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

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

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

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…

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

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

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

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

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

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?

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

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

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

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

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

Interface, Demo, Results Go Source: nsa.gov

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

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

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

Questions?

Extra slides

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.

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

Old Interface

New Interface (1)

New Interface (2)

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

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