2. Outline
Introduction
What are combinatorial problems ?
Challenges
Real life Problems
Solving combinatorial problems
Reducing combinatorial problems
Modeling as
Constraint satisfaction problem
Subset generation problem
Satisfaction problem
Error correcting & Cryptology
Hamming Distance Problem
Logical Cryptanalysis of DES
3. What are combinatorial problems ?
A class of problems that are at least NP:
Enumerative combinatorics
Counting the number of certain combinatorial objects
How many ways are there to make a 2-letter word?
How many ways are there to select 5 integers from {1,..,20} such that the (positive) difference between any 2 of the 5 integers is at least 3?
Design Theory
Study of combinatorial design, which are collection of subsets with certain intersection properties
Ternary Steiner systems
oSelect 3-element subsets from the set {0,1,2,3,4,5,6} such that each object occurs in 3 of the subsets and the intersection of every two subsets has precisely one member.
Graph theory, Order theory, Geometric combinatorics, …etc.
4. What are combinatorial problems ?
Characteristics & Challenges
Most of these problems are classified as NP problems
Huge search space
Real combinatorial problems
Cryptography / Error Correcting
Finding Shortest round trips (TSP)
Boolean satisfaction problems (SAT)
Planning, scheduling & timetabling
Protein structure prediction
Internet packet data routing , …etc.
5. Solving combinatorial design problems
Reducing combinatorial design problems
Need to prune search space before actually searching
Get results in reasonable time
How can this be done ?
Constraint programming
Model problem as a constraint satisfaction problem (CSP)
o(V,D,C)
oV is a set of variables taking values from domain D upon which constraints in set C are satisfied.
Constraint solving
oUse clever algorithms to solve a CSP problem
•Node, Arc & Path consistencies
•Constraint propagation
6. Constraint satisfaction problem (CSP) vs. Generate & Test
15 variables [V1,..,V15] all taking values in the domain {1,..,15}
V1<V2, V2<V3, …, V14<V15
Generate & Test Approach
Enumerating all possible assignments
1515= 437,893,890,380,859,375 possibility
109assignments/second computer would need 14 years to find solution
7. Constraint satisfaction problem (CSP) vs. Generate & Test
Model as a constraint satisfaction problem
Constraint satisfaction Solving
Interleaving generation and checking
After each choice check if constraints have been violated and then fail early
Propagate & distribute
Making tests active rather than passive
oNot just testing whether constraints are violated or not but making them active and propagate constraints
V1 < V2
oV2 must be 1 greater than V1
oV1 = {1,..15} and V2 = {2,..,15}
Repeat this reasoning until V14 < V15
oV15 = {15}
8. Combinatorial Design Problems using sets
Combinatorial Design Problems
Study of combinatorial design, which are collection of subsets with certain intersection properties
Approach
Model variables as sets
Solve using set solvers
9. Combinatorial Design Problems using sets
Ternary Steiner systems
Select 3-element subsets from the set {0,1,2,3,4,5,6} such that each object occurs in 3 of the subsets and the intersection of every two subsets has precisely one member.
Model
Use intersection constraint to state that the intersection of every two 3- element subsets has precisely one member
Use element constraint to state that each element occurs in 3 of the 3- element subsets
Solving
Set solver
10. Combinatorial Design Problems using sets
Ternary Steiner systems
Solving
Set solver
Order in sets doesn’t matter
Symmetry breaking
o{0,1,2}
o={1,0,2}
o={2,1,0}
o={0,2,1}
Solution
{012},{034},{056},{135},{146},{236},{245}
11. Combinatorial Problems as SAT problems
Boolean satisfaction problems (SAT)
The problem of determining if the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to TRUE.
Example
For (A B) (B ¬C) (A ¬B) to be true which variables have to be true to satisfy all clauses ?
Approaches
oExhaustive search in the form of truth table to find the solution
•Number of rows for the truth table is 2# of variables
12. Combinatorial Problems as SAT problems
Model as a SAT problem and use SAT solvers
SAT solvers
Davis-Putnam Algorithm
Splits the clauses using truth assignments
(A B) (B ¬C) (A ¬B) B (B ¬C) ¬B
B ¬C()
()¬C
() A=0
A=1B=0
B=1B=0
B=1C=0
C=1() = Failed Partial Assignment = Successful Partial Assignment
13. Error correcting & Cryptology
In a good cryptographic system, changing one bit in the cipher text changes enough bits in the corresponding plain text to make it unreadable.
What causes the change/error of the cipher text?
Noise from communication channels
Could be an attack
14. Error correcting & Cryptology
Holding a conversation in a noisy room becomes more difficult as the noise become louder.
Data transmission becomes difficult as the communication channel gets noisier.
Solution ?
Raise your voice
Repeat yourself
Adding some redundancy to the transmission in order for the recipient to be able to reconstruct the message.
Replace the symbols in the original message by codewords that have some redundancy built into them.
Code words
Block codes (not varying in length)
15. Hamming Distance Problem
We are trying to find how close the 2 codewords/code vectors are.
The Hamming Distance = d(u,v)
The number of places where the 2 codewords/code vectors differ.
Examples
u = (1,0,1,0,1,0,1,0)
v = (1,0,1,1,1,0,0,0)
d(u,v) = 2
16. Hamming distance optimization problem
As an optimization problem then, the task is to find (and prove) the maximum size (number of codewords = a(n, d,w)) for a binary error correcting code with given parameters (n, d) and optionally fixed weight (w).
Approach
Trying to find increasingly larger codes and proving optimality by failing to find one larger.
17. Hamming Distance Optimization problem
How to model this ?
Using set variables (Si) to represent code words (Ci)
The element x is in the set Si iffthe code Cihas a 1 at position x
[0000]->S={}
[1000]->S={1}
[0100]->S={2}
[1100]->S={1,2}
18. Hamming Distance Optimization problem
Example
a(4,1,1)
How many code words of length 4 , minimal distance of 1 and weight 1 ?
[1000]S = {1}
[0100]S = {2}
[0010]S = {3}
[0001]S = {4}
Answer = 4
19. Logical Cryptanalysis of DES
What is DES ?
Data encryption standard which is a state of the art cipher system used to encrypt/decrypt text.
Cryptanalysis
Trying to break the cipher system and acquire key
Logical cryptanalysis of DES ?
Modeling the DES algorithm as a SAT problem and therefore explicit representation of Plain bits, Key bits, and Cipher bits.
22. Logical Cryptanalysis of DES
ε(P,K,C)
which is true if and only if for the corresponding sequences of bits (P0,P1,C0,C1,… etc) we have that C = Ek(P) holds.
23. Logical Cryptanalysis
Now that we have modeled our problem
We can use the formula to crack the key given a known plain text and cipher text.
We can find out interesting properties of the cipher system
Faithful cipher
for any pair of plaintext and cipher text there is only one key that could have produced them.
