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keywordstyle=\bfseries,
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ndkeywords={class, export, boolean, number, Amount, string, Timestamp, RelativeTime, EddsaPublicKey, BrandtVickreyAuction, BrandtVickreyAuctionMessage, BrandtVickreyAuctionWinner, EddsaSignature, HashCode, throw, implements, import, this, BrandtVickreyReplayOutcome},
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}
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\begin{document}
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\title{AP³\\
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Report for Milestone III\\
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NGI Pointer}
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\author{Özgür Kesim\\
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Christan Grothoff\\
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Florian Dold\\
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Stefan Kügel\\
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Emmanuel Benoist\\[\bigskipamount]
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\normalsize Mentor: Mirko Ross \href{mailto:m.ross@digital-worx.de}{<m.ross@digital-worx.de>}\\[\medskipamount]
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}
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\date{October 15, 2022}
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\maketitle
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\section*{Management summary}
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\begin{abstract}
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The AP³ project presents here the report for the milestone III for NGI Pointer.
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The deliverables for this milestone are:
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\begin{description}
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\item[Anonymous auction] -- PoC for execution of anonymous sealed-bid auctions
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\item[P2P payments] -- PoC UI on Android for P2P transactions in the Taler wallet
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\end{description}
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\end{abstract}
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\vfill
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\hfill {\footnotesize Version: 1.0}
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\thispagestyle{empty}
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\newpage
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\tableofcontents
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\newpage
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\section{Anonymous auctions - PoC}
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We departed from the design of a dedicated escrow service in GNU Taler (see
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report for milestone 2) and identified a more general approach for conditional
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payments---deposits with \textit{policies} attached to them. The design of
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\textit{Deposit Policy Extensions} has been drafted in
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\href{https://docs.taler.net/design-documents/028-deposit-policies.html}{design
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document 28} and is still work-in-progress.
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\subsection{Policy extensions framework for GNU Taler}
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\label{extension}
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The policy-based approach to deposit of coins allows GNU Taler to offer a
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variety of conditional payment types, including: Merchant refunds, escrowed
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payments and Brandt-Vickrey auctions.
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In Brandt-Vickrey auctions, bidders put coins into escrow with the exchange in
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order to participate in an Brandt-Vickrey auction. The deposit confirmation is
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proof to the seller for the escrow and contains a hash of the auction meta-data
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and a deadline.
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After successful execution of the auction, the seller provides a valid
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transcript to the exchange from which the exchange learns which bidder(s) won
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the auction for which price(s). It then transfers the amounts from the winners’
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coins to the seller. The coins' (rest) values can be refreshed by winning and
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losing bidders.
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\begin{quote}
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\itshape Note: Support for Brandt-Vickrey type auctions has been
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implemented as Proof-of-Concept policy-extension for this milestone.
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\end{quote}
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A policy can be in one of the following five states of fulfillment:
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\begin{itemize}
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\item[\itshape Ready:] The policy is funded and ready. The exchange is
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waiting for a proof of fulfillment to arrive before the
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deadline.
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\item[\itshape Insufficient:] The policy lacks funding, that is
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\verb|accumulated_total| $<$ \verb|commitment| (see
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Fig.~\ref{schema}), but has otherwise been accepted. Funding
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can be continued by calling \verb|/deposit| or
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\verb|/batch-deposit| with more coins and the same policy
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details.
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\item[\itshape Success:] The policy is provably fulfilled. The amounts
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for the payout, fees and refresh are transferred/can be
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claimed. Note that a policy fulfillment handler can change the
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values for the amounts for payout, fees and refresh.
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\item[\itshape Timeout:] The policy has timed out. The amounts for
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payout and refresh are transferred/can be claimed.
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\item[\itshape Failure:] The policy is in an failure state. Payouts and
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refreshes are blocked, timeouts are ignored.
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\end{itemize}
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The
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\href{https://git.kesim.org/taler/exchange/src/branch/auction\_brandt/src/exchangedb/exchange-0001-part.sql\#L539-L602}%
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{database-schema has been extended} to include the tables
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\verb|policy_details|, which contains the details to a particular policy
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associated with one or more deposits, and \verb|policy_fulfillments|, which
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keeps proofs of fulfillment when they arrive, associated with one or more
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policies, see Figure~\ref{schema}.
