Transcript Document 7379112

Capstone Project Presentation
A Tool for Cryptography
Problem Generation
CSc 499
Mark Weston
Winter 2006
Introduction
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Idea: Improve Math 121 Problem Generation
Client: Professor Kathryn Lesh
Current system: Excel based
Goal: A better tool for problem generation
Outline
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Purpose 
A Strategy for Problem Generation
Design
Interface, Demo, Results
Purpose
Given a problem type from the course, can we
generate “good” instances of the type?
Outline
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Purpose
A Strategy for Problem Generation 
Design
Interface, Demo, Results
A Strategy for Problem Generation
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How to address goodness?
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Use student work
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Extract measurements: metrics
Algorithms: close relationship
Example metrics
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Steps of problem type algorithm(s)
Maximum, minimum values
Trial Count
And many more…
A Strategy for Problem Generation
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Idea of metrics gives us our strategy
“Generate and Test”
Input
Desired
Metrics
Choose
Problem
Type
Generate
Problem
Type
Instance
Test Unsuccessful
Test Successful
Done
Test
Solution
Metrics
Solve
Instance
Feasibility of Generate and Test
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Random generation
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No guarantee
Initial design planned to improve this
Sufficient?
Yes (!)
Problem List
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Modular Addition, Subtraction, Multiplication
Properties of Divisibility
GCD
Extended Euclidean Algorithm
Linear Combination Theorem
Modular Exponentiation by Repeated Squares and Square-and-Multiply
Chinese Remainder Theorem Applications
Evaluating Jacobi Symbols
Solovay-Strassen Primality Testing
RSA Key Generation
RSA Signatures
Primitive Root Testing
Factoring by Pollard’s p-1
Prime Factorization of a Composite
Cryptographic Coin Toss
Factoring by Dixon’s Random Squares
Problem List
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Modular Addition, Subtraction, Multiplication
Properties of Divisibility
 Not needed
GCD
Extended Euclidean Algorithm
Linear Combination Theorem
Modular Exponentiation by Repeated Squares and Square-and-Multiply
Chinese Remainder Theorem Applications
Evaluating Jacobi Symbols
Solovay-Strassen Primality Testing
RSA Key Generation
RSA Signatures
Primitive Root Testing
Factoring by Pollard’s p-1
Prime Factorization of a Composite
Cryptographic Coin Toss
Factoring by Dixon’s Random Squares
Problem List
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Modular Addition, Subtraction, Multiplication
Properties of Divisibility
 Not needed
GCD
Extended Euclidean Algorithm
 Collapse w/ LCT
Linear Combination Theorem
 Collapse w/ EE
Modular Exponentiation by Repeated Squares and Square-and-Multiply
Chinese Remainder Theorem Applications
Evaluating Jacobi Symbols
Solovay-Strassen Primality Testing
RSA Key Generation
RSA Signatures
 Collapse w/ Mod. Exp.
Primitive Root Testing
Factoring by Pollard’s p-1
Prime Factorization of a Composite
 Collapse w/ Pollard
Cryptographic Coin Toss
Factoring by Dixon’s Random Squares
Problem List
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Modular Addition, Subtraction, Multiplication
Properties of Divisibility
 Not needed
GCD
Extended Euclidean Algorithm
 Collapse w/ LCT
Linear Combination Theorem
 Collapse w/ EE
Modular Exponentiation by Repeated Squares and Square-and-Multiply
Chinese Remainder Theorem Applications
Evaluating Jacobi Symbols
Solovay-Strassen Primality Testing
RSA Key Generation
RSA Signatures
 Collapse w/ Mod. Exp.
Primitive Root Testing
Factoring by Pollard’s p-1
Prime Factorization of a Composite
 Collapse w/ Pollard
Cryptographic Coin Toss
 Feasible?
Factoring by Dixon’s Random Squares
 Feasible?
Problem List (final)
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Modular Addition, Subtraction, Multiplication
GCD
Extended Euclidean Algorithm
Modular Exponentiation by Repeated Squares and Square-andMultiply
Chinese Remainder Theorem Applications
Evaluating Jacobi Symbols
Solovay-Strassen Primality Testing
RSA Key Generation
Primitive Root Testing
Factoring by Pollard’s p-1
Factoring by Dixon’s Random Squares
 Feasible
Cryptographic Coin Toss
 Feasible
Outline
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Purpose
A Strategy for Problem Generation
Design 
Interface, Demo, Results
Design, Requirements
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Design
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Follows from generation strategy
A component that generates problems
A component that solves problems
An interface to provide input
Implementation Choice
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Java
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Java Applet
Other Requirements
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Modular
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Configure for students
Full Output
Data structures
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To deal with number precision
Limit maximum number of digits
Outline
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Purpose
A Strategy for Problem Generation
Design
Interface, Demo, Results 
Interface, Demo, Results
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Go
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Source: nsa.gov
Conclusion
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One tool – many features
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Many problem types
Calculation / Generation
Variable precision, full algorithms
Full output
Refined interface
Students / Professors
Free
No install, lightweight, multiplatform
Support available
Future work
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More problems
Usability / Interface
Other improvements
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New algorithms
Other Crypto-systems
Thanks!
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Client: Professor Kathryn Lesh
Advisor: Professor Brian Postow
Interface Consultants: Professors Chris
Fernandes and Aaron Cass
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Questions?
Extra slides
Configuring an Applet
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Sign it
 Gives permissions to the machine it’s running on
 Don’t want the configuration file there…
Want access to the machine the applet is running on
 File system access here is tricky, once the applet starts running
 Work around
Work around
 Have the applet make a URL Connection to the machine it came
from
 This is legal, even for an unsigned applet
 We can then read a file, and configure from that
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Plain text
XML
Etc.
Generation of complicated problems
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Intelligence
Complexity source
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Algorithm
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Metrics
Composition
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Target sub problems
Old Interface
New Interface (1)
New Interface (2)
Dealing with precision, size of numbers
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Use a number class
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Arithmetic with objects!?
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Vary internal representation independently of the
interface
Limit number of digits
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Watch Number class for add/multiply - cause
growth
Exception?
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Restart the problem
Lower inputs
Try 10 times, give up
An Example
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Greatest Common Divisor (GCD)
A problem type has:
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Inputs -> Instance
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GCD(a, b), vary values a and b
Algorithm -> Metric of “Goodness”
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The Euclidean Algorithm and the number of steps it
takes