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Aniket Kate Purdue University
IOweYou Credit Networks Applications and Privacy
$
$
$$
$
CCS 2016 Tutorial
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Ever Changing Landscape of Communication
Local Marketplaces2
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Ever Changing Landscape of Communication
Local Marketplaces Global2000-2010
2
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Ever Changing Landscape of Communication
Local Marketplaces
Centralized
Global2000-2010
2
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Ever Changing Landscape of Communication
Local Marketplaces
Centralized
Global2000-2010
2
2010 onwards
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Ever Changing Landscape of Communication
Local Marketplaces
Centralized
Decentralized/ Distributed
Global2000-2010
2
2010 onwards
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Ever Changing Landscape of Communication
Local Marketplaces
Centralized
Decentralized/ Distributed
Global2000-2010
2
2010 onwards
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Ever Changing Landscape of Communication
Local Marketplaces
Centralized
Decentralized/ Distributed
Global2000-2010
2
2010 onwards
Blockchain for Everything!
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Ever Changing Landscape of Communication
Local Marketplaces
Centralized
Decentralized/ Distributed
Global2000-2010
Crypto-currencies may or may not
survive, but the concept of distributed
ledgers/blockchains is here to stay
2
2010 onwards
Blockchain for Everything!
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Blockchain can change ... well everything
3
Source: CB Insights
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Blockchain can change ... well everything
4Source: http://startupmanagement.org/blog
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Blockchain can change ... well everything
4Source: http://startupmanagement.org/blog
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This talk proudly aims at leaving you with more interesting questions than mere answers regarding credit networks. :)
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Credit (or IOU settlement) Networks: Basics
6
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Credit (or IOU settlement) Networks: Basics
6
$100
Transactions in the real world
Bob Alice
IOweYou $100
Bob Alice
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Credit (or IOU settlement) Networks: Basics
6
$100 AliceBob
Transactions in the real world A credit network representation
Bob Alice
IOweYou $100
Bob Alice
100
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IOweYou €10
Credit (or IOU settlement) Networks: Basics
6
$100 AliceBob
Transactions in the real world A credit network representation
Bob Alice
IOweYou $100
Bob Alice
€10
Dave Carol
Dave Carol
100
During a hike with Alice & Bob
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IOweYou €10
Credit (or IOU settlement) Networks: Basics
6
$100 AliceBob
Carol
Transactions in the real world A credit network representation
Bob Alice
IOweYou $100
Bob Alice
€10
Dave Carol
Dave Carol
100
Dave
During a hike with Alice & Bob
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IOweYou €10
Credit (or IOU settlement) Networks: Basics
6
$100 Alice
10
Bob
Carol
Transactions in the real world A credit network representation
Bob Alice
IOweYou $100
Bob Alice
€10
Dave Carol
Dave Carol
100
Dave10
110
During a hike with Alice & Bob
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30
15
Payment (or credit) Network: an Example
7
Bob
Dave
Alice
115
10Eve
Carol
05
20
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30
15
Payment (or credit) Network: an Example
7
Bob
Dave
Alice
115
15
10Eve
Carol
05
20
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30
15
Payment (or credit) Network: an Example
7
Bob
Dave
Alice
115
15
10Eve
Carol
05
20
Max-flow Computation
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30
15
Payment (or credit) Network: an Example
7
Bob
Dave
Alice
115
15
10Eve
Carol
05
20
Max-flow Computation
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3020
15
Payment (or credit) Network: an Example
7
Bob
Dave
5
Alice
115
15
100Eve
Carol
0
20
Max-flow Computation
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3020
15
Payment (or credit) Network: an Example
7
Bob
Dave
5
Alice
115
100Eve
Carol
0
20
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Key Questions for Credit Network Designs
✦ Roots in the very old Barter System or Havala
✦ Path Selection ✦ How do we find paths?
-Max flow algorithms may not scale ✦ How do we select paths?
-Social welfare; e.g., allowing many transactions to succeed - NP-hard problem
✦ Liquidity of the network ✦ For randomly chosen pair of nodes, and transaction value,
what is the probability that the transaction succeeds? ✦ More and stronger links, better liquidity; Clique!?
✦ Sybil Tolerance ✦ Number of sybil nodes should not matter ✦ How much IOU credit can we allow the adversary to garner?
-How many sybil links can we manage? -What kind topologies and links values?
8
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Why credit networks matter?
