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Public Key Crypto for all Ages: The Story of Kid Krypto and Polly Cracker
Prof Michael Ralph Fellows
Charles Darwin University, Australia
A talk to Vijyoshi Camp 2012
Indian Institute of Science, Bangalore
December 2012
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Parameterized Complexity,
Rod Downey and Mike Fellows
Springer Publisher, 1999
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Festschrift, presented at Dagstuhl
June 2012
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Goals
(1) Crash course in algorithms and computational
complexity
(2) Basic ideas of modern cryptography
(3) Some of my adventures when sharing these
ideas with 9-year-olds
…with some added advanced commentary
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„If you‟re going to teach, you should either teach
graduate school or 4th grade.”
-Kurt Vonnegut
Famous American writer
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• Model problem for NC, problems solvable in
poly(log n) time on a polynomial number of
processors
• Lower bound of log n depth because
log2 n! ~ n log n
Sorting nets: advanced
commentary
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Sorting nets: advanced
commentary
! Many problems have been shown to be
“inherently” sequential: parallel algorithms are
of no use (modulo a plausible but untouchable
conjecture)
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2-coloring: advanced
commentary O(n2)
Other O(n2) algorithms—school multiplication
6012782
4213612
’5’6 4
etc
---insertion sort
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• These are models for the complexity class P:
problems solvable by a polynomial-time
algorithm.
• System of linear equations O(n3)
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Why efficient algorithms matter
The number 1 lesson
of this crash course
Challenge: Give me a really terrible
algorithm for SORTING
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CHECKING A PROPOSED
3-COLORING SOLUTION
IS O(n2)
Model for the complexity class NP
3-coloring advanced
commentary
P = NP
?
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IWOCA 2009
The “classical” P vs NP
framework is one-dimensional
n = input size
poly(n) 2 poly(n)
vs
“good”
P
positive toolkit of how to
design P-time algorithms
“bad”
NP, etc.
negative toolkit of
NP-hardness, etc.
Unfortunately, almost everything turns out to be NP-hard.
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The parameterized framework is
two-dimensional n = input size
k = a relevant secondary measurement
f(k)nc n g(k) vs
“good”
FPT
“bad”
W-hard, etc.
Complexity frameworks
are driven by contrasting
function classes.
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n = 50 n = 100 n = 150
k = 2 625 2,50 5,625
k = 3 15,625 125,000 421,875
k = 5 390,625 6,250,000 31,640,625
k = 10 1.9 x 1012 9.8 x 1014 3.7 x 1016
k = 20 1.8 x 1026 9.5 x 1031 2.1 x 1035
The Ratio n k+1 for Various Values of n and k 2 k n
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Frameworks in pictures
The classical P vs NP framework
k
nc
n Intrinsic Combinatorial explosion: Most problems are NP-hard or worse.
The parameterized framework
FPT
Try to confine the explosion to the parameter.
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3-SAT Input E = (a + b + c‟ ) (a‟ + c + d)
(b‟ + c‟ + d‟) (a + b‟ + d)
Question Does there exist a truth assignment to the
Boolean variables making E true?
VERTEX COVER
Input
and k = 6
Question Can we choose k vertices to cover all
edges?
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a b c d + - + - + - + -
GE
kE = (2 X # clauses) + ( # variables) = 12
3SAT
Input E = (a + b + c‟ ) (a‟ + c + d)
(b‟ + c‟ + d‟) (a + b‟ + d)
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PERFECT CODE
Ice Cream Stands No one gets
confused!
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PERFECT CODE
Ice Cream Stands No one gets
confused!
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1 4
5 0
3 -1
0 3 2 3 1 7
4 0
2 1 1 5
-1 4
2 3
3 3
4
0 1
1
Private key is
the perfect
code
Public
key
0
4
Message = 13
Step 1: sprinkle numbers
around that add up to 11
(privately)
Step 2: compute local sums
(privately)
Step 3: send the public key
labeled with the local sums
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a b c d
h i j k
e f
g
(a + e + g + h) (c + f + k + i) = ac + af + ak + ai
+ ec + …
1 POLLY CRACKER