Contact Information:

John Franco 

Area Coordinator 

franco@gauss.ececs.uc.edu 

5561817 / 7879960


Carlo Perottino 

Graduate Assistant 

perottca@mail.uc.edu 



Nick Maltbie 

Resource & Assistant 

nick.dmalt@gmail.com 


Handy Information:
Mathematics of Cryptography

Math


Fermat, Chinese Remainder, Mod arithmetic, Z*n, Mod Inverse, Euler's algorithm, Testing for primes, Generating primes 
What is Creativity?

Creativity


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



Arithmetic and logic operations 



Secret Key ciphers: encryption, message integrity, authentication 



Mathematics of cryptography 
Articles, Documentation, Surveys, etc.
Notes

⋄ 

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. 

⋄ 

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 challengebased
unit, and then reevaluate those same attitudes after the unit to try
to measure how challengebased and exploratory/inquiry learning impact
students attitudes toward the subject and their interest in possible
related careers? 

⋄ 

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?

⋄ 

teach math (group theory & stats) with crypto as the motivator?


⋄ 

encourage scientific exploration with math, crypto, logic as motivator?


⋄ 

teach crypto  math is used as needed?


⋄ 

teach a blend of crypto, math, and lab design?


⋄ 

teach lab design with crypto experiments as the motivator?

How are we going to do this?

⋄ 

mainly set up interactive virtual experiments for self discovery?


⋄ 

mainly explain known results and have students verify them with virtual experiments?


⋄ 

create a networked game where students acquire/steal wealth and use crypto algorithms for protection?

What Mathematics would we like the students
to experiment with? (subject to change)

⋄ 

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
 (g^{R} mod M)^{S} mod M = (g^{S} mod M)^{R} mod M
 Chinese Remainder Theorem


⋄ 

Permutations
  
transpositions 
  
products of permutations 
  
conjugated permutations 
  
cyclic structure 
  
degree of permutations 
  
two degree N permutations of disjoint
transpositions have a product containing an even number
of disjoint cycles of the same length. 
  
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. 
  
Two permutations K,L on the same set X
are conjugated if and only if they have the same cyclic
structure. 


⋄ 

Statistics
 mean, standard deviation, moments
 correlation coefficients 
What crypto algorithms would we like the students to experiment with? (subject to change)

⋄ 

Euclid's algorithm
 Find the inverse of M mod N


⋄ 

MillerRabin algorithm
 Generate a (probably) prime number


⋄ 

Chinese Remainder Theorem
 Use in development of public key cryptosystems
 Use in attacking RSA and other cryptosystems
 Use in encrypting a secret

What crypto systems would we like the students to experiment with? (subject to change)
How might a course on math and crypto progress?

⋄ 

Experiment with the Ceaser cipher. See how easily it is cracked. 

⋄ 

Students propose variants like increasing the rotation with every keystroke 

⋄ 

Learn about and experiment with permutation groups (to be developed). 

⋄ 

Break Enigma using the math of permutation groups. 

⋄ 

Students think about the security associated with ciphers based on permutations. 

⋄ 

Message integrity with message digest 
Maybe a project to tie all the concepts together?