

Pedersen Commitments

Hash Commitments are useful, but cryptographers eventually wanted Commitments with additional algebraic properties.

Enter the Pedersen Commitment.

In a suitable cyclic group, a Pedersen commitment can be represented as:

C = g^m · h^r

where:

• m is the message
• r is a random blinding factor
• g and h are group elements
• “·” is the multiplication operation

The important requirement is that nobody knows the discrete logarithm relationship between g and h.

In other words, nobody should know some useful value x such that:

h = g^x

because, knowing that relationship would undermine the binding property.

Why Is Pedersen Interesting?

Because it is Additively Homomorphic.

Suppose:

C1 = g^m1 h^r1

C2 = g^m2 h^r2

Multiply them:

C1 C2 = g^m1 h^r1 · g^m2 h^r2 = g^(m1+m2) h^(r1+r2)

Which is another Pedersen Commitment:

C1 C2 = Commit(m1 + m2, r1 + r2)

We just combined two Commitments without knowing the values inside them.

That is considerably more useful than simply having a cryptographically sealed envelope.

Commitments Become Much More Powerful When You Can Do Math on Them

Imagine a financial system where three parties have committed to confidential values:

Alice → Commit(100) Bob → Commit(200) Charlie → Commit(300)

Nobody knows the values.

But, if the Commitment scheme supports the appropriate algebra, we can calculate a commitment to:

100 + 200 + 300

without opening any of the individual Commitments.

Later, the participants can prove that the result is correct.

This is one of the reasons Commitment schemes appear so frequently in privacy-preserving cryptography.

The Commitment isn’t just hiding information.

It gives us a cryptographic handle on hidden information.

Commitments and Zero-Knowledge Proofs

This is where Commitments stop looking like an obscure cryptographic primitive and start becoming infrastructure.

Suppose I want to prove that I know a secret x.

I could simply tell you x.

Problem solved.

Except you’ve now learned the secret.

Instead:

The Verifier learns that The Prover knows a value satisfying the required conditions without necessarily learning the value itself.

Commitments are therefore a common building block in zero-knowledge protocols.

They allow a Prover to effectively say, “I’ve committed to the value. Now, let me prove something about that value without opening the Commitment.”

This is one of the recurring patterns in modern cryptography:

Which is a pretty good description of zero-knowledge cryptography in general.

Commitments in Sigma Protocols

A classic example is the Sigma protocol.

The details vary depending on what we’re proving, but the general structure is:

The Commitment comes first.

The Verifier then chooses a challenge.

The rover responds based on the secret.

That ordering is crucial.

If the Prover could see the challenge before committing, they might be able to construct a response without actually knowing the secret.

The Commitment forces the Prover to put their cryptographic cards on the table before seeing the challenge.

Again, Commit first. Find out what happens next, second.

Cryptography really likes that rule.

Merkle Trees Are Commitments Too

Here’s a form of Commitment you have probably encountered without thinking of it as a Commitment.

A Merkle tree.

Suppose I have:

Transaction 1 Transaction 2 Transaction 3 … Transaction 1,000,000

I hash the transactions and construct a Merkle tree.

Eventually I get:

The root is a compact cryptographic Commitment to the underlying dataset.

Publish the root.

Later, someone can provide a Merkle proof showing that:

Transaction 723,491

was included.

They don’t need to send all one million transactions.

The Merkle root has committed to the entire structure.

This is why blockchain systems can maintain enormous datasets while putting relatively small cryptographic values into blocks.

The root is effectively saying, “I commit to this entire tree.”

And, if the underlying hash function is secure, changing something deep inside the tree changes the root.

Commitments and Sealed-Bid Auctions

Consider an auction.

If bidders submit plaintext bids:

Alice → $500 Bob → $600

Alice can see Bob’s bid and submit:

Alice → $601

Not terribly exciting.

Instead:

Alice → Commit($500, r1) Bob → Commit($600, r2)

The Commitments are published.

The bidding period closes.

Then everyone reveals their values:

Alice → $500, r1 Bob → $600, r2

The Commitments prove that neither bidder changed their bid after seeing the others.

The Commitment therefore solves a problem that encryption alone doesn’t necessarily solve.

We don’t merely want the bids to be secret.

We want the bids to be secret and immutable until the reveal phase.

Commitments and Randomness

Commitments are also useful for generating randomness between parties.

Suppose, Alice and Bob want to generate a random value together.

Alice chooses:

x = random()

If she simply sends x to Bob, Bob gets to see it before choosing his own value.

