RFC-0180: Bulletproof range proof rewinding

Bulletproof range proof rewinding

Maintainer(s): Hansie Odendaal

Licence

Copyright 2020 The Tari Development Community

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Language

The keywords "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY" and "OPTIONAL" in this document are to be interpreted as described in BCP 14 (covering RFC2119 and RFC8174) when, and only when, they appear in all capitals, as shown here.

Disclaimer

This document and its content are intended for information purposes only and may be subject to change or update without notice.

This document may include preliminary concepts that may or may not be in the process of being developed by the Tari community. The release of this document is intended solely for review and discussion by the community of the technological merits of the potential system outlined herein.

Goals

This Request for Comment (RFC) presents a proposal for Bulletproof range proof rewinding in the Tari blockchain to enable advanced usages like wallet recovery and one‑sided payments.

Introduction

We use dalek-cryptography/bulletproofs in the Tari project and have a need to do wallet recovery from seed values and also to recover the value in the value commitment from the Unspent Transaction Output (UTXO). Pull requests PR#340 for the dalek-cryptography/bulletproofs crate and PR#6 for the zkcrypto/bulletproofs crate were submitted to add Bulletproofs rewinding functionality to the Bulletproofs crate as a user option.

The methodology presented here is closely modelled on Grin's solution as discussed here, but using two private keys instead of one.

Rewind Scheme

Bulletproofs per say are not be discussed in this RFC, only how the rewinding scheme works. Readers who require background information on Bulletproofs can read the excellent documentation created by the Dalek team here. Important to note is that Dalek only implemented the aggregated Multiparty Computation Protocol (MCP) for range proofs and that proving a single range proof is handled a special case.

Constructing a rewindable Bulletproof range proof

Our scheme is discussed with reference to the Party and Dealer's algorithm and using notation defined here.

In this scheme three additional parameters are introduced when creating a range proof for a Pedersen commitment (termed value commitment by Dalek because it is a commitment to the value of the token):

• Private rewind key:   $r_{key}$
• Private blinding key:   $b_{key}$
• Twenty three (23) bytes proof message:   $p_{msg}$.

The 23 bytes worth of proof message can be any message a user wants to embed within the proof. Internally the two private keys, in combination with the value commitment, are converted into two rewind nonces and two blinding nonces:

• Rewind nonce 1:   $r_{n1} = \text{H}( \ \text{H}(r_{key} \cdot \widetilde{B}) \ || \ V_{(j)} \ )$
• Rewind nonce 2:   $r_{n2} = \text{H}( \ \text{H}(b_{key} \cdot \widetilde{B}) \ || \ V_{(j)} \ )$
• Blinding nonce 1:   $b_{n1} = \text{H}( \ \text{H}(r_{key}) \ || \ V_{(j)} \ )$
• Blinding nonce 2:   $b_{n2} = \text{H}( \ \text{H}(b_{key}) \ || \ V_{(j)} \ )$

These four values are seen as nonces due to the fact that each value commitment is unique, whereas the $r_{key}$ and $b_{key}$ can be used over and over without leaking any information.

The value $v_{(j)}$ is an 8 byte word, and $p_{msg}$ is a 23 byte word. The bytes of these two words can be concatenated to form a 32 byte word and when XORed with $r_{n2}$ ,  it can be used to embed the value and proof message. $r_{n2}$ is modified as follows:

\begin{aligned} r^\backprime_{n2} = r_{n2} \ \mathbin{\oplus} \ (v_{(j)_{\ bytes \ 1..8}} \ || \ p_{msg_{\ bytes \ 9..31}} ) \end{aligned} \tag{1}

