Lecture 33 Computer Security

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Lecture 33Computer Security

• Phillip G. Bradford
• Computer Science Department
• The University of Alabama

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Outline

• DSA Signatures
• SHA for DSA
• All For Java Libraries
• A Diversion
• The BiBa Wireless Authentication Protocol
• Originally by Perrig

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Objectives

• Understand DSA for Java Signature Verification
• Understand SHA-1 enough to use in DSA
• Understand the workings and construction of BiBa

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SHA Secure Hash Algorithm

• NIST
• Used in DSA, etc.
• Inputs a message of length lt 264 bits
• Outputs 160 bit message digest

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DSA The Very Basics

• Make the input a multiple of 512 bits
• Append a one followed by up to 511 -64 zeros
• Last 64 bits put in the length of the message
  before the appending started
• Sequentially processes 512 bit chunks
• The Rounds of SHA

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DSA

• NIST proposed for
• DSS (Digital Signature Standard)
• DSA Algorithm that implements DSS
• Key factors considered
• Level of security
• Ease of implementation
• Hardware and Software
• Ease of US export
• Patent restrictions

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DSA

• Royalty Free!
• Economic Benefit to Government and Public
• Even for Smart Cards
• Signing by weak smart cards
• Verification by stronger machine

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DSA

• Controversy over DSA
• What alternatives?
• Just how fast is DSA?
• Same signature generation speed as RSA
• Signature Verification Speed much slower than RSA
• 10 to 40 Times Slower

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DSA How Does It Work?

• Variant of Schnorr and ElGamal Algorithms
• Start with a prime p of L bits
• L 1024 gt L gt 512
• And L is a multiple of 64
• Let q be a 160-bit prime factor of p-1

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DSA How Does It Work?

• Let g h(p-1)/q mod p where h is s.t.
• h p-1 gt h and
• h(p-1)/q mod p gt 1
• Let x lt q and
• y gx mod p
• Public parameters p, q and g
• Public key y
• Private key x

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DSA How Does It Work?

• Alice randomly generates k lt q
• Alive computes
• R (gk mod p) mod q
• S (k-1 (H(m) xr)) mod q
• Send ltR,Sgt as a Signature to Bob

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DSA How Does It Work?

• Bobs verification
• W s-1 mod q
• U1 (H(m)w) mod q
• U2 (rw) mod q
• V ((gu1 yu2) mod p) mod q
• If V R, then Verified !

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DSA Setting the StageWhy it Works?

• Fermats Little Theorem
• If pgt1 is prime, take an integer a s.t. p does
  not divide a, then
• ap-1 1 mod p
• Proof Foundation
• Zp, is the same as
• a mod p, 2a mod p, , (p-1)a mod p
• For any a in Zp

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Fermats Little Theorem

• Why?
• Suppose ia mod p ja mod p for i ! j
• Since each element in Zp has a unique inverse
• This means ij mod p, and since i and j in Zp,
  this is impossible!
• This means
• (p-1)!ap-1 (p-1)! Mod p
• So, ap-1 1 Mod p

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Introduction to BiBa

• By A. Perrig
• Short for Bins and Balls
• Offers fast signature verification
• Different from DSA!
• Though, somewhat expensive signature generation
• Why is inexpensive signature verification
  important?

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Introduction to BiBa

• Basic Idea
• Using the Balls-in-Bins Problem or Birthday
  Problem to our advantage!
• Where did we see the Birthday Attack?
• Start with a well known one-way function F
• Choose a set of Balls B1, , Bt private key
• Say each is represented by an integer
• The value t may be somewhat small
• c1 F(B1), , ct F(Bt) public key

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BiBa Signature Generating

• Given a message m
• Let h H(m)
• The function h chooses another function gh Balls
  ? Bins
• Then select a secret random subset
• Bi1,,Bik, where k lt t
• And i1, , ik subset of 1, ,t
• Post the pattern gh(Bi1), , gh(Bik)

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BiBa Signature Verification

• Verify distinctness of Bi1,,Bik
• Use public key c1, , ct
• Verify F(Bi1), , F(Bik) subset of c1, , ct
• Let h H(m) same H as in sig. generation
• Verify the pattern gh(Bi1), , gh(Bik)

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BiBa Brief Overview

• So, how cheap is signature verification?
• How expensive is signature generation?