✦ A flexible-yet-robust design for distributed (transitive) trust ✦ through pairwise credit allocations
✦ Loss incurred due to misbehaving identities is bounded and (sometimes) localized
9
205
Dave
Alice
1000
30
115
10
Eve
Carol
Bob
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Building trust with credit networks
10
✦ Sybil-resistant applications
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Building trust with credit networks
10
✦ Sybil-resistant applications
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Building trust with credit networks
10
✦ Sybil-resistant applications
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Building trust with credit networks
10
✦ Sybil-resistant applications
30
20
15
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Building trust with credit networks
10
✦ Sybil-resistant applications
30
20
15
Well-behaved nodes Sybil nodes
edge cut
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Building trust with credit networks
10
✦ Sybil-resistant applications
30
20
15
Introducing nodes is much easier than drawing trust from well-behaved nodes
Well-behaved nodes Sybil nodes
edge cut
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Building trust with credit networks
✦ Several Systems✦ Ostra: preventing e-mail spam [NSDI’08]
10
✦ Sybil-resistant applications
30
20
15
Introducing nodes is much easier than drawing trust from well-behaved nodes
Well-behaved nodes Sybil nodes
edge cut
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Building trust with credit networks
✦ Several Systems✦ Ostra: preventing e-mail spam [NSDI’08]✦ Bazaar: strengthening e-commerce [NSDI’11]
10
✦ Sybil-resistant applications
30
20
15
Introducing nodes is much easier than drawing trust from well-behaved nodes
Well-behaved nodes Sybil nodes
edge cut
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Building trust with credit networks
✦ Several Systems✦ Ostra: preventing e-mail spam [NSDI’08]✦ Bazaar: strengthening e-commerce [NSDI’11]✦ SumUp: Sybil-resilient content voting [NSDI’09]
10
✦ Sybil-resistant applications
30
20
15
Introducing nodes is much easier than drawing trust from well-behaved nodes
Well-behaved nodes Sybil nodes
edge cut
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Building trust with credit networks
✦ Several Systems✦ Ostra: preventing e-mail spam [NSDI’08]✦ Bazaar: strengthening e-commerce [NSDI’11]✦ SumUp: Sybil-resilient content voting [NSDI’09]✦ Ripple: A real-life online settlement network
10
✦ Sybil-resistant applications
30
20
15
Introducing nodes is much easier than drawing trust from well-behaved nodes
Well-behaved nodes Sybil nodes
edge cut
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Bazaar: Strengthening Online Marketplaces
11
[NSDI’11]
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Bazaar: Strengthening Online Marketplaces
11
$10
$10
$2
$15
Undirected reputations links!
[NSDI’11]
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Bazaar: Strengthening Online Marketplaces
11
$10
$10
$2
$15
Undirected reputations links!
$10 8
$10 2
$2 0
$15 7
Consider a $10 transaction
[NSDI’11]
![Page 41: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/41.jpg)
Bazaar: Strengthening Online Marketplaces
11
$10
$10
$2
$15
Undirected reputations links!
$10 8
$10 2
$2 0
$15 7
Consider a $10 transaction
$10
$10
$2
$15
$10
After a positive review
[NSDI’11]
![Page 42: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/42.jpg)
Bazaar: Strengthening Online Marketplaces
11
$10
$10
$2
$15
Undirected reputations links!
$10 8
$10 2
$2 0
$15 7
Consider a $10 transaction
$8
$2$7
After a negative review
$10
$10
$2
$15
$10
After a positive review
[NSDI’11]
![Page 43: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/43.jpg)
Ripple Credit (or Settlement) Network
12
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Ripple Credit (or Settlement) Network
12
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Ripple Credit (or Settlement) Network
12
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Ripple Credit (or Settlement) Network
12
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£ 70
$ 100
$ 60
€ 45€
30
Ripple Credit (or Settlement) Network
12
B 5 B 10
€ 10
$ 100
£ 280
$ 40
![Page 48: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/48.jpg)
£ 70
$ 100
$ 60
€ 45€
30
Ripple Credit (or Settlement) Network
12
B 5 B 10
€ 10
$ 100
£ 280
$ 40
~ 1 day
~ 5 seconds
High fees
Tiny fees
Tx timeWorldwide,
inter-currency tx Integrity
Bank only
Public verifiability
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£ 70
$ 100
$ 60
€ 45€
30
Ripple Credit (or Settlement) Network
12
B 5 B 10
€ 10
$ 100
£ 280
$ 40
~ 1 day
~ 5 seconds
High fees
Tiny fees
Tx timeWorldwide,
inter-currency tx Integrity
Bank only
Public verifiability
Ripple can significantly
improve cross-currency
remittance and settlements
![Page 50: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/50.jpg)
Cryptocurrencies vs Credit Networks
Cryptocurrencies IOU Credit Networks
DefinitionMedium of exchange; future productivity of
the public
Credit settlement network: future productivity of a
specific borrower
Transfer of funds
Direct transactions between any two wallets
Transactions only via a path with enough credit
Fungibility Good Restricted by path availability
Scalability Limited transaction rate (<100 tps)
Highly scalable
We already have cryptocurrencies, then why do we need Ripple?
13
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Cryptocurrencies vs Credit Networks
✦ IOU (or Credit) Networks ✦ Combining credit and social trust (still not permissioned)
14
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Cryptocurrencies vs Credit Networks
✦ IOU (or Credit) Networks ✦ Combining credit and social trust (still not permissioned)
✦ Restricted Fungibility of Credit Networks
14
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Cryptocurrencies vs Credit Networks
✦ IOU (or Credit) Networks ✦ Combining credit and social trust (still not permissioned)
✦ Restricted Fungibility of Credit Networks
14
$10$15
Not connectedX
![Page 54: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/54.jpg)
Cryptocurrencies vs Credit Networks
✦ IOU (or Credit) Networks ✦ Combining credit and social trust (still not permissioned)
✦ Restricted Fungibility of Credit Networks
14
$10$15
Not connectedX
Xnot possible
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Cryptocurrencies vs Credit Networks
✦ IOU (or Credit) Networks ✦ Combining credit and social trust (still not permissioned)
✦ Restricted Fungibility of Credit Networks
✦ The Tyranny of Proof of Work ✦ Bitcoin mining could consume as much electricity as Denmark by
2020! ✦ (Distributed) credit networks may not require global consensus!