Consider the start of the protocol where each party $j$ computes three commitments: to the value $v_{(j)}$, to the bits of that value $\mathbf{a}_{L, (j)}, \mathbf{a}_{R, (j)}$, and to the per-bit blinding factors $\mathbf{s}_{L, (j)}, \mathbf{s}_{R, (j)}$:

\begin{aligned} V_{(j)} &\gets \operatorname{Com}(v_{(j)}, {\widetilde{v}_{(j)}}) && = v_{(j)} \cdot B + {\widetilde{v}_{(j)}} \cdot {\widetilde{B}} \\ A_{(j)} &\gets \operatorname{Com}({\mathbf{a}}_{L, (j)}, {\mathbf{a}}_{R, (j)}) && = {\langle {\mathbf{a}}_{L, (j)}, {\mathbf{G}_{(j)}} \rangle} + {\langle {\mathbf{a}}_{R, (j)}, {\mathbf{H}_{(j)}} \rangle} + {\widetilde{a}_{(j)}} {\widetilde{B}} \\ S_{(j)} &\gets \operatorname{Com}({\mathbf{s}}_{L, (j)}, {\mathbf{s}}_{R, (j)}) && = {\langle {\mathbf{s}}_{L, (j)}, {\mathbf{G}_{(j)}} \rangle} + {\langle {\mathbf{s}}_{R, (j)}, {\mathbf{H}_{(j)}} \rangle} + {\widetilde{s}_{(j)}} {\widetilde{B}} \\ \end{aligned} \tag{2}

where $\widetilde{v}_{(j)}, \widetilde{a}_{(j)}, \widetilde{s}_{(j)}$ are sampled randomly from ${\mathbb Z_p}$. (Note that $\widetilde{v}_{(j)}$ is the blinding factor of the value commitment.)

In our scheme:

• blinding factor ${\widetilde{a}_{(j)}}$ is replaced by $r_{n1}$
• blinding factor ${\widetilde{s}_{(j)}}$ is replaced by $r^\backprime_{n2}$

Consider where the party commits to the terms $t_{1, (j)}, t_{2, (j)}$:

\begin{aligned} T_{1, (j)} &\gets \operatorname{Com}(t_{1, (j)}, {\tilde{t}_{(j1}}) && = t_{1, (j)} \cdot B + {\tilde{t}_{1, (j)}} \cdot {\widetilde{B}} \\ T_{2, (j)} &\gets \operatorname{Com}(t_{2, (j)}, {\tilde{t}_{2, (j)}}) && = t_{2, (j)} \cdot B + {\tilde{t}_{2, (j)}} \cdot {\widetilde{B}} \end{aligned} \tag{3}

where $\tilde{t}_{1, (j)}, \tilde{t}_{2, (j)}$ are sampled randomly from ${\mathbb Z_p}$.

In our scheme:

1. blinding factor $\tilde{t}_{1, (j)}$ is replaced by $b_{n1}$
2. blinding factor $\tilde{t}_{2, (j)}$ is replaced by $b_{n2}$

The synthetic blinding factors calculation below is key, as it will be used to extract the data when playing the Bulletproof in reverse:

\begin{aligned} {\tilde{t}}_{(j)}(x) &\gets z^{2} {\tilde{v}}_{(j)} + x {\tilde{t}}_{1, (j)} + x^{2} {\tilde{t}}_{2, (j)} \\ \end{aligned} \tag{4}

\begin{aligned} \tilde{e}_{(j)} &\gets {\widetilde{a}}_{(j)} + x {\widetilde{s}}_{(j)} \end{aligned} \tag{5}

In the end, the complete range proof consists of these elements:

\begin{aligned} \lbrace A, S, T_1, T_2, t(x), {\tilde{t}}(x), \tilde{e}, L_k, R_k, \dots, L_1, R_1, a, b \rbrace \end{aligned} \tag{6}

Note: This scheme has been improved in what has been presented in by Grin after being commented on by Dalek, by not using the same rewind nonce for ${\widetilde{a}_{(j)}}$ and ${\widetilde{s}_{(j)}}$ nor the same blinding nonce for $\tilde{t}_{1, (j)}$ and $\tilde{t}_{2, (j)}$.

Extracting data

Note the presence of ${\tilde{t}}_{(j)}$ and $\tilde{e}$ in (6). The Dalek Bulletproofs are constructed using Merlin Transcripts to automate the Fiat-Shamir transform, so that non-interactive protocols can be implemented as if they were interactive. The prover adds each step of the Bulletproof range proof creation to the protocol transcript, so the verifier has to do the same.

