Socialist millionaire problem: Difference between revisions
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{{short description|Cryptographic problem}} |
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In [[cryptography]], the '''socialist millionaire problem'''<ref name=socialist-millionaire-problem>{{cite conference |
In [[cryptography]], the '''socialist millionaire problem'''<ref name=socialist-millionaire-problem>{{cite conference |
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| author = [[Markus Jakobsson]], [[Moti Yung]] |
| author = [[Markus Jakobsson]], [[Moti Yung]] |
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| title = Proving without knowing: On oblivious, agnostic and blindfolded provers. |
| title = Proving without knowing: On oblivious, agnostic and blindfolded provers. |
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| |
| book-title = Advances in Cryptology – CRYPTO '96, volume 1109 of Lecture Notes in Computer Science |
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| pages = 186–200 |
| pages = 186–200 |
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| year = 1996 |
| year = 1996 |
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| doi = 10.1007/3-540-68697-5_15 |
| doi = 10.1007/3-540-68697-5_15 |
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| url = http://cseweb.ucsd.edu/users/markus/proving.ps |
| url = http://cseweb.ucsd.edu/users/markus/proving.ps |
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| doi-access = free |
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}}</ref> is one in which two millionaires want to determine if their wealth is equal without disclosing any information about their riches to each other. It is a variant of the [[Millionaire's Problem]]<ref name=millionaires-problem>{{cite conference |
}}</ref> is one in which two millionaires want to determine if their wealth is equal without disclosing any information about their riches to each other. It is a variant of the [[Millionaire's Problem]]<ref name=millionaires-problem>{{cite conference |
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| author = [[Andrew Yao]] |
| author = [[Andrew Yao]] |
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| title = Protocols for secure communications |
| title = Protocols for secure communications |
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| |
| book-title = Proc. 23rd IEEE Symposium on Foundations of Computer Science (FOCS '82) |
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| pages = 160–164 |
| pages = 160–164 |
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| year = 1982 |
| year = 1982 |
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| author = [[Andrew Yao]] |
| author = [[Andrew Yao]] |
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| title = How to generate and exchange secrets |
| title = How to generate and exchange secrets |
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| |
| book-title = Proc. 27th IEEE Symposium on Foundations of Computer Science (FOCS '86) |
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| pages = 162–167 |
| pages = 162–167 |
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| year = 1986 |
| year = 1986 |
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== Motivation == |
== Motivation == |
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Alice and Bob have secret values <math> |
Alice and Bob have secret values <math>x</math> and <math>y</math>, respectively. Alice and Bob wish to learn if <math>x = y</math> without allowing either party to learn anything else about the other's secret value. |
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A passive attacker simply spying on the messages Alice and Bob exchange learns nothing about <math> |
A passive attacker simply spying on the messages Alice and Bob exchange learns nothing about <math>x</math> and <math>y</math>, not even whether <math>x = y</math>. |
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Even if one of the parties is dishonest and deviates from the protocol, that person cannot learn anything more than if <math> |
Even if one of the parties is dishonest and deviates from the protocol, that person cannot learn anything more than if <math>x = y</math>. |
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An active attacker capable of arbitrarily interfering with Alice and Bob's communication ( |
An active attacker capable of arbitrarily interfering with Alice and Bob's communication ([[man-in-the-middle attack]]) cannot learn more than a passive attacker and cannot affect the outcome of the protocol other than to make it fail. |
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Therefore, the protocol can be used to authenticate whether two parties have the same secret information. Popular instant message cryptography package [[Off-the-Record Messaging]] uses the Socialist Millionaire protocol for authentication, in which the secrets <math> |
Therefore, the protocol can be used to authenticate whether two parties have the same secret information. Popular instant message cryptography package [[Off-the-Record Messaging]] uses the Socialist Millionaire protocol for authentication, in which the secrets <math>x</math> and <math>y</math> contain information about both parties' long-term authentication public keys as well as information entered by the users themselves. |
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== Off |
== Off-the-Record Messaging protocol == |
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⚫ | |||
[[File:SMP - Socialist Millionaire Protocol.png|thumb|State machine of a socialist millionaire protocol (SMP) implementation.]] |
[[File:SMP - Socialist Millionaire Protocol.png|thumb|State machine of a socialist millionaire protocol (SMP) implementation.]] |
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The protocol is based on [[group theory]]. |
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⚫ | A prime |
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⚫ | A group of prime [[Order (group theory)|order]] <math>p</math> and a generator <math>h</math> are agreed upon ''a priori'', and in practice are generally fixed in a given implementation. For example, in the Off-the-Record Messaging protocol, <math>p</math> is a specific fixed 1,536-bit prime. <math>h</math> is then a generator of a prime-order subgroup of <math>(\mathbb{Z}/p\mathbb{Z})^*</math>, and all operations are performed modulo <math>p</math>, or in other words, in a subgroup of the [[multiplicative group]], <math>(\mathbb{Z}/p\mathbb{Z})^*</math>. |
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⚫ | By <math> |
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⚫ | |||
⚫ | |||
⚫ | |||
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⚫ | |||
⚫ | |||
⚫ | |||
⚫ | |||