- I care only about my links to my friends - I do not even care about links between friends and friends-of-friends -Formalization coming soon … 14
$10$15
Not connectedX
Xnot possible
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Credit Networks: State of the Art
✦ Examples: Ripple, Stellar Settlement Networks ✦ For Ripple,
Trade volume: $800K Payment volume: $400K per day ✦ Several banking systems across the world (US, Canada,
Germany, China, Japan, Singapore,…) are getting involved
✦ Global (Federated!) Consensus ✦ Non-standard atomic broadcast protocol
-An interesting problem to study/improve it ✦ Choice to consensus parties is not convincing
-A major criticism against Ripple in academics and P2P communities
✦ Public verifiability of transactions ✦ Motivated from the Bitcoin success ✦ Same privacy problem!
15
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Attacks on privacy of Ripple links & transactions
Credit GraphTransaction Details
Ripple provides pseudonymity to its users by employing public-key hashes as identities
16
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Attacks on privacy of Ripple links & transactions
Credit GraphTransaction Details
Ripple provides pseudonymity to its users by employing public-key hashes as identities
It is possible to link multiple transactions
and identities belonging to the same user
16
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Is privacy a real problem in Ripple?
Privacy Attacks: Innocent until Proven Guilty
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Is privacy a real problem in Ripple?
Privacy Attacks: Innocent until Proven Guilty
P. Moreno-Sanchez, M. B. Zafar, A. Kate:Linking Wallets and Deanonymizing Transactions in the Ripple Network. Privacy Enhancing Technologies Symposium (PETS) 2016.Ripple Forum Discussion: www.xrpchat.com/topic/1721-linking-wallets-and-deanonymizing-transactions-in-ripple/
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Heuristic 1: The Tale of two Public Logs
18
Bitcoin Ripple
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Heuristic 1: The Tale of two Public Logs
18
Input Output Alice-Bitcoin:
6 BTCDR-Bitcoin:
6 BTCAlice
Bitcoin Ripple
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Heuristic 1: The Tale of two Public Logs
18
Input Output Alice-Bitcoin:
6 BTCDR-Bitcoin:
6 BTCAlice
Bitcoin RippleSender DR-Ripple
Receiver Alice-RippleValue 6 BTC IOUPath Bob —> Alice
Bob
6
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Heuristic 1: The Tale of two Public Logs
18
Input Output Alice-Bitcoin:
6 BTCDR-Bitcoin:
6 BTCAlice
Bitcoin RippleSender DR-Ripple
Receiver Alice-RippleValue 6 BTC IOUPath Bob —> Alice
Bob
6
Alice-Bitcoin Alice-Ripple
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Heuristic 1: The Tale of two Public Logs
18
Input Output Alice-Bitcoin:
6 BTCDR-Bitcoin:
6 BTCAlice
Bitcoin RippleSender DR-Ripple
Receiver Alice-RippleValue 6 BTC IOUPath Bob —> Alice
Bob
6
Alice-Bitcoin Alice-Ripple
DR-Bitcoin DR-Ripple
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Heuristic 1: The Tale of two Public Logs
18
Input Output Alice-Bitcoin:
6 BTCDR-Bitcoin:
6 BTCAlice
✦ How to link these two events?
Bitcoin RippleSender DR-Ripple
Receiver Alice-RippleValue 6 BTC IOUPath Bob —> Alice
Bob
6
Alice-Bitcoin Alice-Ripple
DR-Bitcoin DR-Ripple
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Heuristic 1: The Tale of two Public Logs
✦ Some gateways/exchanges keep public logs of their businesses
✦ This interlog linkability attack possible without public log ✦ timestamps and transaction amounts
19
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Heuristic 2: Hot-Cold Wallets
20
€ 100,000
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Heuristic 2: Hot-Cold Wallets
20
€ 100,000
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Heuristic 2: Hot-Cold Wallets
20
€ 40
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Heuristic 2: Hot-Cold Wallets
20
€ 20
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Heuristic 2: Hot-Cold Wallets
20
€ 20
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Heuristic 2: Hot-Cold Wallets
20
€ 150
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Heuristic 2: Hot-Cold Wallets
20
Ripple Users
€ 5500€ 23
00
€ 150
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Heuristic 2: Hot-Cold Wallets
20
Ripple Users
€ 5500€ 23
00Cold
€ 150
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Heuristic 2: Hot-Cold Wallets
20
Ripple Users
€ 5500€ 23
00
€ 200
Cold
€ 150
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Heuristic 2: Hot-Cold Wallets
20
Ripple Users
€ 5500€ 23
00
€ 200
Cold
Hot€ 150
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Heuristic 2: Hot-Cold Wallets
20
Ripple Users
€ 5500€ 23
00
€ 200
Cold
Hot€ 150
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Heuristic 2: Hot-Cold Wallets
20
Ripple Users
€ 5500€ 23
00
€ 200
Cold
Hot
Link hot and cold wallets!!
