Contact Information:
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John Franco |
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Area Coordinator |
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franco@gauss.ececs.uc.edu |
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556-1817 / 787-9960
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Carlo Perottino |
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Graduate Assistant |
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perottca@mail.uc.edu |
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Nick Maltbie |
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Resource & Assistant |
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nick.dmalt@gmail.com |
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Handy Information:
Mathematics of Cryptography
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Math
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Fermat, Chinese Remainder, Mod arithmetic, Z*n, Mod Inverse, Euler's algorithm, Testing for primes, Generating primes |
What is Creativity?
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Creativity
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Divergent/Convergent thinking, elevator puzzle, bellhop puzzle, Noah's ark puzzle, missing card puzzle, illusions, monk puzzle, Racetrack puzzle, lightbulb puzzle, language puzzle, pennies puzzle, pentagon puzzle, walking puzzle (mod), pipe puzzle, horse puzzle |
Discovery Applets, 2016
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Arithmetic and logic operations |
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Secret Key ciphers: encryption, message integrity, authentication |
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Mathematics of cryptography |
Articles, Documentation, Surveys, etc.
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Curiosity (DAoM)
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Independent learning relies on student curiosity |
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Survey
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Amanda Sopko has contributed a survey that was used in a
recent 6th and 7th grade STEM class |
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Encryption & Math
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From about.com/education |
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Cyber Kill Chain
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Classic Lockheed-Martin anatomy of attack, intel-driven defense |
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Monitoring
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Slide presentation on network security monitoring |
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Logging
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SANS Institute paper on intrusion detection via logging and monitoring |
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Attack Prevention
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SANS Institute paper on attack prevention |
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Common Criteria
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The Common Criteria security evaluation |
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Malware Analysis
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Malware analysis tutorials |
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Security Onion
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Operating System with tools for defense |
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Kali Linux
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Operating System with tools for attack |
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Virtualbox
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Virtualizer for desktop computers - supports above OSes |
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Vulnerability Assessment
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Documentation on many kinds of attacks |
Notes
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Can curiosity be increased by having students build
their own experimental platforms (in this case with java applets) to
test their own theories? This may be a hypothesis we could examine
for this project. |
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Should we give students a survey measuring their
attitudes/dispositions toward mathematics (including their level of
interest in STEM and STEM careers perhaps?) before the challenge-based
unit, and then re-evaluate those same attitudes after the unit to try
to measure how challenge-based and exploratory/inquiry learning impact
students attitudes toward the subject and their interest in possible
related careers? |
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Maxims of cryptography
- never use the same key to encrypt two different messages
- never encrypt the same message twice with two different keys
- always assume the adversary knows the encryption algorithm
- never underestimate the adversary
- only a professional cryptanalyst can judge the security of a cryptosystem
- a cryptographer's error is the cryptanalyst's only hope (David Kahn) |
What are we trying to accomplish?
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teach math (group theory & stats) with crypto as the motivator?
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encourage scientific exploration with math, crypto, logic as motivator?
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teach crypto - math is used as needed?
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teach a blend of crypto, math, and lab design?
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teach lab design with crypto experiments as the motivator?
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How are we going to do this?
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mainly set up interactive virtual experiments for self discovery?
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mainly explain known results and have students verify them with virtual experiments?
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create a networked game where students acquire/steal wealth and use crypto algorithms for protection?
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What Mathematics would we like the students
to experiment with? (subject to change)
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Modular arithmetic
- Fermat's little theorem
- Square roots of N mod M
- Exponentiation of a to the power N mod M
- Inverse of N mod M
- Z*N
- Generators for cyclic groups
- R*S mod M = (R mod M)*(S mod M) mod M
- (gR mod M)S mod M = (gS mod M)R mod M
- Chinese Remainder Theorem
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Permutations
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transpositions |
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products of permutations |
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conjugated permutations |
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cyclic structure |
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degree of permutations |
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two degree N permutations of disjoint
transpositions have a product containing an even number
of disjoint cycles of the same length. |
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If in any permutation of even degree
there appears an even number of disjoint cycles of the
same length, then the permutation can be regarded as a
product of two permutations each of which consists only of
disjoint transpositions. |
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Two permutations K,L on the same set X
are conjugated if and only if they have the same cyclic
structure. |
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Statistics
- mean, standard deviation, moments
- correlation coefficients |
What crypto algorithms would we like the students to experiment with? (subject to change)
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Euclid's algorithm
- Find the inverse of M mod N
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Miller-Rabin algorithm
- Generate a (probably) prime number
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Chinese Remainder Theorem
- Use in development of public key cryptosystems
- Use in attacking RSA and other cryptosystems
- Use in encrypting a secret
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What crypto systems would we like the students to experiment with? (subject to change)
How might a course on math and crypto progress?
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Experiment with the Ceaser cipher. See how easily it is cracked. |
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Students propose variants like increasing the rotation with every keystroke |
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Learn about and experiment with permutation groups (to be developed). |
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Break Enigma using the math of permutation groups. |
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Students think about the security associated with ciphers based on permutations. |
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Message integrity with message digest |
Maybe a project to tie all the concepts together?