The socialist millionaire protocol<ref name=socialist-millionaire-protocol>{{cite journal |
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⚫ | Part of the security also relies on random secrets. However, as written below, the protocol is vulnerable to poisoning if Alice or Bob chooses any of <math> |
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| author = Fabrice Boudot, Berry Schoenmakers, Jacques Traoré |
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| title = A Fair and Efficient Solution to the Socialist Millionaires' Problem |
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| journal = Discrete Applied Mathematics |
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| volume = 111 |
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| number = 1 |
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| pages = 23–36 |
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| year = 2001 |
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| doi = 10.1016/S0166-218X(00)00342-5 |
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| url = https://www.win.tue.nl/~berry/papers/dam.pdf |
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⚫ | |||
⚫ | Part of the security also relies on random secrets. However, as written below, the protocol is vulnerable to poisoning if Alice or Bob chooses any of <math>a</math>, <math>b</math>, <math>\alpha</math>, or <math>\beta</math> to be zero. To solve this problem, each party must check during the [[Diffie-Hellman]] exchanges that none of the <math>h^a</math>, <math>h^b</math>, <math>h^\alpha</math>, or <math>h^\beta</math> that they receive is equal to 1. It is also necessary to check that <math>P_a \neq P_b</math> and <math>Q_a \neq Q_b</math>. |
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{| class="wikitable" style="text-align:center;" |
{| class="wikitable" style="text-align:center;" |
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Q_a &= h^r g^x |
Q_a &= h^r g^x |
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\end{align}</math> |
\end{align}</math> |
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| |
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| <math>\begin{align} |
| <math>\begin{align} |
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P_b &= \gamma^s \\ |
P_b &= \gamma^s \\ |
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|- |
|- |
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| 9 |
| 9 |
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| Test <math>c = |
| Test <math>c = P_a{P_b}^{-1}</math> |
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| |
| |
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| Test <math>c = |
| Test <math>c = P_a{P_b}^{-1}</math> |
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|} |
|} |
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Note that: |
Note that: |
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:<math>\begin{align} |
:<math>\begin{align} |
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P_a{P_b}^{-1} &= \gamma^r \gamma^{-s} = \gamma^{r - s} \\ |
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&= h^{\alpha\beta(r - s)} |
&= h^{\alpha\beta(r - s)} |
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\end{align}</math> |
\end{align}</math> |
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and therefore |
and therefore |
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:<math>\begin{align} |
:<math>\begin{align} |
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c &= \left(Q_aQ_b^{-1}\right)^{\alpha\beta} \\ |
c &= \left(Q_aQ_b^{-1}\right)^{\alpha\beta} \\ |
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&= \left(h^r g^x h^{-s} g^{-y}\right)^{\alpha\beta} |
&= \left(h^r g^x h^{-s} g^{-y}\right)^{\alpha\beta} |
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= \left(h^{r - s} g^{x - y}\right)^{\alpha\beta} \\ |
= \left(h^{r - s} g^{x - y}\right)^{\alpha\beta} \\ |
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&= \left(h^{r - s} h^{ab(x - y)}\right)^{\alpha\beta} |
&= \left(h^{r - s} h^{ab(x - y)}\right)^{\alpha\beta} |
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= h^{\alpha\beta(r - s)} h^{\alpha\beta ab(x - y)} \\ |
= h^{\alpha\beta(r - s)} h^{\alpha\beta ab(x - y)} \\ |
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&= \left( |
&= \left(P_a{P_b}^{-1}\right) h^{\alpha\beta ab(x - y)} |
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\end{align}</math>. |
\end{align}</math>. |
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Because of the random values stored in secret by the other party, neither party can force <math> |
Because of the random values stored in secret by the other party, neither party can force <math>c</math> and <math>P_a{P_b}^{-1}</math> to be equal unless <math>x</math> equals <math>y</math>, in which case <math>h^{\alpha\beta ab(x - y)} = h^0 = 1</math>. This proves correctness. |
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==See also== |
==See also== |
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⚫ | |||
* [[Zero-knowledge proof]] |
* [[Zero-knowledge proof]] |
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==References== |
==References== |
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{{reflist}} |
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<references/> |
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==External links== |
==External links== |
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* [http://www.cypherpunks.ca/otr/Protocol-v2-3.1.0.html Description of the OTR-Messaging Protocol version 2] |
* [http://www.cypherpunks.ca/otr/Protocol-v2-3.1.0.html Description of the OTR-Messaging Protocol version 2] |
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* |
* {{Webarchive|url=https://web.archive.org/web/20221225214935/http://twistedoakstudios.com/blog/Post3724_explain-it-like-im-five-the-socialist-millionaire-problem-and-secure-multi-party-computation|date=December 25, 2022|title=Explain it like I’m Five: The Socialist Millionaire Problem and Secure Multi-Party Computation}} |
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* [http://goldbug.sf.net Goldbug Messenger, which uses an implementation the Socialist Millionaire Protocol] |
* [http://goldbug.sf.net Goldbug Messenger, which uses an implementation the Socialist Millionaire Protocol] |
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Latest revision as of 20:35, 24 July 2024
In cryptography, the socialist millionaire problem[1] is one in which two millionaires want to determine if their wealth is equal without disclosing any information about their riches to each other. It is a variant of the Millionaire's Problem[2][3] whereby two millionaires wish to compare their riches to determine who has the most wealth without disclosing any information about their riches to each other.