The extraction procress starts by adding the values $A$ and $S$ are to the protocol transcript to obtain challenge scalars $z$ and $x$ from the transcript.

There after, ${\widetilde{s}_{(j)}}$ is extracted from (5) by replacing ${\widetilde{a}_{(j)}}$ with $r_{n1}$  :

\begin{aligned} {\widetilde{s}}_{(j)} = ( \tilde{e}_{(j)} - r_{n1} ) \cdot \frac{1}x \end{aligned} \tag{7}

Next, the value and proof message are extracted from ${\widetilde{s}_{(j)}}$ when XORed with $r_{n2}$ :

\begin{aligned} v_{(j)} &= ( r_{n2} \ \mathbin{\oplus} \ {\widetilde{s}}_{(j)} ) | _{\ bytes \ 1..8} \\ p_{msg} &= ( r_{n2} \ \mathbin{\oplus} \ {\widetilde{s}}_{(j)} ) | _{\ bytes \ 9..31} \end{aligned} \tag{8}

Finally, the blinding factor is extracted from (4):

\begin{aligned} \widetilde{v} = \frac{1}{z^2} \cdot ( {\tilde{t}}(x) - x \cdot \tilde{t}_{1, (j)} - x^2 \cdot \tilde{t}_{2, (j)} ) \end{aligned} \tag{9}

Some notes on usage and use cases

Rewinding a Bulletproof can take place according to one or both of these steps:

• Peak value only: Using this step returns the value and proof message only, but returning garbage data if the wrong rewind nonces are provided, or,
• Rewind fully: Using this step returns the value, blinding factor and proof message, returning an error if the wrong rewind and blinding nonces are provided. Note that this step is independent from peaking the value only, thus do not have ot be preceded by it. If many range proofs need to be scanned to uncover those that belong to a particuler wallet, peaking the value only before fully rewinfing it will provide a performance benefit.

The main use case has to do with wallet recovery. A user would normally have a backup of their unique wallet seed words somewhere, but could more easily lose their entire wallet without having made any backups or only having old backups. If a wallet can derive one or more sets of private keys from the seed words and use them in every UTXO construction as proposed, it can enable wallet recovery using Bulletproof rewinding.

A secondary use case would be for trusted 3rd parties to identify spending, by only having access to the public rewind key and the embedded proof message. The public rewind keys can be shared with a 3rd party out of band. The owner and/or delegated 3rd party can then use these keys in conjunction with a specific value commitment to calculate candidate rewind nonces for its proof. The returned proof message from the peak value only rewind step can be used to narrow down the probability that the particular proof belongs to a specific collection. In this mode the owner alone will be able to use both sets of pub-pvt key pairs in conjunction with a specific value commitment to calculate candidate rewind and blinding nonces for its proof. The rewind fully step will reveal the details of the value commitment and proof message if successful.

The use for this protocol, as opposed to simply revealing the original value along with the blinding factor to whoever wants the plain value, is to protect the UTXO. In Mimblewimble, if the value commitment can be opened, it can be spent without the owners knowledge.

The proof message is private or can be shared with a trusted 3rd party in the same way one would share the public rewind keys, but not common public knowledge. It is totally arbitrary, but known data, to enable identifying beyond a doubt if the returned value $v_{(j)}$ is from a specific collection of value commitments $V_{(j)}$.

Implementation

Using the Application Programmers Interface (API) it is possible to:

• create a rewindable Zero-knowledge (ZK) proof with up to 23 bytes of additional embedded proof message $p_{msg}$  ;
• extract the value $v_{(j)}$ and 23 bytes proof messsage $p_{msg}$ only;
• extract the value $v_{(j)}$  ,   blinding factor $\widetilde{v}$ and 23 bytes proof messsage $p_{msg}$  .

Credits

• @jaspervdm for his improved bulletproof rewind scheme, used as precurser.
• @cathieyun for provifing valuable feedback to improve this scheme.