public key crypto for all ages: the story of kid krypto...

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

Parameterized Complexity,

Rod Downey and Mike Fellows

Springer Publisher, 1999

http://csunplugged.org

Tim Bell, Michael Fellows, Ian Witten

Festschrift, presented at Dagstuhl

June 2012

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

„If you‟re going to teach, you should either teach

graduate school or 4th grade.”

-Kurt Vonnegut

Famous American writer

• 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

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)

2-coloring: advanced

commentary O(n2)

Other O(n2) algorithms—school multiplication

6012782

4213612

’5’6 4

etc

---insertion sort

• These are models for the complexity class P:

problems solvable by a polynomial-time

algorithm.

• System of linear equations O(n3)

Why efficient algorithms matter

The number 1 lesson

of this crash course

Challenge: Give me a really terrible

algorithm for SORTING

CHECKING A PROPOSED

3-COLORING SOLUTION

IS O(n2)

Model for the complexity class NP

3-coloring advanced

commentary

P = NP

?

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.

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.

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

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.

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?

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)

PERFECT CODE

Ice Cream Stands No one gets

confused!

PERFECT CODE

Ice Cream Stands No one gets

confused!

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

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

Thank you

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