1. Ciphering and the Theory of Secrecy Systems
Network Security
Design Fundamentals
ET-IDA-082
Lecture-6
Ciphering and the Theory of Secrecy Systems
v29
Prof. W. Adi

2. And Fundamentals of ciphering
Secrecy Theory
And Fundamentals of ciphering
Outlines
Historical Overview
Basic Definitions
Shannon’s Secrecy Theorem
Perfect Secrecy „Vernam Cipher“
Unicity Distance
Secret Key Cipher Principle

3. Two interesting statements around security!
„The only system which is truly secure is one which is switched off , unplugged, locked in a titanium lined safe, buried in a concrete bunker, surrounded by nerve gas and very highly paid armed guards.
Even then, I wouldn't stake my life on it ....“
Gene Spafford - Computer Operations, Audit and Security Technology (COAST), Purdue University
„If I am asked to stake my life on a cryptographic function, I would not trust any function related to known mathematics“
Ulli Maurer – ETH Zürich, Swisserland

4. Scientific History of Cryptography
Cryptography (Greek: hidden word): is it Art or Science ?
The scientific story has three epochs:
I. Conventional Cryptography as Art till 1949
Julius Caesar Cipher
Kaisiski “ The Art of Deciphering” 1863, ... Gauss
Vernam (AT&T) 1926, first unbreakable system
II world ware 1945, Enigma, Hagelin .... Alan Turing
يقوب بن اسحق الكندي
رسائل في استخراج المعمى
Not widely known in convectional public literature:
Alkindi Bagdad (801–873): "father of Islamic or Arabic philosophy“, Mathematician physician and musician. Presented mathematical tools for cryptanalysis:
“Treaties on Extracting Cryptograms”, (Doc 4832 Sulaimania Library Istambul Turky)

5. Akindi’s Cipher Classification (استخراج المعمى)
(873 AD)
From the year 873

6. Akindi’s Contribution in Crytoanalysis (استخراج المعمى)
(scientific treatment 1000 years before Shannon’s modern entropy techniques)
Introduced for the first time
Determining the statistics of the frequency of letters in a particular language
and made use of letter statistics to break a cipher by comparing
the letter frequency in the cryptogram (showed an example of a text with 3667 letters)
He indicated the importance that the cryptogram must be sufficiently long to get more
accuracy.
Classified different ciphering/deciphering techniques using a tree diagram
Showed techniques for hiding information in an existing natural text as Steganography
Alkindi’s treatise were dedicated to the government (Calif. In Bagdad Abu Al-abbas)
Use of Cryptography was mainly to transfer confidential reports to the Caliph from the different wide spread areas of the Islamic rules territories (called “Bareed “ service)
Ref.:

7. Scientific Epoch and Breakthrough in Cryptography
2. Modern scientific epoch: (1949)
Shannon (AT&T) “A Mathematical Theory of Communication”
Shannon’s Breakthrough in Communication:
Error-free transmission is possible on noisy channels! C= B log2 (1 + S/N)
Shannon (AT&T) 'Communication Theory of Secrecy Systems‘
proved mathematically that Vernam cipher is unbreakable
!Just new Scientific methodology but NO breakthrough in Cryptography!
3. Breakthrough to Modern Cryptology in 1976
Diffie and Hellman 1976 Public key Cryptography (Stanford University)
Diffie Hellman ‘s Breakthrough in Cryptography:
Secured transmission is possible on unsecured channels!
Any new Breakthrough expected ? !

8. 2. Is Security a Science, Art or Magic ?
BIG OPEN QUESTIONS
1. Is Security Measurable?
2. Is Security a Science, Art or Magic ?
Question raised by James L. Massey
Cryptography - Science or Magic?
MIT, October 1, 2001, Running Time: 00:57:10

9. Two Major Security Tasks
Authentication
Securely identify an entity
Secrecy
Keep data safe against illegal users
Security tasks require to deploy Cryptographic mechanisms to be realized
Cryptography is the science dealing with hiding information and data security questions

10. Secret Key Cryptography in Use
Conventional
Secret Key Cryptography in Use
Fundamental Concepts

11. Cryptography : Basic definetions
Attacker
Crypto-analyser
Channel
Y = E (Ze,X)
Cryptogram/Cipher text
Message
Clear text
D (Zd,Y)
Decryption Key
Deciphering X
Receiver
Message X
Sender
Clear text
Ciphering
Encryption-Key
E (Ze,X)
Intruder
Attacker Ze Zd

12. Cryptographic Attacks: Basic definetions
Type of attacks:
Cipher text only attack
Known plaintext attack
Chosen plaintext attack
Chosen ciphertext attack
Chosen Cipher/Plain Text Attack: is mostly assumed as a basis to evaluate the quality of a cipher
Kerckhoff’s Principle: A General system security evaluation assumption:
Attacker knows everything but not the key !

13. Secret Key Crypto-System : mechanical analog
Key = Z Secret key agreement Key = Z Z
SENDER
RECEIVER Z
Lock
Message
Message Z

14. Conventional Cryptography till 1976 : Secret Key systems
Known locks as Standard Ciphers
Security rests on the Cipher
Ciphering
De-Ciphering
Sender
Receiver
Y = E (Z,X) X
E ( Z,X ) X
D ( Z,Y )
Message
Channel
Message
Secret Key = Z Z
Secret Key Channel Z

15. Is Cryptographic Security Measurable ?
Yes and No !
Security measures adopted today:
System is unconditionally secure (perfect) :
System impossible to break with any means (whatever)
One not very practical system is known !
Non-sceientific statement
System is practically secure:
System possible to break but with very huge means
All modern practical systems fall under this category!

