LibraryPrivacy2015Design paperCorpus record
Confidential Transactions
Confidential Transactions. Gregory Maxwell.
Hide the amounts on a Bitcoin-style transaction with Pedersen commitments, and use range proofs so a commitment cannot stand for a negative number. The graph of who paid whom stays visible.
Confidential Transactions hide amounts inside Pedersen commitments and prove the amounts are in range, so the ledger can check that inputs still equal outputs without learning the numbers.
- Evidence
- Primary paper
- Re-measured
- No
- Assumptions
- 4
- Records linked
- 8
Code
Source, repository, and what is actually hosted
- Elements Projecthttps://elementsproject.org/features/confidential-transactions
- The source URL is not a GitHub repository. No code remote is invented.
- Full text is not mirrored. Public availability is not a licence to copy the file onto this site.
- Add to a reading list
01 Claim ledger
What the paper is allowed to say
Each row is a sentence already in the study. The status is the same on every row: a model claim, not a live measurement.
Claim 01 · Paper model
Only the amount is hidden
The note does not hide the graph of outputs. An observer can still see that this input was spent to those outputs. They cannot see how much moved, apart from a public fee if there is one.
Claim 02 · Paper model
Commitments must balance
Pedersen commitments add. The verifier checks that output commitments sum to input commitments. A mismatch is an inflation bug. A match reveals no values.
Claim 03 · Paper model
Range proofs stop the underflow
A commitment to a negative output would mint value and still balance. The range proof is what makes the hiding safe. Omit it and the scheme is broken.
Claim 04 · Paper model
Later systems split the idea
Liquid used confidential amounts on a Bitcoin-like graph. Monero's RingCT combined the commitments with ring signatures. Neither deployment is the 2015 note.
02 Three cuts
Observation, model, falsifier
A desk does not stop at the summary. Each claim is cut three ways, using only this study's own assumptions and checks. Nothing here is a new figure.
01 Only the amount is hidden
Observation
What the study says
The note does not hide the graph of outputs. An observer can still see that this input was spent to those outputs. They cannot see how much moved, apart from a public fee if there is one.
Model
What has to hold
Blinding factors are actually secret. A reused or leaked blind reveals the amount.
Falsifier
What would retire it
Are amounts hidden, or only counterparties?
02 Commitments must balance
Observation
What the study says
Pedersen commitments add. The verifier checks that output commitments sum to input commitments. A mismatch is an inflation bug. A match reveals no values.
Model
What has to hold
The discrete logarithm problem is hard in the group used for the commitments.
Falsifier
What would retire it
Does the balance equation include the fee?
03 Range proofs stop the underflow
Observation
What the study says
A commitment to a negative output would mint value and still balance. The range proof is what makes the hiding safe. Omit it and the scheme is broken.
Falsifier
What would retire it
Is there a range proof on every committed value?
04 Later systems split the idea
Observation
What the study says
Liquid used confidential amounts on a Bitcoin-like graph. Monero's RingCT combined the commitments with ring signatures. Neither deployment is the 2015 note.
Model
What has to hold
The discrete logarithm problem is hard in the group used for the commitments.
Falsifier
What would retire it
Are amounts hidden, or only counterparties?
03 Sequence
One action, as an operating tape
- 01The sender picks amounts and random blinds, and publishes a commitment for each output.
- 02A range proof is attached so each committed amount is in a permitted interval.
- 03The sender proves they know the openings of the inputs they are spending.
- 04Nodes check the proofs and the homomorphic balance.
- 05The recipient, who knows the blind and the amount, can later spend the output. The network never needed the amount in the clear.
04 Load-bearing
The argument, and where a pitch drops it
Hiding and binding are different
The cut
Hiding means the observer does not learn the amount. Binding means the sender cannot open the same commitment to two amounts. A scheme that is only hiding is an inflation bug waiting for a second opening.
Why it carries weight
If this cut is skipped, the paper's name is being used without the mechanism that makes the name mean anything.
Where it is dropped
Hiding amounts does not hide counterparties.
Privacy marketing collapses three papers
The cut
CryptoNote hides the signer among decoys and leaves amounts public. Zerocash hides more of the graph. Confidential Transactions hide the number. A white paper that says privacy without saying which of the three is not yet a design.
Why it carries weight
If this cut is skipped, the paper's name is being used without the mechanism that makes the name mean anything.
Where it is dropped
Hiding amounts does not hide counterparties.
05 Register
What has to be true
Model · Not re-measured
The discrete logarithm problem is hard in the group used for the commitments.
Cryptography · Not re-measured
Range proofs are sound. A forged range proof is a licence to print.
Model · Not re-measured
Blinding factors are actually secret. A reused or leaked blind reveals the amount.
Model · Not re-measured
Fees, if public, are accounted for in the balance equation. Forgetting the fee breaks conservation.
06 Divergence
What happened after the paper
Bulletproofs replaced the bulky range proofs this note relied on. Liquid shipped confidential amounts. RingCT took the commitment into a ring. The 2015 design remains the right citation for amount hiding on a visible graph, and the wrong citation for unlinkability.
A later client, parameter set, or reward formula is a different object. Cite this paper for the mechanism. Cite a primary release for the network. This desk has not re-run the proof.
07 Pre-mortem
What to check before you use the idea
- 0 of 4 marked on this browser. A mark is a reading note, not a pass, a rating, or a recommendation.
08 Anatomy
The paper, in the order a builder needs
The problem it names
A public amount lets an observer price a payment, target a wallet, and match deposits. Maxwell's note hides the number while still letting the network check that inputs and outputs balance.
What the design proposes
- Commit to each amount. Publish the commitment, not the amount.
- Homomorphism: the commitments of the outputs should sum to the inputs, plus an explicit fee if the fee is public.
- A range proof stops a commitment to a huge number that underflows.
How the mechanism is specified
- Pedersen commitments are binding and hiding under the discrete-log assumption. The binding is what stops you changing the amount later.
- The verifier checks a linear relation on commitments. It never learns the values if the proofs are honest.
- Addresses and the transaction graph are still on the page. This paper is not CryptoNote and not Zerocash.
What this page does not treat as proven
- Hiding amounts does not hide counterparties.
- Range proofs in the original form are large. Bulletproofs are a later compression.
- Liquid and RingCT borrowed the idea. Neither is this note.
Why the desk still reads it
If a product says confidential, ask whether the amount, the sender, or the recipient is hidden. This paper answers only the first.
09 Lexicon
Terms, opened into the record
- Pedersen commitment
- A commitment to a number that can be added to other commitments without opening them.
- Range proof
- A proof that the hidden number sits between two bounds, so it cannot be negative or absurd.
10 Repository
Every linked record on this page
Underlined words open a page that already exists: a concept, a protocol profile, a failure record, or another paper. If a word is not underlined, this desk does not have a record for it.
This is Blockchain Lab's reading of a public design paper. It is not the paper, not a copy of it, and not an offer of tokens, equity, custody or a partnership. Later network behaviour can diverge from the text. Nothing here is investment, legal or technical advice.
Research status: Design paper. Last reviewed: 1 October 2026. This is a reading of a public paper, not investment, legal or security advice.