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Capstone Project Presentation A Tool for Cryptography Problem Generation CSc 499 Mark Weston Winter 2006

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Introduction Idea: Improve Math 121 Problem Generation Client: Professor Kathryn Lesh Current system: Excel based Goal: A better tool for problem generation

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Outline Purpose A Strategy for Problem Generation Design Interface, Demo, Results

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Purpose Given a problem type from the course, can we generate “good” instances of the type?

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Outline Purpose A Strategy for Problem Generation Design Interface, Demo, Results

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A Strategy for Problem Generation How to address goodness? Use student work Extract measurements: metrics Algorithms: close relationship Example metrics Steps of problem type algorithm(s) Maximum, minimum values Trial Count And many more…

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A Strategy for Problem Generation Idea of metrics gives us our strategy “Generate and Test” Choose Problem Type Input Desired Metrics Generate Problem Type Instance Solve Instance Test Solution Metrics Test Successful Test Unsuccessful Done

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Feasibility of Generate and Test Random generation No guarantee Initial design planned to improve this Sufficient? Yes (!)

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Problem List 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

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Problem List 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

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Problem List 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

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Problem List 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?

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Problem List (final) Modular Addition, Subtraction, Multiplication GCD Extended Euclidean Algorithm Modular Exponentiation by Repeated Squares and Square-and- Multiply 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

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Outline Purpose A Strategy for Problem Generation Design Interface, Demo, Results

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Design, Requirements Design Follows from generation strategy A component that generates problems A component that solves problems An interface to provide input Implementation Choice Java Java Applet

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Other Requirements Modular Configure for students Full Output Data structures To deal with number precision Limit maximum number of digits

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Outline Purpose A Strategy for Problem Generation Design Interface, Demo, Results

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Interface, Demo, Results Go Source: nsa.gov

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Conclusion One tool – many features Many problem types Calculation / Generation Variable precision, full algorithms Full output Refined interface Students / Professors Free No install, lightweight, multiplatform Support available

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Future work More problems Usability / Interface Other improvements New algorithms Other Crypto-systems

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Thanks! Client: Professor Kathryn Lesh Advisor: Professor Brian Postow Interface Consultants: Professors Chris Fernandes and Aaron Cass

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Questions?

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Extra slides

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Configuring an Applet 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 Plain text XML Etc.

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Generation of complicated problems Intelligence Complexity source Algorithm Metrics Composition Target sub problems

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Old Interface

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New Interface (1)

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New Interface (2)

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Dealing with precision, size of numbers Use a number class Arithmetic with objects!? Vary internal representation independently of the interface Limit number of digits Watch Number class for add/multiply - cause growth Exception? Restart the problem Lower inputs Try 10 times, give up

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An Example Greatest Common Divisor (GCD) A problem type has: Inputs -> Instance GCD(a, b), vary values a and b Algorithm -> Metric of “Goodness” The Euclidean Algorithm and the number of steps it takes

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