A Kid Krypto System based on DIRECTED CYCLE COVER

FPT CHALLENGE
The first FPT implementation challenge PACE: Parameterized Algorithms and Computational
Experiments Challenge is respectfully dedicated to David Johnson, creator of the DIMACS
algorithms challenges and a leader and advocate for algorithms and theoretical computer science.
Website https://pacechallenge.wordpress.com for details.
ENTER YOUR TEAM NOW.
Track A: Tree Decompositions: optimal solutions, heuristics, generating hard instances,
and collecting real-world instances. The tree decomposition validator is available at
https://github.com/holgerdell/td-validate/
Track B: Feedback Vertex Set: fixed-parameter algorithms.
IMPORTANT DATES
* Benchmark instances available NOW.
* 1 June 2016: Register participation. Track A for TreeWidth send email to Holger Dell at
pace16@holgerdell.com and for Track B Feedback Vertex Set send email to
Christian.Komusiewicz@uni-jena.de
* 1 August 2016: DEADLINE TO SUBMIT Implementations
* 22–26 August 2016: Results announced at the International Symposium on Parameterized
and Exact Computation (IPEC 2016).
A Kid Krypto System based on
DIRECTED CYCLE COVER
Frances Rosamond
Department of Informatics
University of Bergen, Norway
Scottish Combinatorics Meeting 26—27 April 2016
•  Perfect Code Kid Crypto
Computer Science Unplugged
www.csunplugged.org
•  Directed Disjoint-Cycle Packing Crypto
•  Polly Cracker
•  Creative Mathematical Sciences Communication
www.tcs.uni-luebeck.de/cmsc/
4—7 October Rudiger Reischuk
Neal Koblitz, University of
Washington
An inventor of elliptic curve
cryptography
Cryptography as a Teaching Tool.
https://www.math.washington.edu/
~koblitz/
Neal and Ann Hibner Koblitz, author of A Convergence of
Lives. Sofia Kovalevskaia: Scientist, Writer, Revolutionary.
Founders of the Kovalevskaia Prize
Michael Fellows, Univ Bergen
Parameterized Complexity
This is MEGA-Mathematics
http://www.c3.lanl.gov/ captors/
mega-math, (Los Alamos Natl Labs)
1992, with Nancy Casey.
Neal and Mike created a kid crypto system based on a
special kind of dominating set in a graph called a Perfect
Code.
Def: A set of vertices V’ ⊆ V in a graph G = (V, E) is said
to be a perfect code if for every vertex u ∈ V the
neighborhood N[u] contains exactly one vertex of V’ .
Perfect Code is an NP-hard problem.
Computer Science Unplugged!
•  http://csunplugged.com/ book, teacher
guides and support
•  http://video.google.com
videos
COMPUTER SCIENCE Unplugged!
Michael Fellows Michael.Fellows@cdu.edu.au
Frances Rosamond Frances.Rosamond@cdu.edu.au
Charles Darwin University, Parameterized Complexity Research Unit, Faculty of Engineering
and IT, Northern Territory, 0909 Australia
REFERENCES: More information and buy the book: “Computer Science Unplugged!” by Tim
Bell, Michael Fellows and Ian Witten, available at http://www.csunplugged.org and
http://www.cosc.canterbury.ac.nz/tim.belltour2006.
THEMES of the DEMONSTRATIONS: ALGORITHMS and COMPLEXITY
THEME
ACTIVITIES
1) P vs NP
How difficult is it to solve a
problem?
A million-dollar prize
Unsolved problems
2) Examples of polynomial
time sorting and bad sorting
(parallel is good, logn).
2-coloring versus 3-coloring
Sorting Networks
CONNECTIONS
Concept of an “algorithm” Time complexity table
Number sense
Modeling
Computational thinking
Concept of “minimum” Parallel versus serial
Permutations
Combinatorial objects
Universal quantification
Factorial
Cooperative learning
3) More P vs NP
Muddy City (Minimum
Spanning Tree) versus
Ice Cream Stands (Steiner)
Arithmetic in action
Weighted paths
“Greedy” method Careful checking
Scale and layout
4) Public Key Cryptography
Kid Krypto
Coin Flip over the Phone
Perfect codes
One-way functions
Linear algebra
Public key/private key
Proofs
Color the graph
with as few
colors as
possible.
Two vertices
connected by an
edge must get
different colors.
3-coloring
Minimum Weight
panning Tree
Discrete Steiner
Place ice cream
stands so that no
matter which
corner you might
be standing on,
you need walk at
most one block to
get an ice-cream.
Tourist Town
village map
How do we know we can do it with six?
Do you want to create a graph where you know
the solution but it will be really hard for mom or
dad to solve?
The idea of a one-way function.
Create a graph with a
Perfect Code.
Take care that no two stars
share a common vertex.
Add extra edges to confuse
the Adversary. Add
additional disguising edges
only between vertices not
in the dominating set.
Create a graph with a
Perfect Code
Add extra edges to confuse
the Adversary
A Perfect Code may not be
unique.
Create a graph with a
Perfect Code
Add extra edges to confuse
the Adversary
The Perfect Code is not
unique.