16. The Theory of “System Secrecy”
“The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth.” Niels Bohr ( )
InformationSOURCE
Output
Example:
P : Probability function

17. The Theory of “System Secrecy”
Based on the Concept of Information Entropy: Shannon 1949
The measure of information I(p) is a function which fulfils the following properties:
Non- negative quantity:
If an event has probability 1 , we get no information
In case of two independent events, then
In general,
Defined,
b is called the base.
Log2 units are bits (from “Binary”) and Log3 units are trits (from “trinary”)

18. The Theory of “System Secrecy”
Example for calculating information entropy
The probability of Head is 0.5 and the probability of Tail is 0.5 too.
If you get either head or tail you will get 1 bit of information :
For instance, 0 denotes Head and 1 denotes Tail : vice versa
The experience of flipping a fair coin

19. The Theory of “System Secrecy”
Based on the Concept of Information Entropy: Shannon 1949
Amount of information in a message M is defined by Shannon as “Entropy” H(M) where:
over all possible messages Mi
Example 1:
M = months of the year
H(M) = log2 12  3.6 bits
Example 2:
2k equally probable keys
H(M) = log2 2k = k bits
If there are n equally probable messages, then Prob(Mi) = 1/n for any i, that is:
= n  ( 1/n log2 n )
= log2 n
where n is the number of possible meanings

20. 0  H (X)  log2 t (for t possibilities for message X)
The Theory of Secrecy Systems
Shannon 1949
If the entropy function for information X: H (X) = -  pr(xi) . log2 pr(xi)
{x}
0  H (X)  log2 t (for t possibilities for message X)
Shannon Perfect Security condition is H (Z)  H (X)
The key entropy (H(Z) for a key with k-bits if all key combinations are equally used:
H(Z) = - 2k [ ( 1/2k ) . Log2 (1/2k) ]
H (Z) = k
In that case the necessary condition for Perfect Security is : k  H (X)

21. The only known Perfect Cipher: One-time-Pad OTP (Vernam Cipher)
Invented by Vernam (AT&T 1926)
Proved later to be impossible to break by Shannon (AT&T 1949)
Gilbert Vernam 1890 – 1960
(AT&T)
Cipher Text X+Z
Clear Text X Z +
Clear Text X+Z+Z=X + Z
!Addition in GF(2)!
key-tape
Use key just one time!!
Random One Time secret Key
Unconditional Secrecy if : Key length = Clear text length (Shannon 1949)
H(z) > H(X)

22. Generalised One-Time Pad
System is also perfect for any Group < G, * >
Cipher Text X*Z
Clear Text X * Z * Z-1 =X
Z-1
Clear Text X * * Z
Key-tape
One Time secret Key
Key length = Clear text length !!!
No key is repeatedly used!!

23. Improving Security by Message Compression
Enhance security by reducing redundancy
Perfect security always possible iff: Key length = Clear text length that is K=N
Idea: make K=N if N > K by reducing N trough data compression
Compressed to k Bits
Clear text X
N Bits /
N > K K / K /
Ideal
Compressor
Cipher
Cipher text
/ K
Unconditionally Secure!
Key

24. Unicity Distance: Minimum required ciphertext to break a cipher
Key equivocation function: H (Z|y1, y2, y3...yn) for non-randomized cipher
Cipher Unicity Distance
Information Redundancy
Unicity Distance nu : is the minimum amount of ciphertext symbols which in principle
can determin the secret key in a ciphertext only attack
Or: Expected minimum amount of ciphertext needed for brute-force to break a cipher

25. (by increasing the unicity distance) Original Clear text N-Bits
Plain Text Padding technique to improve security
(by increasing the unicity distance)
As larger uniucity distance means higher security, therefore a technique is proposed to increase the unicity distance by appending random unknown clear text as follows:
Original Clear text N-Bits
How to reduce r without compression?
N Bits
Random pattern
L Random Bits
New clear text N+L Bits
H(X)
N‘= N +L
New unicity distance: u n N
N + L =

26. Compute the cipher‘s unicity distance nu and the clear text entropy.
Example: A block cipher having a key size of 128 bits is encrypting a clear text with a block length of 128 bits. The clear text redundancy is r=0.8.
Compute the cipher‘s unicity distance nu and the clear text entropy.
The „unicity distance was doubled by appending L random bits to the clear text block. Compute L and the new clear text entropy.
After all the above cipher changes an observer was able to watch 1000 cipher text bits. Would the observer with unlimited resources theoretically be able to uniquely break the cipher in that case ? Give a reasoning for your answer.
SDF2009

27. Solution: K= 128 Bits, H(x)=? Bits, r = 0.8 K
before
128 bits Data
K= 128 Bits, H(x)=? Bits, r = 0.8
after
128 bits Data
128 random bits K r
1. Unicity distance nu = = 128/0.8 = 160 Bits (the cipher can be theoretically broken after 160 cipher bits)
As r = [ N – H(x) ] / N
=> N . r = N – H(x) => entropy H(x) = N · (1-r) => H(x) = 128 · (1-0.8) = 25.6 Bits
n‘u = [ ( N + L ) / N ] · nu
2 ∙ 160 = [( L ) / 128] · 160
=> L = 128 Bits
H‘(x) = L + H(x) = ,6 = bits
3. The observer can theoretically break the cipher as the number of the observed cryptogram bits (1000 bits) is more than the unicity distance (320 bits) of the cipher. 27