Instead:

The final random value might be:

x XOR y

Because, Alice committed to x before seeing y, she can’t simply choose x afterward to force a particular result.

The Commitment prevents one participant from waiting to see the other participant’s choice before deciding their own.

That pattern appears in various multi-party protocols.

Commitments in SNARKs

Now, we get into the modern cryptographic machinery.

Many SNARK constructions use polynomial Commitments.

Instead of committing to a simple number:

Commit(x)

we might commit to an entire polynomial:

Commit(P(x))

The Prover can then demonstrate properties about that polynomial without revealing the entire polynomial.

For example:

“This polynomial evaluates to this value at this point.”

or:

“This polynomial satisfies these constraints.”

Polynomial Commitments are one of the important mechanisms that allow succinct cryptographic proofs to exist.

Rather than sending a massive computation transcript to the Verifier, the Prover can commit to mathematical objects and then provide short proofs concerning those Commitments.

The Verifier checks the proof.

The Verifier doesn’t need the entire underlying object.

KZG Commitments

One particularly famous polynomial Commitment construction is the Kate-Zaverucha-Goldberg Commitment, commonly called a KZG Commitment.

Very roughly, given a polynomial:

P(x) = a0 + a1x + a2x² + … + anxⁿ

a KZG scheme creates a compact Commitment using elliptic-curve pairings.

Conceptually:

The Prover can then provide a proof that:

P(z) = y

without revealing the entire polynomial.

KZG Commitments have become important in modern proof systems and blockchain protocols.

But, there is a catch.

They rely on pairing-friendly elliptic curves and associated cryptographic assumptions.

And, that brings us to everybody’s favorite question.

“What happens when the quantum computers arrive?”

Are Cryptographic Commitments Post-Quantum?

It depends.

There is no such thing as “a Commitment scheme” with one universal security assumption.

Different schemes depend on different primitives.

A Commitment based on discrete logarithms:

C = g^m h^r

inherits security assumptions associated with the underlying group.

A sufficiently capable quantum computer running Shor’s algorithm can solve discrete logarithm problems efficiently.

So, classical discrete-log-based Commitment schemes are not considered post-quantum secure.

That includes the assumptions underlying many elliptic-curve constructions.

Hash-Based Commitments

Hash-Based Commitments are a different story.

A sufficiently strong cryptographic hash function isn’t broken by Shor’s algorithm in the same catastrophic way that RSA and discrete logarithms are.

Quantum computers do, however, affect brute-force search through Grover’s algorithm.

Very roughly, Grover gives a quadratic speedup for generic search.

So if a classical attack costs approximately:

2^n

a generic quantum attack might cost approximately:

2^(n/2)

That doesn’t mean SHA-256 is broken.

It means that quantum security levels need to account for the reduced effective security against generic search.

This is one reason larger hash outputs can be useful when designing long-lived post-quantum systems.

KZG Has a Different Problem

KZG Commitments are based on elliptic-curve pairings.

That’s fantastic for compact proofs.

It’s not fantastic for post-quantum security.

A sufficiently powerful quantum computer can attack the underlying discrete-log assumptions.

So, if you’re building a system intended to survive into the post-quantum era, you can’t simply say_,_ “We’re using Commitments, therefore we’re post-quantum.”

You have to ask which Commitment scheme?

And then, what assumption provides its binding and hiding properties?

And, then, does a quantum adversary break that assumption?

Welcome to cryptographic engineering.

STARKs Take a Different Approach

This is one reason STARK-style proof systems are interesting in the post-quantum discussion.

STARK constructions generally rely heavily on hash functions and error-correcting-code techniques rather than elliptic-curve pairings.

The result is generally considered much more compatible with post-quantum assumptions than pairing-based systems such as KZG-based constructions.

But, “post-quantum” doesn’t mean “invulnerable to quantum computers.”

The security parameters still need to account for quantum attacks.

And, the underlying hash functions and proof-system assumptions still need careful analysis.

There is no magical checkbox labeled:

☑ Quantum Proof

Unfortunately.

Computational vs. Statistical Security

Another important distinction is that hiding and binding don’t necessarily have to have the same security character.

A Commitment can be:

Computationally binding Statistically hiding

or:

Statistically binding Computationally hiding

Those are different security guarantees.

Computational security means that breaking the property is believed to be computationally infeasible.

Statistical security means that the property holds based on the mathematical probability distribution, rather than depending primarily on computational limitations.

Pedersen Commitments are a particularly nice example because their hiding property can be information-theoretic under the usual construction, while their binding property depends on the discrete-log relationship between the generators remaining unknown.

So, the security properties aren’t simply, “This Commitment is secure.”