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A new stored-procedure
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\href{https://git.kesim.org/taler/exchange/src/branch/auction\_brandt/src/exchangedb/procedures.sql\#L2190-L2301}%
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{\ttfamily insert\_or\_update\_policy\_details} has been added to handle policy
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details, including the accumulation of the total amount that has been put into
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deposit for an existing policy.
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\begin{figure}
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\centering
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\includegraphics[width=0.95\textwidth]{pics/schema.png}
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\caption{Tables and their relationships for policy extensions}
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\label{schema}
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\end{figure}
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The
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\href{https://git.kesim.org/taler/exchange/src/branch/auction\_brandt/src/include/taler\_extensions.h}{extensions-API}
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has been extended to include the following function pointers:
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\begin{description}
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\item[\href{https://git.kesim.org/taler/exchange/src/branch/auction\_brandt/src/include/taler\_extensions.h\#L167-L185}%
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{\ttfamily create\_policy\_details}] When a deposit request
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contains a
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\href{https://docs.taler.net/core/api-exchange.html#deposit}%
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{policy object} that refers to this extension, this handler
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will be called before the deposit transaction. The policy
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extension is responsible to generate the necessary data to fill
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the \verb|policy_details| table and provide a
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\verb|policy_hash_code| as unique reference to these details.
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\item[\href{https://git.kesim.org/taler/exchange/src/branch/auction\_brandt/src/include/taler\_extensions.h\#L187-L202}%
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{\ttfamily policy\_post\_handler}] Handler for POST-requests to
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the \verb|/extensions/policy_$name| endpoint, responsible for
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validating incoming proofs of fulfilment of a policy with name
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\verb|$name|.
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\end{description}
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\subsection{Transscript and Replay for libbrandt}
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When we started our project, the most recent implementation of Brandt-Vickrey
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auctions was from Markus Teich (\url{https://git.gnunet.org/libbrandt.git/}).
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In our fork at \url{https://git.kesim.org/oec/libbrandt}, we added the
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following functionality:
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\begin{description}
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\item[Transscript generation] The existing unit-test file
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\href{https://git.kesim.org/oec/libbrandt/src/branch/transcript/test\_brandt.c}{\ttfamily test\_brandt.c}
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has been extended to generate and print a transcript in JSON
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encoding for each auction, containing parameters of the
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auction---such as number of bidders, prices and auction
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type---and the list of all messages that the seller has
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received during the protocol execution.
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The definition of the transcript structure is given in appendix
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\ref{transcript}.
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\item[Replay of transscript] The new file
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\label{replay}
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\href{https://git.kesim.org/oec/libbrandt/src/branch/transcript/replay.c}{\ttfamily replay.c}
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implements a standalone program that reads a transcript from
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\verb|stdin|, parses it and executes an auction, replaying all
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messages from the transcript.
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On success, it prints a result in
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\href{https://docs.taler.net/design-documents/032-brandt-vickrey-auctions.html#transcripts}{JSON form}
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to \verb|stdout|:
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\begin{lstlisting}[language=typescript]
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interface BrandtVickreyReplayOutcome {
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// If the replay of the transcript was successfull, will contain the
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// the list of winners and prices.
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winner?: BrandtVickreyAuctionWinner[];
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// If an error occured during replay, will contain an error message.
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error?: string;
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}
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interface BrandtVickreyAuctionWinner {
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// The index of the bidder into the
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// `BrandtVickreyAuctionTranscript`.bidder array.
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bidder: number;
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// The index of the winning price into the
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// `BrandtVickreyAuction`.prices array.
