Dynamics of Some Maps Associated with Number Theoretic Functions. ITEC809 Project Report. Semester 2, 2011 Jeffrey Smith – Supervisor : Igor Shparlinski

Presentation Outline Project Objectives ◦ Pseudorandom numbers and applications ◦ Iterative maps for Number Theoretic Functions (NTFs) ◦ Building an analysis tool ◦ Data Analysis Background ◦ Iterative processing of NTFs ◦ Cycle and Tail length analysis for pseudorandom number generation ◦ Cycle detection ◦ The NTFs implemented with the tool Analysis tool development ◦ Development choices ◦ Calculations with NTL and lessons learned ◦ Implementing NTFs ◦ Using the Analysis tool Performance Enhancements ◦ Checking for previously calculated maps ◦ Enhancement results and future experimentation Analysis Results ◦ Sample Results data ◦ Sample Frequency distribution Conclusion 2

Project Objectives The need for Pseudorandom numbers in computing. ◦ Sources of random numbers with complex algorithms ◦ Used to create hashing functions for cryptographic methods to provide security ◦ Pseudorandom numbers need complex methods of generation to prevent prediction ◦ Would be useful to find a simple method of producing a highly complex sequences of numbers 3

Project Objectives Iterative maps of Number Theoretic Functions. ◦ Generated by taking relatively simple number theoretic functions and produce iterative sequences of numbers. ◦ Generates a sequence of seemingly random numbers for some functions. ◦ Simple method of generating complex output. ◦ Possible source of pseudorandom numbers for computing applications. 4

Project Objectives Building an Analysis Tool ◦ Project goal is to build a tool that can produce iterative maps of Number Theoretic Functions ◦ Tool implements several Number Theoretic Functions and allows analysis over ranges of values to study the behaviour of these maps. ◦ Tool will produce data for range analysis on variance and dispersion characteristics for the length of maps generated to assess how effective the technique is. 5

Project Objectives Data Analysis ◦ The analysis tool should allow us to investigate how Maps of Number Theoretic Function behave. ◦ Investigation of implemented NTFs and assessment of pseudorandom behaviour with respect to the dispersion of the tail lengths. ◦ Tool allows us to investigate characteristic behaviour including average tail and cycle lengths, distinct cycles and variance. 6

Background Iterative maps of Number Theoretic Functions ◦ Iterative maps are created by resubmitting the result of the function on successive iterations. 7

Background Cycle and tail length analysis ◦ Looking at the cycle length and the length of the ‘tail’ before the sequence add complexity. 8

Background Cycle detection ◦ Floyd’s Algorithm vs Brent’s Algorithm 9 Source:

Background Number Theoretic Functions ◦ Collatz function: 10

Background Number Theoretic Functions ◦ Collatz function modulus p: 11

Background Number Theoretic Functions ◦ Fermat Quotient (Type 1): 12

Background Number Theoretic Functions ◦ Fermat Quotient (Type 2): 13

Analysis Tool Development Development Choices ◦ Visual Studio C++  Useful as already knew VS development environment  Compatible with the C++ libraries required for NTL. Calculations with NTL and lessons learned ◦ NTL – Library for number theoretic calculations – Victor Shoup. ◦ Required for large integer calculations. ◦ NTL library allows accurate and efficient calculations. 14

Analysis Tool Development Implementing Number Theoretic Functions ◦ Functions must be broken down to algorithms, using the NTL operators and big integer variables. 15

Analysis Tool Development Using the Analysis tool Console application Takes variables command line variables 16

Analysis Tool Development Using the Analysis tool ◦ Console output for progress and debug: 17

Analysis Tool Development Using the Analysis tool ◦ Range Analysis Output as CSV or as summary of analysis information. 18

Performance Enhancements Tool takes days to produce some of the data sets. Improvements were investigated by keeping track of previously generated paths and using this to greatly reduce the processing overhead by back filling new data rather than computing complete maps. Leverages the finding of rather small numbers of unique cycles. Processing time comparison: 19 PFermat Quotient (Type 2)Fermat Quotient (Type 2) Enhanced :37:490:23: :20:39Error processing :37:160:31: :34:55Error processing :07:35 0:47: More than 72:00:000:51:20

Performance Enhancements ◦ Enhanced range analysis processing time vs number of maps. Future work using this enhanced method would allow a much greater range of data to be analysed at once. On discussion with faculty staff, this would be well suited to distributed computing approaches for future implementations. 20

Analysis Results Sample Analysis data: Collatz Mod P data 21 P Average CycleAverage TailLargest TailTail VarianceDispersion IndexDistinct Cycles No. Stopping Points

Analysis Results Sample Analysis data: Fermat Quotient (Type 2). 22 P Average Cycle Average Trajectory Largest Trajectory Trajectory VarianceDispersion IndexDistinct Cycles No. Stopping Points

Conclusion Analysis tool built to produce maps of number theoretic functions and compile useful results for analysis. Performance enhancements investigated to alleviate enormous processing overhead and reduce processing time from days to minutes. Brief analysis of large primes found that we can produce pseudorandom numbers with number theoretic function maps. More analysis would give us better insight into selecting the right functions and values to do this. 23