€ 150
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Heuristic 2: Hot-Cold Wallets
✦ Correlation between network topology and transactions
21
€ 200
€ 5500€ 20
00A
B
CD
Sender Receiver AmountA B €275B D €30D C €10B C €45
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Heuristic 2: Hot-Cold Wallets
✦ Correlation between network topology and transactions
21
€ 200
€ 5500€ 20
00A
B
CD
Sender Receiver AmountA B €275B D €30D C €10B C €45
✦ Cold wallet only issues credit
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Heuristic 2: Hot-Cold Wallets
✦ Correlation between network topology and transactions
21
€ 200
€ 5500€ 20
00A
B
CD
Sender Receiver AmountA B €275B D €30D C €10B C €45
✦ Cold wallet only issues credit
Cold
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Heuristic 2: Hot-Cold Wallets
✦ Correlation between network topology and transactions
21
€ 200
€ 5500€ 20
00A
B
CD
Sender Receiver AmountA B €275B D €30D C €10B C €45
✦ Cold wallet only issues credit
Cold
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Heuristic 2: Hot-Cold Wallets
✦ Correlation between network topology and transactions
21
€ 200
€ 5500€ 20
00A
B
CD
Sender Receiver AmountA B €275B D €30D C €10B C €45
✦ Cold wallet only issues credit✦ Cold wallet must top off hot wallet
Cold
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Heuristic 2: Hot-Cold Wallets
✦ Correlation between network topology and transactions
21
€ 200
€ 5500€ 20
00A
B
CD
Sender Receiver AmountA B €275B D €30D C €10B C €45
✦ Cold wallet only issues credit✦ Cold wallet must top off hot wallet
Cold
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Heuristic 2: Hot-Cold Wallets
✦ Correlation between network topology and transactions
21
€ 200
€ 5500€ 20
00A
B
CD
Sender Receiver AmountA B €275B D €30D C €10B C €45
✦ Cold wallet only issues credit✦ Cold wallet must top off hot wallet
Cold
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Heuristic 2: Hot-Cold Wallets
✦ Correlation between network topology and transactions
21
€ 200
€ 5500€ 20
00A
B
CD
Sender Receiver AmountA B €275B D €30D C €10B C €45
✦ Cold wallet only issues credit✦ Cold wallet must top off hot wallet✦ Hot wallet used to fund client wallets
Cold
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Heuristic 2: Hot-Cold Wallets
✦ Correlation between network topology and transactions
21
€ 200
€ 5500€ 20
00A
B
CD
Sender Receiver AmountA B €275B D €30D C €10B C €45
✦ Cold wallet only issues credit✦ Cold wallet must top off hot wallet✦ Hot wallet used to fund client wallets
Cold
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Heuristic 2: Hot-Cold Wallets
✦ Correlation between network topology and transactions
21
€ 200
€ 5500€ 20
00A
B
CD
Sender Receiver AmountA B €275B D €30D C €10B C €45
✦ Cold wallet only issues credit✦ Cold wallet must top off hot wallet✦ Hot wallet used to fund client wallets
Cold
Hot
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Heuristic 2: Hot-Cold Wallets
✦ Correlation between network topology and transactions
21
€ 200
€ 5500€ 20
00A
B
CD
Sender Receiver AmountA B €275B D €30D C €10B C €45
✦ Cold wallet only issues credit✦ Cold wallet must top off hot wallet✦ Hot wallet used to fund client wallets
Cold
Hot
A, B belong to the same user
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Deanonymization of several gateways
22
Known Deanonymized
Unknown transactions
Sharing the same owner
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Towards privacy-preserving transactions credit networks
P. Moreno-Sanchez, A. Kate, M. Maffei, and K. Pecina:Privacy Preserving Payments in Credit Networks. NDSS 2015
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Defining privacy for a credit network
24
Transaction sender privacy can be defined similarly
Transaction value privacy
≈
10
30
Bob
Bob
Carol
Carol
Transaction receiver privacy
10
10
≈
Bob
Bob Carol
Dave
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Transaction Value Privacy: Definition (II)
✦ A credit network satisfies value privacy if:
25
≈
Pr
Pr
-30
+20 Balancing transaction
Challenge transaction
-10
+0Balancing transaction
Challenge transaction
-30Challenge transaction is
-30Challenge transaction is
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Towards credit network privacy
26
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Towards credit network privacy
✦ A decentralized or centralized architecture?
26
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Towards credit network privacy
✦ A decentralized or centralized architecture?
✦ Centralized setting: the network is maintained by a server ✦ The service provider can trivially break the privacy
-The routing computation can be performed privately, but any modifications to the edges not -Use of pseudonyms and anonymous channels
(e.g, Tor) is not sufficient ✦ In our NDSS’15 paper, we resolve this issue using minimally
trusted hardware and oblivious algorithms
26
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Towards credit network privacy
✦ A decentralized or centralized architecture?
✦ Centralized setting: the network is maintained by a server ✦ The service provider can trivially break the privacy
-The routing computation can be performed privately, but any modifications to the edges not -Use of pseudonyms and anonymous channels
(e.g, Tor) is not sufficient ✦ In our NDSS’15 paper, we resolve this issue using minimally
trusted hardware and oblivious algorithms
✦ Decentralized setting: edges are maintained locally ✦ A transaction passing through a node requires its
active involvement ✦ We will consider this later during the talk
26
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Towards credit network privacy
✦ Centralized setting ✦ The network is maintained by a service provider
✦ Threat Model ✦ The service provider is honest-but-curious ✦ Some users are controlled by the service provider
✦ The service provider can trivially break the privacy ✦ The routing computation can be performed privately,
but any modifications to the edges cannot
✦ We resolve this feasibility issue using minimally trusted hardware
27
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Our centralized approach: PrivPay
✦ Threat Model ✦ The service provider is honest-
but-curious ✦ Some users are controlled by
the service provider
✦ A service-side trusted hardware module maintains the network graph in the untrusted server memory
✦ Correctness of the hardware module can be verified using remote code attestation
✦ Encryption by itself prevents an attacker from learning the database entry but monitoring memory accesses is still possible
✦ We develop oblivious algorithms for routing to solve this problem28
Untrusted Server
Environment
TLS
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Privacy Preserving Payments in Credit Networks
Routing: max-flow computation
✦ Routing challenge:Known max-flow algorithms are not scalable: O(V3) or O(V2log(E))
✦ We employ landmark routing: Calculate only a subset of all possible routes through intermediary nodes called landmarks [Tsuchiya SigComm’88] [Vishanath et al. Eurosys’12]
29NDSS'15
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Privacy Preserving Payments in Credit Networks
Routing: max-flow computation
✦ Routing challenge:Known max-flow algorithms are not scalable: O(V3) or O(V2log(E))
✦ We employ landmark routing: Calculate only a subset of all possible routes through intermediary nodes called landmarks [Tsuchiya SigComm’88] [Vishanath et al. Eurosys’12]
29NDSS'15
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Privacy Preserving Payments in Credit Networks
Routing: max-flow computation
✦ Routing challenge:Known max-flow algorithms are not scalable: O(V3) or O(V2log(E))
✦ We employ landmark routing: Calculate only a subset of all possible routes through intermediary nodes called landmarks [Tsuchiya SigComm’88] [Vishanath et al. Eurosys’12]
29
. . .
