1. Randomness and Statistical Tests
Sayed Mahdi Mohammad Hasanzadeh
Spring 2004

2. Goals in cryptography
1) Cryptography remove the redundancy from sequence
Pr(1)=Pr(0)=1/2
2) Cryptography is a mapping in the sequence space
3)Cryptography provide security in some cases: data integrity, Authentication and identification ,…

3. Stream Cipher Pr(Pi=0) Pr(Pi=1) 1/2 Pr(ai=0) =Pr(ai=1) =1/2
Pr(Ci=0) =Pr(Ci=1) =1/2

4. Generator Properties Randomness Provable security Bit rate Key length
Complexity of algorithm
Memory
Resistant against every attack

5. Randomness
Golomb rules
Statistical tests
Linear Complexity

6. Golomb rules Maximum period Pseudorandom 01010001101111100111
#’1’ - #’0’ =0,1
#run1=R/2^1, #run2=R/2^2,…,#runi=R/2^I
t : C(t)=
#run1=5,#run2=2,#run3=2,#run4=0,#run5=1
#gap=5 , #block=5

7. Randomness
Golomb rules
Statistical tests
Linear Complexity

8. Why Statistical tests
impossible to give a mathematical proof that a generator is indeed a random bit generator, the tests help detect certain kinds of weaknesses the generator may have.
This is accomplished by taking a sample output sequence of the generator and subjecting it to various statistical tests.

9. Statistical tests
the term “accepted” should be replaced by “not rejected”
six Basic Test (Using Chi-square analysis)
Frequency Test: # of 0 and 1
Serial Test: # of 00, 01, 10, 11
Poker-k Test: # of each k-bit string
Run Test: comparing with expected run length
Autocorrelation test: correlations between s and shifted version
Binary derivation

10. Working with a Software for
Statistical Tests