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price_idx: number;
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// The winning price
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price: Amount;
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}
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\end{lstlisting}
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\end{description}
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\subsection{Brandt-Vickrey-auction extension for GNU Taler}
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Based on the policy extensions framework for GNU Taler
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(section~\ref{extension}), we implemented the
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\href{https://git.kesim.org/taler/exchange/src/branch/auction\_brandt/src/extensions/policy\_brandt\_vickrey\_auction/policy\_brandt\_vickrey\_auction.c}%
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{policy extension \ttfamily policy\_brandt\_vickrey\_auction} to handle
|
|
|
|
|
|
deposits of coins that are put into escrow for the participation in an auction
|
|
|
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|
|
and the transfer or release of coins after successful proof of fulfillment,
|
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|
|
which in this cases involves the replay of a transcript of the corresponding
|
|
|
|
|
|
Brandt-Vickrey auction. In particular, we implemented the following handlers:
|
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|
|
\begin{description}
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|
\item[\href{https://git.kesim.org/taler/exchange/src/branch/auction\_brandt/src/extensions/policy\_brandt\_vickrey\_auction/policy\_brandt\_vickrey\_auction.c\#L767-L821}{\ttfamily auction\_create\_policy\_details}]
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|
|
This implements the {\ttfamily create\_policy\_details}
|
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|
|
function of the policy extension interface and generates a
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|
|
hash-code for incoming policy in JSON form, by calculating the
|
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|
|
hash $H(h_a || p_b)$, where $h_a$ is the hash of the auction
|
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|
|
meta data and $p_b$ is the public key of the bidder. Both
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|
parameters are part of the policy structure provided during a
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|
deposit.
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\item[\href{https://git.kesim.org/taler/exchange/src/branch/auction\_brandt/src/extensions/policy\_brandt\_vickrey\_auction/policy\_brandt\_vickrey\_auction.c\#L694-L764}{\ttfamily auction\_policy\_post\_handler}]
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|
|
This implements the {\ttfamily policy\_post\_handler} function
|
2022-10-17 19:11:08 +02:00
|
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|
|
of the policy extension interface. The POST-handler receives
|
2022-10-17 19:02:36 +02:00
|
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|
the transcript and the list of policy-details of former
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|
|
deposit-operations. It parses the transcript and calls the
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|
|
external program (sec.~\ref{replay}) to replay the auction and
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|
|
determine the winners and winning price.
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|
\end{description}
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|
\subsection{Future work}
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To a fully functional and maintainable solution, further development is needed
|
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|
in the following areas:
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|
|
\begin{description}
|
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|
|
\item[libbrandt upgrade] The code base of this library is from 2017 and doesn't
|
2022-10-17 19:25:20 +02:00
|
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|
|
compile with the latest version of
|
2022-10-17 19:02:36 +02:00
|
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|
|
\href{https://git.gnunet.org/gnunet.git}{GNUnet}. It also
|
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|
|
uses \href{https://gnupg.org/software/libgcrypt/index.html}%
|
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|
|
{\ttfamily libgcrypt} as cryptographic library, which is quite
|
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|
|
slow compared to its modern alternative
|
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|
|
\href{https://doc.libsodium.org/}{\ttfamily libsodium}, at
|
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|
|
least with respect to the primitives used in libbrandt.
|
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|
|
For future work we plan to refactor libbrandt to use libsodium
|
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|
|
and the current version of GNUnet.
|
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|
2022-10-17 19:11:08 +02:00
|
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|
|
\item[Brandt-Vickrey-auction continuation] The current code is a
|
2022-10-17 19:02:36 +02:00
|
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|
|
proof-of-concept. The following known problem exist:
|
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|
|
\begin{itemize}
|
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|
|
\item Signature verification of the transcript and its messages
|
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|
is missing
|
2022-10-17 19:25:20 +02:00
|
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|
|
\item Unit- and integration-tests with the exchange are missing
|
2022-10-17 19:02:36 +02:00
|
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|
|
\item A fully featured auction(er) system, including billboard
|
2022-10-17 19:11:08 +02:00
|
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|
|
and client software, is missing.
|
2022-10-17 19:02:36 +02:00
|
|
|
|
\end{itemize}
|
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|
|
We will address at least the first two issues and plan to
|
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|
|
address the lack of an auction system once we find volunteers
|
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|
|
and funding for it.
|
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|
|
\item[Policy framework continuation] Based on the current design of policy
|
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|
|
|
extensions for deposit---aka conditional payments---we plan to add
|
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|
|
|
following policy extensions to the GNU Taler exchange:
|
|
|
|
|
|
\begin{itemize}
|
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|
|
\item{\itshape Merchant refunds:} Merchant can grant customers refundable
|
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|
|
payments. In this case, the amount of the deposit is put into
|
|
|
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|
|
escrow by the exchange for a certain period until which the
|
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|
|
|
|
customer can claim a refund.
|
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|
|
Right now, this policy is implicit and optional in the usual
|
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|
|
|
deposit-flow. Future work on Taler will lift this into a
|
|
|
|
|
|
policy-extension.