NDSS'15
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Privacy Preserving Payments in Credit Networks
Routing: max-flow computation
✦ Routing challenge:Known max-flow algorithms are not scalable: O(V3) or O(V2log(E))
✦ We employ landmark routing: Calculate only a subset of all possible routes through intermediary nodes called landmarks [Tsuchiya SigComm’88] [Vishanath et al. Eurosys’12]
29
. . .
. . .
NDSS'15
Landmark Universe
![Page 105: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/105.jpg)
Privacy Preserving Payments in Credit Networks
Routing: max-flow computation
✦ Routing challenge:Known max-flow algorithms are not scalable: O(V3) or O(V2log(E))
✦ We employ landmark routing: Calculate only a subset of all possible routes through intermediary nodes called landmarks [Tsuchiya SigComm’88] [Vishanath et al. Eurosys’12]
29
. . .
. . .
NDSS'15
?
Landmark Universe
![Page 106: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/106.jpg)
Privacy Preserving Payments in Credit Networks
Routing: max-flow computation
✦ Routing challenge:Known max-flow algorithms are not scalable: O(V3) or O(V2log(E))
✦ We employ landmark routing: Calculate only a subset of all possible routes through intermediary nodes called landmarks [Tsuchiya SigComm’88] [Vishanath et al. Eurosys’12]
29
. . .
. . .
NDSS'15
?
Landmark Universe
![Page 107: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/107.jpg)
Privacy Preserving Payments in Credit Networks
Routing: max-flow computation
✦ Routing challenge:Known max-flow algorithms are not scalable: O(V3) or O(V2log(E))
✦ We employ landmark routing: Calculate only a subset of all possible routes through intermediary nodes called landmarks [Tsuchiya SigComm’88] [Vishanath et al. Eurosys’12]
29
. . .
NDSS'15
?
Path Stitching
![Page 108: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/108.jpg)
PrivPay architecture
30
Untrusted Server
Environment
Universe CreatorTransaction
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PrivPay architecture
✦ Landmark universe creator module ✦ Oblivious BFS computation for selected landmark nodes
30
Untrusted Server
Environment
Universe CreatorTransaction
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PrivPay architecture
✦ Landmark universe creator module ✦ Oblivious BFS computation for selected landmark nodes
30
Untrusted Server
Environment
Universe CreatorTransaction
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PrivPay architecture
✦ Landmark universe creator module ✦ Oblivious BFS computation for selected landmark nodes
✦ Transaction (path stitcher) module ✦ Given a sender and a receiver, traverse the BFS trees in
an oblivious manner for the overlapping landmark nodes
30
Untrusted Server
Environment
Universe CreatorTransaction
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PrivPay architecture
✦ Landmark universe creator module ✦ Oblivious BFS computation for selected landmark nodes
✦ Transaction (path stitcher) module ✦ Given a sender and a receiver, traverse the BFS trees in
an oblivious manner for the overlapping landmark nodes
✦ Privacy properties are formally proven30
Untrusted Server
Environment
Universe CreatorTransaction
![Page 113: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/113.jpg)
PrivPay architecture
✦ Landmark universe creator module ✦ Oblivious BFS computation for selected landmark nodes
✦ Transaction (path stitcher) module ✦ Given a sender and a receiver, traverse the BFS trees in
an oblivious manner for the overlapping landmark nodes
✦ Privacy properties are formally proven30
Untrusted Server
Environment
Universe CreatorTransaction
![Page 114: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/114.jpg)
PrivPay architecture
✦ Landmark universe creator module ✦ Oblivious BFS computation for selected landmark nodes
✦ Transaction (path stitcher) module ✦ Given a sender and a receiver, traverse the BFS trees in
an oblivious manner for the overlapping landmark nodes
✦ Privacy properties are formally proven30
Untrusted Server
Environment
Universe CreatorTransaction
![Page 115: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/115.jpg)
PrivPay architecture
✦ Landmark universe creator module ✦ Oblivious BFS computation for selected landmark nodes
✦ Transaction (path stitcher) module ✦ Given a sender and a receiver, traverse the BFS trees in
an oblivious manner for the overlapping landmark nodes
✦ Privacy properties are formally proven30
Two modules are not synchronized
Untrusted Server
Environment
Universe CreatorTransaction
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Applying PrivPay to the Ripple network
✦ We have implemented PrivPay as a C++ library ✦ We employed real-world Ripple transactions over a period of
four months (Oct'13 – Jan'14)
31
Time in msec Non-Private [Eurosys’12]
PrivPay
Payment 0.078 1510
Change link 0.005 95
Oblivious BFS 50 22000
Coverage 97% 95%
Ripple takes 5 sec. to confirm a
transaction
Background Process
No false positive, only false negative
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PrivPay: Deployment Challenges
✦ Ripple is currently focusing on their business growth ✦ The privacy concerns was secondary to them ✦ Trusted hardware-based solutions require investment
-Ripple is not ready for the challenge yet!
