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CSC 3130: Automata theory and formal languages
Interaction, randomness,
and zero-knowledge
MELJUN P. CORTES MBA MPA BSCS ACS
MELJUN CORTES
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Authentication
What happens when you type in your password?
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Nave authentication
The server knows your password
So they can impersonate you at other web sites
where you use the same password
login: me
password: opensesame
OK
you server
acme.com
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Zero-knowledge authentication
acme.com
Can you convince the server that you knowyour password,
without revealing it (or any other information)?
I know the password
Can you prove it?
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What is knowledge?
Example 1: Tomorrows lottery numbers
We are ignorant of them because they are
random
What is ignorance?
(lack of knowledge)
2 31 12 7 28 11
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What is ignorance?
Example 2: A difficult math problem
We are ignorant because it takes a lot of work tofigure out the answer
Questions of this type include Finding satisfying assignments to Boolean formulas
Finding cliques in graphs
All NP-hard problems
Prove that P NP
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Using ignorance to our advantage
acme.com
I know the password
Can you prove it?
We want to convincethe server that we knowthe password,
while keeping it ignorantof the password itself
The server is convinced, but gains zero-knowledge!
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A protocol for non-color-blindness
You want to convince me you are not color-blind
I pull at random either a red ball
or a blue ball and show it to you
You say red or blue
We repeat this 10 times
If you got all the answers right,I am convinced you know red from blue
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Interaction and knowledge
What knowledgedid I gain from this interaction?
I learned that you can tell blue from red
But I also learned the colors of the ballsin each glass
Suppose I was color-blind
Then I used you to gain some knowledge!
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A different protocol
box 1
box 2
I pull at random either two balls from same box
or one ball from box 1 and one from box 2
You say same color or different color
We repeat 10 times
If you got all the answers right,
I am convinced you know red from blue
But I did not gain any other knowledge!
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Zero-knowledge
Suppose I am color-blind but you are not
In the first protocol, I cannot predictyour answer
ahead of time
In the second protocol, I knowwhat you will say,
so I do not gain knowledgewhen you say it
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Zero-knowledge password
authentication
acme.com
Oded Goldreich Silvio Micali Avi Wigderson
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Graph coloring
Theorem
Task: Assign one of 3colors tothe vertices so that no edge has
both endpoints of same color
3COL = {G: Ghas a valid 3-coloring}
3COLis NP-complete
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NP-hardness of 3COL
Proof sketch of NP-hardness: Reduce from 3SAT
We describe G:
3CNF formulaf graphG
Ghas a valid 3-coloring
R
fis satisfiable
Part I: 3 special verticesT (true), F (false), and X
T F
X
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NP-hardness of 3COL
Part 2: For each variable xi
xi
X
xi
Either xihas color of Tand xihas color of F
Or xihas color of F
and xihas color of T
Part 3: For each clause, e.g. x1x2x3
Can be 3-colored
iff clause is satisfied
x1
x2
x3
T
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Password authentication via 3-coloring
Step 0: When you register for the web service,
choose your password to be a valid 3-coloring of
some (suitable) graph
acme.comregistration
password:1
2
3 4
5
6 G
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Password authentication via 3-coloring
When the server asks for your password
do not send the password, but send the graph G
instead (without the colors)
acme.com
password:1
2
3 4
5
6 G
password?
G
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Intuition about registration phase
Because 3-coloring is hard, the server will not beable to figure out your password (coloring) from G
Later, when you try to log in, you will convincethe server that you knowhow to color G, without
revealing the coloring itself
The server will be convinced you know yourpassword but remain ignorant about what it is
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The login phase
password: You randomly permute the colors
You lock each of the colors in a
box
and send the boxes to the server
1
2
3 4
5
6
The server chooses an edge at
random and asks for the keys tothe boxes at the endpoints
You send the requested keys
The server unlocks the two
boxesand checks the colors are
differentRepeat this 1000 times. Login
succeeds if colors always different
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Analysis in the login phase
If you are an impostor, you wont know how tocolor the graph, so at least one of the edges will
have endpoints of the same color, and the server
is likely to catch this
If you are honest, the server remains ignorant
about your password, because all he sees are
two random different colors
But how do we send locked boxesand keysover
the internet?
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