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|
|
\item{\itshape Escrowed payments:} A trustor puts coins into escrow
|
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|
|
with the exchange. It can be claimed by a beneficiary until a
|
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|
|
certain deadline, when the claim is signed by
|
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|
|
both, the beneficiary’s and the trustor’s keys.
|
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|
|
This was the basic idea behind the design we presented in
|
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|
|
Milestone 2. It will become a policy-extension in future work.
|
|
|
|
|
|
\end{itemize}
|
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|
\end{description}
|
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|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
|
\newpage
|
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|
|
\section{P2P payments}
|
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|
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|
|
After the implementation of Peer-to-Peer payments on the exchange for milestone
|
|
|
|
|
|
2, we continued to implement the UI's for Android and Web extensions, both as
|
|
|
|
|
|
Proof-of-Concepts.
|
|
|
|
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|
|
|
|
|
|
|
\subsection{Wallet Core}
|
|
|
|
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|
|
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|
|
|
|
The core functionality for the P2P payment in the wallet is located at {
|
|
|
|
|
|
|
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|
|
|
|
\centering \url{https://git.taler.net/wallet-core.git}
|
|
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|
|
|
|
|
|
|
|
|
}
|
|
|
|
|
|
\noindent under
|
|
|
|
|
|
\href{https://git.taler.net/wallet-core.git/tree/packages/taler-wallet-core/src}{/packages/taler-wallet-core/src},
|
|
|
|
|
|
in particular:
|
2022-10-16 11:59:11 +02:00
|
|
|
|
|
2022-10-16 11:49:51 +02:00
|
|
|
|
\begin{itemize}
|
2022-10-17 19:02:36 +02:00
|
|
|
|
\item Peer-to-peer payment operations:
|
|
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|
|
|
|
|
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|
|
\hspace{3em}\href{https://git.taler.net/wallet-core.git/tree/packages/taler-wallet-core/src/operations/pay-peer.ts}%
|
|
|
|
|
|
{\ttfamily operations/pay-peer.ts}
|
|
|
|
|
|
|
|
|
|
|
|
\item Database schema with p2p state in it:
|
|
|
|
|
|
|
|
|
|
|
|
\hspace{3em}\href{https://git.taler.net/wallet-core.git/tree/packages/taler-wallet-core/src/db.ts\#n2065}%
|
|
|
|
|
|
{{\ttfamily src/db.ts}, starting at line 2065}
|
|
|
|
|
|
|
|
|
|
|
|
\item Peer-to-peer transactions in the transaction list type declarations:
|
|
|
|
|
|
|
|
|
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|
|
\hspace{3em}\href{https://git.taler.net/wallet-core.git/tree/packages/taler-util/src/transactions-types.ts#n251}%
|
|
|
|
|
|
{{\ttfamily src/transactions-types.ts}, starting at line 251}
|
|
|
|
|
|
|
2022-10-16 11:49:51 +02:00
|
|
|
|
\end{itemize}
|
|
|
|
|
|
|
|
|
|
|
|
|
2022-10-17 19:02:36 +02:00
|
|
|
|
\subsection{Android App}
|
2022-10-16 11:59:11 +02:00
|
|
|
|
|
2022-10-17 19:02:36 +02:00
|
|
|
|
\begin{figure}
|
|
|
|
|
|
\centering
|
|
|
|
|
|
\includegraphics[width=0.24\linewidth]{pics/photo_2022-10-17_11-52-30.jpg}
|
|
|
|
|
|
\includegraphics[width=0.24\linewidth]{pics/photo_2022-10-17_11-52-32.jpg}
|
|
|
|
|
|
\includegraphics[width=0.24\linewidth]{pics/photo_2022-10-17_11-52-33.jpg}
|
|
|
|
|
|
\includegraphics[width=0.24\linewidth]{pics/photo_2022-10-17_11-52-34.jpg}
|
2022-10-17 19:25:20 +02:00
|
|
|
|
\caption{Screenshots of the UI for P2P payments in the Android version
|
|
|
|
|
|
of the GNU Taler wallet}
|
2022-10-17 19:02:36 +02:00
|
|
|
|
\label{android}
|
|
|
|
|
|
\end{figure}
|
2022-10-16 11:59:11 +02:00
|
|
|
|
|
2022-10-17 19:02:36 +02:00
|
|
|
|
The implementation of the UI for P2P on the Android app is located at {
|
2022-10-16 11:59:11 +02:00
|
|
|
|
|
2022-10-17 19:02:36 +02:00
|
|
|
|
\centering \url{https://git.taler.net/taler-android.git/}
|
2022-10-16 11:59:11 +02:00
|
|
|
|
|
2022-10-17 19:02:36 +02:00
|
|
|
|
} \noindent under
|
|
|
|
|
|
\href{https://git.taler.net/taler-android.git/tree/wallet/src/main/java/net/taler/wallet/peer}%
|
|
|
|
|
|
{/wallet/src/main/java/net/taler/wallet/peer}. The resulting UI is shown in
|
|
|
|
|
|
the screenshots in Fig.~\ref{android}.