They are more like this particular security property of this particular ommitment scheme depends on this particular assumption.

Cryptography loves footnotes.

Commitments and Digital Signatures

Commitments and signatures also solve different problems.

A digital signature means the holder of this signing key authorized this message.

A commitment means that this value was fixed without being revealed.

You can combine them:

Now, you have something that means Alice cryptographically committed to this value.

This combination can be useful in protocols involving legal or financial Commitments, distributed systems, elections, randomness, and multi-party computation.

The signature gives you authenticity.

The Commitment gives you binding over time.

Commitments and Confidential Transactions

Commitments are also central to privacy-preserving financial systems.

Suppose I want to prove:

Input amount = Output amount

without revealing the amounts.

A system might commit to:

Commit(input) Commit(output)

and then provide a zero-knowledge proof demonstrating that the committed values satisfy the required relationship.

The observer doesn’t necessarily learn:

$1,237.42

but can still verify:

input = output

or perhaps:

input >= output

or:

amount is within an allowed range

This is the broader pattern:

Hide the data. Expose proofs about the data.

Commitments are one of the primitives that make that possible.

The Most Important Property: You Can’t Change History

There is a philosophical reason Commitments show up in so many protocols.

Distributed systems have a fundamental problem:

Time.

At time T1, I make a decision.

At time T2, you make a decision.

At time T3, I reveal my decision.

If I can see your decision at T2 before fixing mine, I may cheat.

A Commitment changes the sequence:

The commitment forces the decision to exist before the information that could influence it becomes available.

That’s the deeper reason the primitive is so useful.

It’s not really about hiding.

It’s about committing the future to the past.

A Useful Mental Model

You can think about several cryptographic primitives this way:

And, modern cryptographic systems frequently stack these primitives together.

For example:

At that point, a single tiny cryptographic message can be making several different claims simultaneously:

• This value exists
• It hasn’t been changed
• I know it
• It satisfies some mathematical relationship
• I am authorized to make this assertion

Which is considerably more useful than putting a piece of paper in an envelope.

Although the envelope metaphor did get us surprisingly far.

This Matters for Identity

Commitments also fit naturally into modern identity systems.

Consider a credential containing:

Name Date of Birth Nationality Employee Number Department Security Clearance

Maybe the Verifier only needs to know:

Security Clearance >= SECRET

A privacy-preserving credential system can use Commitments and zero-knowledge proofs to allow the holder to prove that statement without revealing every other attribute.

The Verifier gets:

Proof: clearance >= SECRET

rather than:

Here is my entire identity record. Please don’t put it in a database forever.

This is one of the major themes behind modern selective-disclosure and zero-knowledge credential systems.

Instead of transferring all the underlying data, the holder can transfer cryptographic evidence about the data.

Summary

Cryptographic Commitments are deceptively simple.

At their core:

Commit(value, randomness)

creates a cryptographic object that means “I have chosen something.”

Later:

Open(commitment, value, randomness)

says, “Here is what I chose.”

The security properties ensure that:

Before opening: You shouldn’t learn the value.

After committing: I shouldn’t be able to change the value.

From that tiny idea we get building blocks for:

• Sealed-bid auctions
• Distributed randomness
• Voting systems
• Merkle trees
• Zero-Knowledge Proofs
• Multi-party computation
• Confidential transactions
• SNARKs
• STARKs
• Polynomial Commitments
• Privacy-preserving credentials
• Blockchain protocols

And, importantly, different Commitment schemes have radically different properties.

A Hash Commitment, a Pedersen Commitment, and a KZG Polynomial Commitment aren’t interchangeable just because they all have the word Commitment in their names.

They rely on different mathematical assumptions.

They provide different properties.

And they have different post-quantum implications.

So the next time somebody tells you:

“We’re using a cryptographic Commitment.”

the appropriate response isn’t, “Cool.”

It is “Which one, and what are its security assumptions?”

Because, in cryptography, the interesting part is almost always hiding in the second sentence.

Notes

• AI / GenAI / ChatGPT / etc were not used to generate the text of this article.
• ChatGPT was used to generate the images.
• I used em dashes in my writing before the current GenAI wave was a thing. Not planning on changing now.
• Names have been changed to protect the guilty.
• None of the hostnames or users used in examples actually exist.
• Feel free to post any comments or suggestions below.
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• Do not attempt to use any ideas, code, or programs discussed here in a production environment without first testing it in a non-production environment.
• I’m not responsible for the spontaneous combustion of the known universe or anything else bad that happens to you today as a result of your having read this blog post.