✦ Scalability of (background) Oblivious BFS algorithm as number of users have increased ten holds ✦ the coverage will reduces
✦ Question: Can we find some solution that is compatible with the current Ripple architecture? ✦ Yes! but with a caveat
32
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PathShuffle
33
[In Submission]
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PathShuffle
✦ Idea: Perform several transactions simultaneously enables privacy-preserving transactions
✦ Similar to Conjoin or CoinShuffle for Bitcoin
33
12/2215/05
30/20 10/20
20/3060/50
[In Submission]
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PathShuffle
✦ Idea: Perform several transactions simultaneously enables privacy-preserving transactions
✦ Similar to Conjoin or CoinShuffle for Bitcoin
✦ Ripple only allows single sender/receiver per transaction ✦ Employ threshold signature techniques overcome the problem ✦ Pathjoin!
33
12/2215/05
30/20 10/20
20/3060/50
[In Submission]
![Page 121: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/121.jpg)
PathShuffle
✦ Idea: Perform several transactions simultaneously enables privacy-preserving transactions
✦ Similar to Conjoin or CoinShuffle for Bitcoin
✦ Ripple only allows single sender/receiver per transaction ✦ Employ threshold signature techniques overcome the problem ✦ Pathjoin!
✦ 100% Compatible with Ripple. We tested it on the real Ripple network!33
12/2215/05
30/20 10/20
20/3060/50
[In Submission]
![Page 122: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/122.jpg)
Towards Secure Distributed Credit Networks
A. Kate, M. Maffei, G. Malavolta, and P. Moreno-Sanchez:SilentWhispers: Enforcing Security and Privacy in Decentralized Credit Networks To appar at NDSS 2017TechReport: http://crypsys.mmci.uni-saarland.de/projects/DecentralizedPrivPay/
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35
A Distributed Credit Network
✦ Each user maintains her own credit links
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35
A Distributed Credit Network
✦ Each user maintains her own credit links
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35
A Distributed Credit Network
✦ Each user maintains her own credit links
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Local Knowledge is Sufficient!
✦ Credit links of a user determine his credit in the network
36
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Local Knowledge is Sufficient!
✦ Credit links of a user determine his credit in the network
36
45015
25
In-flow = 450 Out-flow = 40 Net-flow = 410Bob
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Local Knowledge is Sufficient!
✦ Credit links of a user determine his credit in the network
36
✦ A user checks net-flow does not change
45015
25
In-flow = 450 Out-flow = 40 Net-flow = 410Bob
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Local Knowledge is Sufficient!
✦ Credit links of a user determine his credit in the network
36
✦ A user checks net-flow does not change
45015
25
In-flow = 450 Out-flow = 40 Net-flow = 410
15
25
In-flow = 450 Out-flow = 40 Net-flow = 410
450
Bob
Bob
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Local Knowledge is Sufficient!
✦ Credit links of a user determine his credit in the network
36
✦ A user checks net-flow does not change
45015
25
In-flow = 450 Out-flow = 40 Net-flow = 410
15
25
In-flow = 450 Out-flow = 40 Net-flow = 410
5
450
Bob
Bob
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Local Knowledge is Sufficient!
✦ Credit links of a user determine his credit in the network
36
✦ A user checks net-flow does not change
45015
25
In-flow = 450 Out-flow = 40 Net-flow = 410
25
10 In-flow = 450 Out-flow = 40 Net-flow = 410
5
450
Bob
Bob
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Local Knowledge is Sufficient!
✦ Credit links of a user determine his credit in the network
36
✦ A user checks net-flow does not change
45015
25
In-flow = 450 Out-flow = 40 Net-flow = 410
25
10 In-flow = 450 Out-flow = 40 Net-flow = 410
5
445
Bob
Bob
![Page 133: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/133.jpg)
Local Knowledge is Sufficient!
✦ Credit links of a user determine his credit in the network
36
✦ A user checks net-flow does not change
45015
25
In-flow = 450 Out-flow = 40 Net-flow = 410
25
10 In-flow = 450 Out-flow = 40 Net-flow = 410
5
44544535
Bob
Bob
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Challenges
✦ How to find paths between a sender and a receiver?
✦ How to find the IOU credit available in the path?
✦ How to ensure credit links form a path?
✦ And maintaining strong privacy, availability, and accountability guarantees…
37
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38
Credit in a Path
30 15 25 10
[x]: Secret sharing of x
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38
Credit in a Path
30 15 25 10
[x]: Secret sharing of x
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38
Credit in a Path
30 15 25 10
[30]
[30]
[30]
[x]: Secret sharing of x
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38
Credit in a Path
30 15 25 10
[30]
[30]
[30]
✦ Given [x] it is not possible to know x
[x]: Secret sharing of x
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38
Credit in a Path
30 15 25 10
[30]
[30]
[30]
✦ Given [x] it is not possible to know x✦ How to ensure that [x] comes from a user in a path?
[x]: Secret sharing of x
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38
Credit in a Path
30 15 25 10
[30]
[30]
[30]
✦ Given [x] it is not possible to know x✦ How to ensure that [x] comes from a user in a path?
