Cryptographic Commitments: I Promise I’m Not Changing My Answer

There are a few cryptographic primitives that sound considerably more complicated than they actually are.

Cryptographic Commitments are one of them.

The basic idea is straightforward enough: I want to commit to a value now, without revealing the value, and later prove that I committed to that exact value.

Think of putting a piece of paper in a sealed envelope.

You write “ 42” on the paper, put it in the envelope, seal it, and hand the envelope to somebody.

So, long, and thanks for all the fish.

Later, you open the envelope.

Everyone can now see that you wrote 42 when you sealed it. More importantly, you can’t credibly claim that you originally wrote 73.

The cryptographic version does the same thing without requiring an actual envelope.

And, because this is cryptography, we get to replace the envelope with mathematics.

The Two Properties That Matter

A cryptographic commitment generally needs two fundamental properties:

Hiding: The commitment should not reveal the value that you committed to.

Binding: Once you’ve committed to a value, you shouldn’t be able to later produce a convincing opening showing that you committed to a different value.

In other words:

The commitment hides X.

The opening demonstrates what X was.

And the binding property prevents you from deciding that X was actually something else after seeing how the game turned out.

This gives us a rather useful cryptographic equivalent of: “I said what I said.” If you are Popeye, that’s all you said. If you are the Mandalorian’s friend, you have spoken.

Why Not Just Encrypt It?

At this point someone will inevitably ask, “Why don’t we just encrypt the value?”

You can.

But, encryption and commitment solve different problems.

Encryption is primarily about confidentiality: Nobody without the key should be able to see this.

A commitment is about locking in a value: Nobody should be able to change what I committed to, and later I should be able to prove what that value was.

A commitment doesn’t necessarily need a secret key. It is an integrity-and-verifiability primitive, rather than simply a confidentiality mechanism.

That distinction becomes important when you have thousands or millions of independent parties that need to verify the commitment.

The Envelope Has Two Locks

The traditional cryptographic terminology is worth remembering.

A commitment scheme generally tries to provide:

Hiding

Given the commitment: C

you shouldn’t be able to determine: X

with meaningful probability.

Binding

Given: C

you shouldn’t be able to produce:

X1, R1

and:

X2, R2

where:

X1 != X2

but both produce the same commitment.

In other words:

The scheme should make it computationally infeasible for those two openings to contain different values.

That’s the binding property.

The fact that the commitment doesn’t tell you what the value is provides hiding.

Those two properties are the heart of the primitive.

There Isn’t Just One Commitment Scheme

Cryptographers, naturally, could not leave well enough alone.

There are several families of commitment schemes with different properties and assumptions.

One particularly important construction is a Pedersen commitment.

In a suitable Elliptic-Curve or finite-Group setting, a Pedersen commitment has the general form:C = C = gˣhʳ

where:

• x is the committed value
• r is a random blinding factor
• g and h are appropriate group generators

The useful property is that the random value r hides x.

Pedersen commitments are particularly important in zero-knowledge systems because they provide strong hiding properties while allowing useful algebraic operations over commitments.

And, that brings us to where commitments get considerably more interesting.

Commitments Can Be Added Without Opening Them

Some commitment schemes allow us to construct new commitments.

It’s a small mathematical party trick. We’ll cover it in more detail in the next post where we cover the math.

This algebraic property is one of the reasons commitment schemes are such useful building blocks for privacy-preserving cryptographic protocols.

Zero-Knowledge Proofs Love Commitments

Commitments show up constantly in Zero-Knowledge Proofs.

Suppose, I want to prove: I know a secret value x satisfying some mathematical relationship.

But, I don’t want to reveal x.

A commitment gives me a way to first lock in the value:

I can then participate in a protocol that proves knowledge of the committed value without exposing it.

This pattern appears throughout modern zero-knowledge systems.

Commitments are therefore often one of the hidden pieces underneath the phrase: “Prove something about this secret without revealing the secret.

There is usually a considerable amount of mathematics hiding underneath that innocent sentence.

Commitments and Digital Signatures Are Not the Same Thing

Another useful distinction:

A digital signature generally answers: Who authorized this data?

A commitment answer: What value did I lock myself into?

You can combine them, of course.

For example:

The signature establishes the identity of the signer.

The commitment establishes the value that was locked in.

This combination is useful when you need both authenticity and non-equivocation.

Sealed-Bid Auctions

Here’s a very intuitive application.

Suppose ten people are bidding on something.

If everyone submits their bid in plaintext, later bidders can simply offer more.

Instead, each bidder commits to their bid:

Bidder A → Commit($500)
Bidder B → Commit($700)
Bidder C → Commit($600)

Everyone publishes the commitments.

Nobody knows the actual bids.

Once bidding closes, everyone opens their commitment:

A → $500
B → $700
C → $600

The commitments prevent someone from saying, “Oh, I totally bid $900” after discovering that $700 was the winning bid.

You can make the protocol considerably more sophisticated, but the basic cryptographic mechanism is the same.

Elections

Commitments can also be used in voting protocols.

A voter can commit to a choice without immediately revealing it: Commit(“Alice”)

The commitment can be published during one phase of the protocol.