Public-Key Cryptography (Asymmetric encryption)
Three players
Alice – publishes her public key in “phonebook”, a
trusted, neutral source.
She has a private key to decode messages sent
to her using her public key.
Bob – has an “encryption method” for using
Alice’s public key to send her a message.
Adversary – tries to crack their communication
system, knowing how it works (in general).
Alice wants to be able
to receive an encrypted
bit from Bob. She
constructs a graph
G(V, E) with a perfect
code.
The public key is the
graph G.
Alice’s private key is
the perfect code.
-4
2
3
4
6
5
-5
The Encryption Method
(2 steps)
Privately and secretly
Step 1. Bob puts
numbers on vertices that
sum to the Message.
The Message is 11.
Step 2.
-4
2 7
4
3 10
6 8
5
-5
The Encryption Method
(2 steps)
Privately and secretly
Step 1. Bob puts
numbers on vertices that
sum to the Message.
The Message is 11.
Step 2. Sum the solid
neighborhood of each
vertex (the numbers in
red).
-4 8
2 7
4 1
3 10
6 8
5 3
-5 4
The Encryption Method
(2 steps)
Privately and secretly
Step 1. Bob puts
numbers on vertices that
sum to the Message.
The Message is 11.
Step 2. Sum the solid
neighborhood of each
vertex (the numbers in
red).
Bob erases all traces of
his calculations.
8
7
10
1
8
3
4
Bob returns the graph to
Alice annotated only
with the red numbers.
8
7
10
1
8
3
4
To decipher the
message, Alice takes
the sum over the
perfect code.
8
7
10
1
8
3
4
To decipher the
message, Alice takes
the sum over the
perfect code.
Sharing secrets is very exciting for kids. Research in
cryptography has been awarded the Turing Award
three times.
Whitfield Diffie and Martin Hellman were awarded
the Turing Award in 2015.
The 2012 Turing Award was awarded to Shafi
Goldwasser and Silvio Micali.
Ron Rivest, Adi Shamir and Leonard Adleman won
the 2002 Turing Award.
New kid crypto system based on directed cycles
Directed Disjoint Cycle Cover problem asks, for
input a digraph D = {V, A} whether there exists a
family F = {C1, ..., Cm} vertex-disjoint directed
cycles that “spans D”, that is, for every v in V
there is a unique directed cycle of F that passes
through V.
Directed Disjoint Cycle Cover is an NP-hard
problem.
Disjoint directed cycles
The cycles with disguising arcs.
The message is 11.
Labels the vertices
to add to 11.
Encryption: Label the edges
(twice the tail minus the
head).
8
7
10
1
Both systems are secure
up to smart high school
students. Gaussian
elimination.
8
3
4
Processor for each local sum. Large
graph maybe 10,000 vertices.
Equations with 10,000 variables.
Linear algebra sequential, O(n3).
New memory heirarchies.
To decipher the
message, Alice takes the
sum over the perfect
code.
Graph Coloring Kid Crypto
Mathematics communication is a two-way street
Polly Cracker system
Lay our best at the feet of the children,
including the frontiers of what we know
– Elementary school students deserve to experience
profound and imaginative mathematical ideas.
– Open unsolved problems are the creative drivers
for mathematical activity.
– Mathematics is an “interdisciplinary powerhouse.”
– Mathematics popularization is a research area of
basic interest.
Fellows, M.: Computer SCIENCE in the elementary schools. (1991)
3rd CREATIVE MATHEMATICAL SCIENCES
COMMUNICATION CONFERENCE (CMSC)
The Creative Mathematical Sciences Communication
conference (CMSC) explores new ways of popularizing the
rich mathematics underlying computer science including
outdoor activities, art, dance, drama and all forms of
storytelling. Hang out with people who develop creative new
ways to explain your research to your colleagues down the
hall, in different disciplines, government, your kids, mom.
Date: 4-7 October 2016
Location: Lübeck, Germany
Abstracts due: 10 June 2016
Submissions due: 8 July 2016
Early Registration: 15 August 2016
Website http://www.tcs.uni-luebeck.de/cmsc/
Thank you
educators and artists (theatre, dance, graphic arts).
The CMSC has several aspects.
• Involve and support researchers to share the frontiers of computer science and
mathematics with children and the general public.
• Research communication is a “two-way street”. Explaining your research can inspire new
research questions. For example, Mike Fellows describes how “Kid Crypto” inspired the new
research of Polly Cracker crypto systems.
• Future directions of Computer Science Unplugged! Discuss with Tim Bell, Mike Fellows
and others future directions of this grass-roots movement, which is now translated into 19
languages.
• Storyfull, whole-body, kinesthetic math activities. Demonstrate and design new whole
body activities that connect math with the inner self and community. Design activities that
include cultural understanding and relevance. Demonstrate activities that foster curiosity,
enthusiasm and perseverance.
• Expand computational thinking across the curriculum, and explore how mathematical
thinking strategies nurture 21st Century competencies.
• Policy makers in government, business and industry. What are the issues and unanswered
questions of executives and policy makers?