|
2022-10-16 11:59:11 +02:00
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2022-10-17 19:02:36 +02:00
|
|
|
|
\subsection{Web extensions UI}
|
|
|
|
|
|
WebExtension UI stuff related to peer-to-peer is in repository {
|
2022-10-16 11:49:51 +02:00
|
|
|
|
|
2022-10-17 19:02:36 +02:00
|
|
|
|
\centering \url{https://git.taler.net/wallet-core.git/}
|
|
|
|
|
|
|
|
|
|
|
|
} \noindent under
|
|
|
|
|
|
\href{https://git.taler.net/wallet-core.git/tree/packages/taler-wallet-webextension/src/cta/}%
|
|
|
|
|
|
{\ttfamily /packages/taler-wallet-webextension/src/cta/}, in directories
|
|
|
|
|
|
\href{https://git.taler.net/wallet-core.git/tree/packages/taler-wallet-webextension/src/cta/InvoiceCreate}%
|
|
|
|
|
|
{\ttfamily InvoiceCreate} and
|
|
|
|
|
|
\href{https://git.taler.net/wallet-core.git/tree/packages/taler-wallet-webextension/src/cta/InvoicePay}%
|
|
|
|
|
|
{\ttfamily InvoicePay}. The resulting UI is shown in the screenshots in Fig.~\ref{web}
|
|
|
|
|
|
|
|
|
|
|
|
\begin{figure}
|
|
|
|
|
|
\centering
|
|
|
|
|
|
\framebox{\includegraphics[width=0.225\textwidth]{pics/Selection_276.png}}
|
|
|
|
|
|
\framebox{\includegraphics[width=0.225\textwidth]{pics/Selection_277.png}}
|
|
|
|
|
|
\framebox{\includegraphics[width=0.225\textwidth]{pics/Selection_278.png}}
|
|
|
|
|
|
\framebox{\includegraphics[width=0.225\textwidth]{pics/Selection_279.png}}
|
|
|
|
|
|
\caption{Screenshots of the P2P payments in the Web extension}
|
|
|
|
|
|
\label{web}
|
|
|
|
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
2022-10-16 11:49:51 +02:00
|
|
|
|
|
|
|
|
|
|
\newpage
|
|
|
|
|
|
\part*{Appendix}
|
|
|
|
|
|
\addcontentsline{toc}{part}{Appendix}
|
|
|
|
|
|
\appendix
|
|
|
|
|
|
|
2022-10-17 19:11:08 +02:00
|
|
|
|
\section{Definition of a Transcript of Brandt-Vickrey-auctions}
|
2022-10-16 15:57:37 +02:00
|
|
|
|
\label{transcript}
|
|
|
|
|
|
|
|
|
|
|
|
The following definition is provided in TypeScript and taken from
|
|
|
|
|
|
\url{https://docs.taler.net/core/api-exchange.html#tsref-type-BrandtVickreyAuction}.
|
|
|
|
|
|
|
|
|
|
|
|
\begin{lstlisting}[language=typescript]
|
|
|
|
|
|
// This structure defines an auction of Brandt-Vickory kind.
|
|
|
|
|
|
// It is used for the PolicyBrandtVickreyAuction.
|
|
|
|
|
|
interface BrandtVickreyAuction {
|
|
|
|
|
|
// Start date of the auction
|
|
|
|
|
|
time_start: Timestamp;
|
|
|
|
|
|
|
|
|
|
|
|
// Maximum duration per round. There are four rounds in an auction of
|
|
|
|
|
|
// Brandt-Vickrey kind.
