Modern Cryptography: Digital Envelopes, Zero Knowledge, and Computational Complexity, Lecture notes of Cryptography and System Security

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Digital Envelopes,

Zero Knowledge,

and other wonders of

modern cryptography

(How computational complexity

enables digital security & privacy)

Attribution

• These slides were prepared for the New Jersey Governor’s School course “The Math Behind the Machine” taught in the summer of 2011 by Grant Schoenebeck
• Large parts of these slides were copied or modified from a presentation by Sanjeen Arora who adapted them from a presentation by the original author Avi Wigderson.

Cryptography pre-

(before computational complexity)

Secret communication

Assuming shared information

which no one else has

Tasks Traditional method

Encryption

Identification Driver License

Money transfer Notes, checks

Public bids Sealed envelopes

Code books

Elections Secret ballots

Need to be done online!

Qs. Why do you think this a problem???

What are we assuming

here??

Axiom 1: Agents are computationally

limited.

Consequence 1: Only tasks having efficient

algorithms can be performed

Impossible

p,q pq

Easy

Axiom 1: Agents are computationally limited

Axiom 2: Factoring is hard

Easy

Impossible

“One way

function”

Fact: Axioms -- > “Digital Envelope”

x E(x)

• Easy to insert x (any value, even 1 bit)
• Hard to compute content (even partial info)
• Impossible to change content (E(x) defines x)
• Easy to verify that x is the content

Theorem:  Cryptography

OPEN CLOSED

El Gamal

• Assumption: Given x computing log 4 x mod Q is hard.
  • Given 4y^ computing y is hard.
• Alice
  • Has message x
  • Send y = 4x^ to Bob
• Alice
  • Send x
• Bob
  • Check that 4x^ = y
• Why is this secure?

The power of the

digital envelope

Examples of increasing difficulty

Mind games of the 1980’s – before

Internet & E-commerce were imagined

Public Lottery (on the phone)

Alice Bob

Bob: flipping... You lost!

Theorem: ^ Symmetry breaking

Alice: if I get the car (else you do)

Bob: flipping... What did you pick?

Blum

Identification / Passwords

Public password file

Name E (pswd)

alice Palice =E (…)

grant Pgrant=E (haha)

bob Pbob =E (…)

Computer: 1 checks if E (pswd)= Pgrant

2 erases password from screen

login: grant password: haha

Copyrights

Dr. Alice: I can prove Riemann’s Hypothesis

Dr. Alice: Lemma…Proof…Lemma…Proof...

Prof. Bob: Impossible! What is the proof?

Prof. Bob: Amazing!! I’ll recommend tenure

Amazing!! I’ll publish first

Zero-Knowledge Proof

“Claim”

Bob Alice (“proof”)

Accept/Reject

“Claim” false  Bob rejects

“Claim” true  Bob accepts

Bob learns nothing

with high probability

Goldwasser-Micali -Rackoff 1984