Later, the election system can perform another phase in which votes are opened, combined, or proven to satisfy particular conditions.

Modern electronic voting systems typically require considerably more than simple commitments, but commitments are useful building blocks for preventing premature disclosure and manipulation.

Blockchains Use Commitments Everywhere

The word “commitment” also shows up constantly in blockchain technology.

One obvious example is a Merkle tree.

Suppose, I have a large collection of transactions:

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

I can construct a Merkle tree and publish only the root:

The root acts as a compact cryptographic commitment to the entire underlying dataset.

Later, someone can provide a Merkle proof demonstrating that a particular transaction was included.

They don’t need to send the entire million-transaction dataset.

The root committed to the structure.

This is one reason a tiny hash can represent a very large amount of underlying information.

Commitments Are About Time

There’s a deeper pattern here.

Commitments are particularly useful when a protocol has multiple phases separated in time.

For example:

That separation is incredibly useful.

It lets you make a decision without immediately revealing it, while still preventing yourself from changing the decision later.

This appears in:

• Auctions
• Voting
• Multi-party computation
• Zero-Knowledge Proofs
• Blockchain protocols
• Randomness generation
• Cryptographic games
• Distributed protocols
• Privacy-preserving applications

The common theme is that someone needs to make a cryptographically binding statement before everyone knows what the eventual outcome will be.

Commitments Can Also Commit to Randomness

Here’s another useful application.

Suppose Alice and Bob need to generate a random number together.

Alice could simply announce:

Alice: My random number is 17.

Bob can then choose something that exploits that information.

Instead: Alice → Commit(17)

Bob responds: Bob → 83

Alice then reveals: 17

The resulting value could be: 17 XOR 83

Because, Alice committed before seeing Bob’s contribution, she couldn’t simply choose a value designed to produce her preferred result.

This technique appears in various forms in distributed randomness protocols.

Again, commit first, reveal later.

That pattern keeps coming back.

Commitments in Zero-Knowledge Systems

Modern zero-knowledge protocols take this much further.

A Prover might have private information:

The verifier can learn things like:

• “The committed value is in this range.”
• “The committed values add up correctly.”
• “The prover knows the committed secret.”
• “The committed value satisfies this equation.”

without learning the underlying secret itself.

This is one of the fundamental ingredients behind many privacy-preserving systems.

And, this is why cryptographic commitments are considerably more important than the phrase “sealed envelope” initially makes them sound.

Commitment Schemes Have Assumptions

There is, naturally, a catch.

Cryptographic Commitments don’t provide some mystical mathematical guarantee that survives the heat death of the universe.

Their security depends on assumptions.

A hash-based commitment might depend on properties such as:

• Preimage resistance
• Second-preimage resistance
• Collision resistance

A Pedersen Commitment relies on assumptions about the underlying mathematical group and the inability to solve particular discrete-log relationships.

And, that matters for Post-Quantum Cryptography.

A Commitment scheme based on discrete logarithms is not automatically post-quantum secure merely because someone put it inside an otherwise modern protocol.

Likewise, a hash-based commitment doesn’t automatically become unsafe just because someone mentions quantum computers.

The security properties and required security levels need to be analyzed individually.

Binding and Hiding Can Even Depend on Different Assumptions

One particularly useful detail is that hiding and binding aren’t necessarily obtained from the same underlying property.

For example, a scheme might be:

Computationally binding
Statistically hiding

or:

Statistically binding
Computationally hiding

Those are very different security statements.

“Statistical” generally means the property holds regardless of computational power within the mathematical model.

“Computational” means we believe breaking the property is infeasible for an adversary with realistic computational resources.

So, when someone says that “This is a secure commitment scheme” the next question should probably be “Secure in what sense?”

Cryptography has an impressive ability to turn a three-word sentence into a forty-page standards document.

Commitments Are Not Encryption

It is worth repeating because the distinction matters.

Encryption says: Here is information that only someone with the appropriate key should be able to read.

A commitment says: Here is information that I have cryptographically locked myself into, but I’m not telling you what it is yet.

Digital signatures say: I authorize this particular piece of information.

Zero-knowledge proofs say: I can demonstrate that something is true without necessarily revealing the information that makes it true.

And cryptographic protocols routinely combine all four.

That’s the fun part.

Summary

Cryptographic Commitments are one of those primitives that look almost trivial when first explained:

But, that simple pattern gives protocols something extremely valuable: the ability to make a statement before revealing the information behind it.

That enables protocols where participants must commit before they know what everyone else is doing.

• It prevents changing answers after the fact.
• It provides compact representations of large datasets.
• It helps build zero-knowledge proofs.
• It supports privacy-preserving computation.

And, it provides a mathematical foundation for a whole collection of protocols that would otherwise have to rely on everyone behaving themselves.

Which, as anyone who has ever designed a distributed system knows, is generally not a viable security strategy.

The fundamental idea is beautifully simple:

Commit now. Reveal later. And make changing your mind computationally expensive.

That turns out to be an extraordinarily useful thing to be able to do with mathematics. You can learn more about the mathematics behind Commitments in this post.

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.
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• None of the hostnames or users used in examples actually exist.
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• 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.