[x]: Secret sharing of x
![Page 141: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/141.jpg)
Proof of Credit Links in a Path
39
30
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Proof of Credit Links in a Path
39
30
sk1, vk1 sk2, vk2
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Proof of Credit Links in a Path
39
30
sk1, vk1 sk2, vk2
σ1 := Sig(sk1, ([30], vk1, vk2))σ2 := Sig(sk2, ([30], vk1, vk2))
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Proof of Credit Links in a Path
39
30
sk1, vk1 sk2, vk2
σ1 := Sig(sk1, ([30], vk1, vk2))σ2 := Sig(sk2, ([30], vk1, vk2))
[30], vk{1,2}, σ{1,2}
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Proof of Credit Links in a Path
39
30
sk1, vk1 sk2, vk2
σ1 := Sig(sk1, ([30], vk1, vk2))σ2 := Sig(sk2, ([30], vk1, vk2))
[30], vk{1,2}, σ{1,2}
Correct proof for a path
(vk1, vk2), (vk2, vk3), (vk3, vk4), …
![Page 146: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/146.jpg)
Proof of Credit Links in a Path
39
30
sk1, vk1 sk2, vk2
σ1 := Sig(sk1, ([30], vk1, vk2))σ2 := Sig(sk2, ([30], vk1, vk2))
[30], vk{1,2}, σ{1,2}
Correct proof for a path
(vk1, vk2), (vk2, vk3), (vk3, vk4), …
![Page 147: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/147.jpg)
Proof of Credit Links in a Path
39
30
sk1, vk1 sk2, vk2
σ1 := Sig(sk1, ([30], vk1, vk2))σ2 := Sig(sk2, ([30], vk1, vk2))
[30], vk{1,2}, σ{1,2}
Correct proof for a path
(vk1, vk2), (vk2, vk3), (vk3, vk4), …
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Proof of Credit Links in a Path
39
30
sk1, vk1 sk2, vk2
σ1 := Sig(sk1, ([30], vk1, vk2))σ2 := Sig(sk2, ([30], vk1, vk2))
[30], vk{1,2}, σ{1,2}
Fresh keys per transaction
Correct proof for a path
(vk1, vk2), (vk2, vk3), (vk3, vk4), …
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40
Privacy-preserving Credit in a Path
30 15 25 10
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40
Privacy-preserving Credit in a Path
30 15 25 10
[30], v
k{1,2},
σ {1,2}
[30], vk{1,2} , σ{1,2}
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40
Privacy-preserving Credit in a Path
30 15 25 10
[30], v
k{1,2},
σ {1,2}
[30], vk{1,2} , σ{1,2}
[15]
, vk {
2,3}
, σ{2
,3}
[15]
, vk {
2,3}
, σ{2
,3}
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40
Privacy-preserving Credit in a Path
30 15 25 10
[30], v
k{1,2},
σ {1,2}
[30], vk{1,2} , σ{1,2}
[15]
, vk {
2,3}
, σ{2
,3}
[15]
, vk {
2,3}
, σ{2
,3}
[25], vk{3,4}, σ{3,4}
[25], v
k {3,4},
σ {3,4}
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40
Privacy-preserving Credit in a Path
30 15 25 10
[30], v
k{1,2},
σ {1,2}
[30], vk{1,2} , σ{1,2}
[15]
, vk {
2,3}
, σ{2
,3}
[15]
, vk {
2,3}
, σ{2
,3}
[25], vk{3,4}, σ{3,4}
[25], v
k {3,4},
σ {3,4}
[10], vk{4,5}, σ{4,5}
[10], vk{4,5}, σ{4,5}
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40
Privacy-preserving Credit in a Path
30 15 25 10
[30], v
k{1,2},
σ {1,2}
[30], vk{1,2} , σ{1,2}
[15]
, vk {
2,3}
, σ{2
,3}
[15]
, vk {
2,3}
, σ{2
,3}
[25], vk{3,4}, σ{3,4}
[25], v
k {3,4},
σ {3,4}
[10], vk{4,5}, σ{4,5}
[10], vk{4,5}, σ{4,5}
✦ Landmarks perform SMPC min computation over the shared link values
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40
Privacy-preserving Credit in a Path
30 15 25 10
[30], v
k{1,2},
σ {1,2}
[30], vk{1,2} , σ{1,2}
[15]
, vk {
2,3}
, σ{2
,3}
[15]
, vk {
2,3}
, σ{2
,3}
[25], vk{3,4}, σ{3,4}
[25], v
k {3,4},
σ {3,4}
[10], vk{4,5}, σ{4,5}
[10], vk{4,5}, σ{4,5}
[min(30, 15, …)]
[min(30, 15, …)]
✦ Landmarks perform SMPC min computation over the shared link values
![Page 156: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/156.jpg)
40
Privacy-preserving Credit in a Path
30 15 25 10
[30], v
k{1,2},
σ {1,2}
[30], vk{1,2} , σ{1,2}
[15]
, vk {
2,3}
, σ{2
,3}
[15]
, vk {
2,3}
, σ{2
,3}
[25], vk{3,4}, σ{3,4}
[25], v
k {3,4},
σ {3,4}
[10], vk{4,5}, σ{4,5}
[10], vk{4,5}, σ{4,5}
[min(30, 15, …)]
[min(30, 15, …)]
✦ Landmarks perform SMPC min computation over the shared link values ✦ Given enough “copies” of [x] it is possible to recover x for Alice
![Page 157: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/157.jpg)
Transaction Execution
41
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Transaction Execution
✦ Sequential friend-to-friend communication
41
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Transaction Execution
✦ Sequential friend-to-friend communication✦ Two-step transaction: on hold (or block) and settle
41
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Transaction Execution
✦ Sequential friend-to-friend communication✦ Two-step transaction: on hold (or block) and settle✦ Example:
41
15 20
5
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Transaction Execution
✦ Sequential friend-to-friend communication✦ Two-step transaction: on hold (or block) and settle✦ Example:
41
10 20
(5)
5
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Transaction Execution
✦ Sequential friend-to-friend communication✦ Two-step transaction: on hold (or block) and settle✦ Example:
41
10 20
Incentive
(5)
5
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Transaction Execution
✦ Sequential friend-to-friend communication✦ Two-step transaction: on hold (or block) and settle✦ Example:
41
2510
Incentive
(5) (5)
5
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Transaction Execution
✦ Sequential friend-to-friend communication✦ Two-step transaction: on hold (or block) and settle✦ Example:
41
2510
Incentive
Ok, received!