|
|
|
|
|
|
time_round: RelativeTime;
|
|
|
|
|
|
|
|
|
|
|
|
// This integer m refers to the (m+1)-type of the Brandt-Vickrey-auction.
|
|
|
|
|
|
// - Type 0 refers to an auction with one highest-price winner,
|
|
|
|
|
|
// - Type 1 refers to an auction with one winner, paying the second
|
|
|
|
|
|
// highest price,
|
|
|
|
|
|
// - Type 2 refers to an auction with two winners, paying
|
|
|
|
|
|
// the third-highest price,
|
|
|
|
|
|
// - etc.
|
|
|
|
|
|
auction_type: number;
|
|
|
|
|
|
|
|
|
|
|
|
// The vector of prices for the Brandt-Vickrey auction. The values MUST
|
|
|
|
|
|
// be in strictly increasing order.
|
|
|
|
|
|
prices: Amount[];
|
|
|
|
|
|
|
|
|
|
|
|
// The type of outcome of the auction.
|
|
|
|
|
|
// In case the auction is declared public, each bidder can calculate the
|
|
|
|
|
|
// winning price. This field is not relevant for the replay of a
|
|
|
|
|
|
// transcript, as the transcript must be provided by the seller who sees
|
|
|
|
|
|
// the winner(s) and winning price of the auction.
|
|
|
|
|
|
outcome_public: boolean;
|
|
|
|
|
|
|
|
|
|
|
|
// The public key of the seller.
|
|
|
|
|
|
pubkey: EddsaPublicKey;
|
|
|
|
|
|
|
|
|
|
|
|
// The seller's account details.
|
|
|
|
|
|
payto_uri: string;
|
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
// This structure defines the transcript of an auction
|
|
|
|
|
|
// of Brandt-Vickrey kind.
|
|
|
|
|
|
interface BrandtVickreyAuctionTranscript {
|
|
|
|
|
|
// The auction definition.
|
|
|
|
|
|
auction: BrandtVickreyAuction;
|
|
|
|
|
|
|
|
|
|
|
|
// The public keys of the bidders, in Crockford Base32 encoding.
|
|
|
|
|
|
bidders: EddsaPublicKey[];
|
|
|
|
|
|
|
|
|
|
|
|
// Signatures of the auction in Crockford Base32 encoding.
|
|
|
|
|
|
// One signature per bidder.
|
|
|
|
|
|
signatures: EddsaSignature[];
|
|
|
|
|
|
|
|
|
|
|
|
// List of policy hash codes that identify policy details associated with
|
|
|
|
|
|
// each bidder. Those codes were generated by the policy extension
|
|
|
|
|
|
// policy_brandt_vickrey_auction during the deposit of coins for this
|
|
|
|
|
|
// auction.
|
|
|
|
|
|
policy_hash_codes: HashCode[];
|
|
|
|
|
|
|
|
|
|
|
|
// The transcript of all messages received by the seller.
|
|
|
|
|
|
transcript: BrandtVickreyAuctionMessage[];
|
|
|
|
|
|
|
|
|
|
|
|
// Optionally, the seller can provide the winners it had calculated.
|
|
|
|
|
|
winners?: BrandtVickreyAuctionWinner[];
|
|
|
|
|
|
|
|
|
|
|
|
// The signature over the hash of this JSON object, without the
|
|
|
|
|
|
// key ``sig`` and in normalized form, basically over
|
|
|
|
|
|
// H(auction, bidders, signatures, transcripts, winners?)
|
|
|
|
|
|
// It is signed by the private key that corresponds to the public key
|
|
|
|
|
|
// in `BrandtVickreyAuction`.``pubkey``.
|
|
|
|
|
|
// This signature is in Crockford Base32 encoding.
|
|
|
|
|
|
sig: EddsaSignature;
|
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
interface BrandtVickreyAuctionMessage {
|
|
|
|
|
|
// The index of the bidder into the
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// `BrandtVickreyAuctionTranscript`.``bidders`` array.
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bidder: number;
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// The raw message in Crockford Base32 encoding.
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msg: string;
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// The signature over the message. The signature is in Crockford Base32
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// encoding. It must be signed by the private key corresponding to the
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// bidder's public key in `BrandtVickreyAuctionTranscript`.``bidders``.
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sig: EddsaSignature;
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}
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\end{lstlisting}
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2022-10-16 11:49:51 +02:00
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\end{document}
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