Pseudorandom Bit Generation

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Pseudorandom Bit Generation

• Artur Gadomski
• Piero Giammarino
• Henrik Goldman
• Massimo Giulio Caterino

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Definitions

• A random bit generator is a device or algorithm
  which outputs a sequence of statistically
  independent and unbiased binary digits.
• A pseudorandom bit generator(PRBG) is a
  deterministic algorithm which, given a truly
  random binary sequence of length k, outputs a
  binary sequence of length lk which appears to
  be random. The input to the PRBG is called the
  seed, while the output of the PRBG is called a
  pseudorandom bit sequence.

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Definitions

• A pseudorandom bit generator is said to pass all
  polynomial-time statistical tests if no
  polynomial-time algorithm can correctly
  distinguish between an output sequence of the
  generator and a truly random sequence of the same
  length with probability significantly greater
  that 1/2.
• A pseudorandom bit generator is said to pass the
  next-bit test if there is no polynomial time
  algorithm which, on input of the first l bits of
  an output sequences, can predict the l1 bit of s
  with probability significantly greater than 1/2

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Definitions

• A PRBG that passes the next-bit test is called a
  cryptographically secure pseudorandom bit
  generator (CSPRBG)

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Random bit generation
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Hardware based generators

• elapsed time between emission of particles during
  radioactive decay
• thermal noise from a semiconductor diode or
  resistor
• the frequency instability of a free running
  oscilator
• the amount a metal insulator semiconductor
  capacitor is charged during a fixed period of
  time
• air turbulence within a sealed disk drive which
  causes random fluctuations in disk drive sector
  read latency times
• sound from a microphone or video input from a
  camera.

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Software based generators

• the system clock
• elapsed time between kaystrokes or mouse
  movement
• content of input/output buffers
• user/system/hardware/network serial numbers
  and/or addresses
• user input
• operating system values such as system load and
  network statistics.

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Mixing functions

• A strong mixing function is one which combines
  two or more inputs and produces an output where
  each output bit is a different complex
  non-linearfunction of all the input bits.

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Example

• A trivial example for single bit inputs is the
  Exclusive Or function.
• DES is an example of a strong mixing function for
  multiple bit quantities.
• Cryptographic hash function such as SHA-1 or MD5.
  
• Diffie-Hellman expotential key exchange is
  another example. If initial values are random,
  then the shared secret contains the combined
  randomness of them both, assuming they are
  uncorelated.

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De-skewing

• Suppose in an output sequence the probability of
  1 is p. Then lets group the output bits into
  pairs and lets treat each 01 as 1 and 10 as 0. We
  discard 00 and 11 pairs. The resulting sequence
  is both unbiased and uncorelated.

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Pseudo Random Bit Generators
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Matematics Model Of PRBG

• INPUT?X0 seed
• Xi1f(i,X0,X1,X2,X3,...) i0,1,2,3,...
• OUTPUT? X1 X2 X3 X4 ... Pseudorandom sequence

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Pseudorandom Generators

• Linear Congruential Generator
• J-Bit Output Feedback
• Ansi X9.17
• Blum Blum Shub Pseudorandom Bit Generator
• RSA Pseudorandom Bit Generator

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Linear Congruential Generator

• Nowadays the most used technique for
• Pseudorandom generator
• Lehmer 1951
• X0Seed
  mgt0
• 
  0altm
• Xi1a(Xib) mod m
  0bltm

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Example of LCG

• a 7
• b 0
• m 32
• Xi1 7 Xi mod 32 7, 17, 23, 1, 7,
  17, 23, X0 1
• period 4
• a 5
• b 0
• m 32
• Xi1 5 Xi mod 32 5, 25, 29, 17, 21, 9,
  13, 1, X0 1
• period 8

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Linear Congruential Generator

• Xi1 75 Xi mod 231-1
• a 75
• b 0
• m231-1 (Prime number convient for 32 bits)
• Used for IBM 3601969

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J-Bit Output Feedback
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ANSI X9.17 Generator

• Ad-hoc construction which is not proved to be
  cryptographicly secure,
• though it should be sufficient for most
  applications

• U.S. Federal Information Processing Standard
  (FIPS) approved method

• Makes use of 2 key tripple DES algorithm

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Algorithm
Def Ek is 3DES encryption under key k is XOR
Input s 64 bit secret seed m interger
(counter) k 3DES key
1. Get 64 bit representation of computer
date/time, D
2. Calc I Ek(D)
3. for (i 0 i lt m i) xi Ek(I s)
// Calc next 64 bit string s Ek(xi I) //
Update seed
4. Return Xis
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Blum blum shub PRBG

• Generate p and qtwo big blum primes
• Npq
• Choose s?1,n-1 The Seed
• X0s2(mod n)
• The sequence is defined as xixi-12(mod n) and
  ziparity(xi)
• The output is z1,z2,z3.....

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Example

• Let npq719133
• S100
• X01002(mod 133)25
• X1252(mod 133)93
• X2932(mod 133)42
• X3422(mod 133)16
• X4162(mod 133)123
• The OUTPUT1,0,0,1

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RSA generator

• It is a pseudorandom bit generation and is
  cryptographically secure pseudorandom bit
  generation under the assumption that factoring a
  large number n composed of two large prime p and
  q is intractable!

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RSA generator
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Algorithm

1. Generate p and q
2. npq
3. Pich a random integer e 1lteltf and gcd(e, f)1
4. Select a random integer x0 (the seed) in the
   interval 1,n-1
5. For i1 to l
6. Xi xi-1e mod n
7. ZiLSB of xi
8. Return z1,...,zl

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RSA generator

• p and q -gtprime
• n-gtpq
• einteger in 3,?(n)gcd(e,?(n))1

Z i -gtz i-1 e (mod n)
z0
zi
xi
C log log n bit less significative
ii1
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Statistical tests
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Frequency test (monobit test)

• The purpose of this test is to determine whether
  the number of 0s and 1s in a genrator output
  sequence are approximately the same, as would be
  expected for a random sequence.

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Serial test (two-bit test)

• The purpose of this test is to determine whether
  the number of occurrences of 00, 01, 10, and 11
  as subsequences of s are approximately the same,
  as would be expected for a random sequence.

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Poker test

• Lets divide s into k non-overlaping parts each
  of length m. The poker test determines whether
  the sequences of length m each appear
  approximately the same number of times in s, as
  would be expected for a random sequence. Note
  that this test is a generalization of the
  frequency test setting m 1 in the poker test
  yields the frequency test.

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Runs test

• The purpose of the runs test is to determine
  whether the number of runs (of either zeros or
  ones) of various lengths in the sequence s is as
  expected for a random sequence.

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Autocorrelation test

• The purpose of this test is to check for
  correlations between the sequence s and
  (noncyclic) shifted versions of it.

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References

• Handbook Of Applied Cryptography
• A. Menezes
• P. van Oorschot
• S. Vanstone
• www.cacr.math.uwaterloo.ca/hac
• www.ietf.org/rfc/rfc1750.txt

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Thats all folks...