It is often used as a cryptographic protocol that allows two parties to verify the identity of the remote party through the use of a shared secret, avoiding a man-in-the-middle attack without the inconvenience of manually comparing public key fingerprints through an outside channel. In effect, a relatively weak password/passphrase in natural language can be used.
Motivation
[edit]Alice and Bob have secret values and , respectively. Alice and Bob wish to learn if without allowing either party to learn anything else about the other's secret value.
A passive attacker simply spying on the messages Alice and Bob exchange learns nothing about and , not even whether .
Even if one of the parties is dishonest and deviates from the protocol, that person cannot learn anything more than if .
An active attacker capable of arbitrarily interfering with Alice and Bob's communication (man-in-the-middle attack) cannot learn more than a passive attacker and cannot affect the outcome of the protocol other than to make it fail.
Therefore, the protocol can be used to authenticate whether two parties have the same secret information. Popular instant message cryptography package Off-the-Record Messaging uses the Socialist Millionaire protocol for authentication, in which the secrets and contain information about both parties' long-term authentication public keys as well as information entered by the users themselves.
Off-the-Record Messaging protocol
[edit]The protocol is based on group theory.
A group of prime order and a generator are agreed upon a priori, and in practice are generally fixed in a given implementation. For example, in the Off-the-Record Messaging protocol, is a specific fixed 1,536-bit prime. is then a generator of a prime-order subgroup of , and all operations are performed modulo , or in other words, in a subgroup of the multiplicative group, .
By , denote the secure multiparty computation, Diffie–Hellman–Merkle key exchange, which, for the integers, , , returns to each party:
- Alice calculates and sends it to Bob, who then calculates .
- Bob calculates and sends it to Alice, who then calculates .
as multiplication in is associative. Note that this procedure is insecure against man-in-the-middle attacks.
The socialist millionaire protocol[4] only has a few steps that are not part of the above procedure, and the security of each relies on the difficulty of the discrete logarithm problem, just as the above does. All sent values also include zero-knowledge proofs that they were generated according to protocol.
Part of the security also relies on random secrets. However, as written below, the protocol is vulnerable to poisoning if Alice or Bob chooses any of , , , or to be zero. To solve this problem, each party must check during the Diffie-Hellman exchanges that none of the , , , or that they receive is equal to 1. It is also necessary to check that and .
Alice | Multiparty | Bob | |
---|---|---|---|
1 | Message Random |
Public | Message Random |
2 | Secure | ||
3 | Secure | ||
4 | Test , | Test , | |
5 | |||
6 | Insecure exchange | ||
7 | Secure | ||
8 | Test , | Test , | |
9 | Test | Test |
Note that:
and therefore
- .
Because of the random values stored in secret by the other party, neither party can force and to be equal unless equals , in which case . This proves correctness.
See also
[edit]References
[edit]- ^ Andrew Yao (1982). "Protocols for secure communications" (PDF). Proc. 23rd IEEE Symposium on Foundations of Computer Science (FOCS '82). pp. 160–164. doi:10.1109/SFCS.1982.88.
- ^ Andrew Yao (1986). "How to generate and exchange secrets" (PDF). Proc. 27th IEEE Symposium on Foundations of Computer Science (FOCS '86). pp. 162–167. doi:10.1109/SFCS.1986.25.