(5) (5)
5
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Transaction Execution
✦ Sequential friend-to-friend communication✦ Two-step transaction: on hold (or block) and settle✦ Example:
41
2510
Incentive
Ok, received!
(5)
5
![Page 166: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/166.jpg)
Transaction Execution
✦ Sequential friend-to-friend communication✦ Two-step transaction: on hold (or block) and settle✦ Example:
41
2510
Incentive
Ok, received!
5
![Page 167: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/167.jpg)
SilentWhispers: Characteristics/Limitations
✦ Distributed credit network transactions are possible without requiring ✦ a blockchain ledger ✦ a proof-of-work
✦ SilentWhispers can be modified by using landmarks as distributed stores [more details in the paper]
✦ In case of disputes, this leaves task of proving links to the users
✦ It is blocking solution, and deadlocks are possibles ✦ Problem: designing non-blocking solutions in the asynchronous
communication setting -distributed max-flow computation and atomic broadcast
42
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In the Future
✦ Payment Channels and lighting network https://lightning.network
✦ Designing distributed solutions for lighting network
✦ The Interledger Protocol https://www.w3.org/community/interledger
✦ Several distributed/decentralized/centralized ledger solutions are coming up
✦ Performing transactions across different ledgers
43
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Thanks to My Collaborators
44
Muhammad Bilal Zafar
Srivatsan RaviKim Pecina
Matteo MaffeiPedro Moreno-Sanchez
Giulio Malavolta
Sonia Fahmy
Tim Ruffing
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Thanks to My Collaborators
44
Muhammad Bilal Zafar
Srivatsan RaviKim Pecina
Matteo MaffeiPedro Moreno-Sanchez
Giulio Malavolta
Sonia Fahmy
Tim Ruffing
![Page 171: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/171.jpg)
Thanks to My Collaborators
44
Muhammad Bilal Zafar
Srivatsan RaviKim Pecina
Matteo MaffeiPedro Moreno-Sanchez
Giulio Malavolta
Sonia Fahmy
Tim Ruffing
To make credit networks great
again!
![Page 172: IOweYou Credit Networks - Purdue University...Cryptocurrencies vs Credit Networks IOU (or Credit) Networks Combining credit and social trust (still not permissioned) Restricted Fungibility](https://reader034.vdocuments.mx/reader034/viewer/2022051907/5ffa2a417b06b30ba0670f33/html5/thumbnails/172.jpg)
Take home message
✦ Credit networks have interesting properties and can be used in multiple scenarios
45
Why Credit Networks?
9
✦ Several applications: ✦ Ostra: preventing e-mail spam [NSDI’08] ✦ Bazaar: strengthening e-commerce [NSDI’11] ✦ SumUp: Sybil-resilient content voting [NSDI’09] ✦ Ripple: A real-life online settlement network
✦ Sybil-resistant applications
30
20
15
Introducing nodes is much easier than drawing trust from well-behaved nodesWell-behaved nodes Sybil nodes
edge cut
✦ Ledgers although provide accountability, it makes privacy a real problem in credit networks
✦ SlientWhispers: a decentralized architecture for providing accountability and privacy for credit networks
✦ Several questions remain unanswered leaving lots of open problems
The tale of two Public Logs
13
Input Output Alice-Bitcoin:
6 BTCDR-Bitcoin:
6 BTCAlice
✦ How to link these two events?
Bitcoin RippleSender DR-Ripple
Receiver Alice-RippleValue 6 BTC IOUPath Bob —> Alice
Bob
6
Alice-Bitcoin Alice-Ripple
DR-Bitcoin DR-Ripple
In the Future
✦ Payment Channels and lighting network https://lightning.network
✦ Designing distributed solutions for lighting network
✦ The Interledger Protocol https://www.w3.org/community/interledger
✦ Several distributed/decentralized/centralized ledger solutions are coming up
✦ Performing transactions across different ledgers
27
24
Privacy-preserving Credit in a Path
30 15 25 10
[30]
, vk{1
,2}, σ
{1,2
}
[30], vk{1,2} , σ{1,2}
[15]
, vk {
2,3}
, σ{2
,3}
[15]
, vk {
2,3}
, σ{2
,3}
[25], vk{3,4}, σ
{3,4}
[25], v
k {3,4}
, σ{3
,4}
[10], vk{4,5}, σ{4,5}
[10], vk{4,5}, σ{4,5}
[min(30, 15, …
)]
[min(30, 15, …)]
✦ Landmarks perform SMPC min computation over the shared link values ✦ Given enough “copies” of [x] it is possible to recover x for Alice
Thanks!