1. Lecture 6 Overview

2. The minimum requirements
A symmetric-key cryptosystem
A block cipher
Capable of supporting a block size of 128 bits
Capable of supporting key length of 128, 192, and 256 bits
Available on a worldwide, non-exclusive, royalty-free basis
CS 450/650 Lecture 6: AES

3. Criteria for Evaluation
Security
Soundness of the mathematical basis for an algorithm’s claimed strength
Research community search for flaws
Computational Efficiency
Memory Requirements
Flexibility
Simplicity
CS 450/650 Lecture 6: AES

4. Advanced Encryption Standard
10, 12, 14 rounds for 128, 192, 256 bit keys
Regular Rounds (9, 11, 13)
Final Round is different (10th, 12th, 14th)
Each regular round consists of 4 steps
Byte substitution (BSB)
Shift row (SR)
Mix column (MC)
Add Round key (ARK)
CS 450/650 Lecture 6: AES

5. AES Overview 9 rounds Plaintext (128) ARK Subkey0 BSB SR
Ciphertext (128)
ARK
Subkey10
CS 450/650 Lecture 6: AES

6. Round i operations 128-bit substitution boxes confusion
transposition step of circular shift
confusion
Left shift and XOR of bits
diffusion and confusion
portion of key is XORed
confusion
Subkeyi
CS 450/650 Lecture 6: AES

7. Shift Row (128-bit) b0 b4 b8 b12 b1 b5 b9 b13 b2 b6 b10 b14 b3 b7 b11
CS 450/650 Lecture 6: AES

8. Mix Column = * Multiplying by 1  no change2 3 1
S0,i
S1,i
S2,I
S3,i
S’0,I
S’1,I
S’2,I
S’3,i = *
Multiplying by 1  no change
Multiplying by 2  shift left one bit
Multiplying by 3  shift left one bit and XOR with original value
More than 8 bits  is subtracted
CS 450/650 Lecture 6: AES

9. Add Key = b’x bx kx XOR b0 b4 b8 b12 b1 b5 b9 b13 b2 b6 b10 b14 b3 b7
CS 450/650 Lecture 6: AES

10. Circular left shift 1byte
Key Generation
4 bytes
Circular left shift 1byte
S-box
XOR
XOR
Round constant
XOR
XOR
4 bytes
CS 450/650 Lecture 6: AES

11. DES vs AES DES AES Date 1976 1999 Block size 64 bits 128 bits
Key length
56 bits
128, 192, 256, … bits
Encryption primitives
Substitution and permutation
Substitution, shift, bit mixing
Cryptographic primitives
Confusion and diffusion
Design
Open
Design rationale
Closed
Selection process
Secret
Secret (accepted public comment)
Source
IBM, enhanced by NSA
Belgian cryptographers
CS 450/650 Lecture 6: AES

12. Lecture 8 Algorithm Background
CS 450/650
Fundamentals of
Integrated Computer Security
Slides are modified from Hesham El-Rewini

13. Analysis of Algorithms
Time Complexity
Space Complexity
An algorithm whose time complexity is bounded by a polynomial is called a polynomial-time algorithm.
An algorithm is considered to be efficient if it runs in polynomial time.
CS 450/650 Lecture 8: Algorithm Background

14. Time and Space Should be calculated as function of problem size (n)
Sorting an array of size n,
Searching a list of size n,
Multiplication of two matrices of size n by n
T(n) = function of n (time)
S(n) = function of n (space)
CS 450/650 Lecture 8: Algorithm Background

15. Growth Rate
We Compare functions by comparing their relative rates of growth.
1000n vs. n2
CS 450/650 Lecture 8: Algorithm Background

16. Definitions T(n) = O(f(n)): T is bounded above by f
The growth rate of T(n) <= growth rate of f(n)
T(n) = W (g(n)): T is bounded below by g
The growth rate of T(n) >= growth rate of g(n)
T(n) = Q(h(n)): T is bounded both above and below by h
The growth rate of T(n) = growth rate of h(n)
T(n) = o(p(n)): T is dominated by p
The growth rate of T(n) < growth rate of p(n)
CS 450/650 Lecture 8: Algorithm Background

17. Time Complexity O(2log n) C O(n) O(log n) O(nlogn) O(n2) … O(nk) O(2n)
O(kn)
O(nn)
Polynomial
O(2log n)
Exponential
CS 450/650 Lecture 8: Algorithm Background

18. P, NP, NP-hard, NP-complete
A problem belongs to the class P if the problem can be solved by a polynomial-time algorithm
A problem belongs to the class NP if the correctness of the problem’s solution can be verified by a polynomial- time algorithm
A problem is NP-hard if it is as hard as any problem in NP
Existence of a polynomial-time algorithm for an NP-hard problem implies the existence of polynomial solutions for every problem in NP
NP-complete problems are the NP-hard problems that are also in NP
CS 450/650 Lecture 8: Algorithm Background

19. Relationships between different classesNP
NP-hard
NP-complete P
CS 450/650 Lecture 8: Algorithm Background

20. Partitioning Problem
Given a set of n integers, partition the integers into two subsets such that the difference between the sum of the elements in the two subsets is minimum
13, 37, 42, 59, 86, 100
CS 450/650 Lecture 8: Algorithm Background

21. Bin Packing Problem Suppose you are given n items of sizes
s1, s2,..., sn
All sizes satisfy 0  si  1
The problem is to pack these items in the fewest number of bins,
given that each bin has unit capacity
CS 450/650 Lecture 8: Algorithm Background

22. Bin Packing Problem
Example (Optimal; Solution) for 7 items of sizes: 0.2, 0.5, 0.4, 0.7, 0.1, 0.3, 0.8.
CS 450/650 Lecture 8: Algorithm Background