Blum Blum Shub

Blum Blum Shub (B.B.S.) is a pseudorandom number generator proposed in 1986 by Lenore Blum, Manuel Blum and Michael Shub[1] that is derived from Michael O. Rabin's one-way function.

Blum Blum Shub takes the form

x_{n+1}=x_{n}^{2}{\bmod {M}}

,

where M = pq is the product of two large primes p and q. At each step of the algorithm, some output is derived from x_n+1; the output is commonly either the bit parity of x_n+1 or one or more of the least significant bits of x_n+1.

The seed x₀ should be an integer that is co-prime to M (i.e. p and q are not factors of x₀) and not 1 or 0.

The two primes, p and q, should both be congruent to 3 (mod 4) (this guarantees that each quadratic residue has one square root which is also a quadratic residue), and should be safe primes with a small gcd((p-3)/2, (q-3)/2) (this makes the cycle length large).

An interesting characteristic of the Blum Blum Shub generator is the possibility to calculate any x_i value directly (via Euler's theorem):

x_{i}=\left(x_{0}^{2^{i}{\bmod {\lambda }}(M)}\right){\bmod {M}}

,

where $\lambda$ is the Carmichael function. (Here we have $\lambda (M)=\lambda (p\cdot q)=\operatorname {lcm} (p-1,q-1)$ ).

Security

There is a proof reducing its security to the computational difficulty of factoring.[1] When the primes are chosen appropriately, and O(log log M) lower-order bits of each x_n are output, then in the limit as M grows large, distinguishing the output bits from random should be at least as difficult as solving the Quadratic residuosity problem modulo M.

Example

Let $p=11$ , $q=23$ and $s=3$ (where $s$ is the seed). We can expect to get a large cycle length for those small numbers, because ${\rm {gcd}}((p-3)/2,(q-3)/2)=2$ . The generator starts to evaluate $x_{0}$ by using $x_{-1}=s$ and creates the sequence $x_{0}$ , $x_{1}$ , $x_{2}$ , $\ldots$ $x_{5}$ = 9, 81, 236, 36, 31, 202. The following table shows the output (in bits) for the different bit selection methods used to determine the output.

Parity bit	Least significant bit
0 1 1 0 1 0	1 1 0 0 1 0

The following Common Lisp implementation provides a simple demonstration of the generator, in particular regarding the three bit selection methods. It is important to note that the requirements imposed upon the parameters p, q and s (seed) are not checked.

(defun get-number-of-1-bits (bits)
  "Counts and returns the number of 1-valued bits in the BITS."
  (declare (integer bits))
  (loop for bit-index from 0 below (integer-length bits)
        when (logbitp bit-index bits) sum 1))

(defun get-even-parity-bit (number)
  (declare (integer number))
  (mod (get-number-of-1-bits number) 2))

(defun get-least-significant-bit (number)
  (declare (integer number))
  (ldb (byte 1 0) number))

(defun make-blum-blum-shub (&key (p 11) (q 23) (s 3))
  "Returns a function of no arguments which represents a simple
   Blum-Blum-Shub pseudorandom number generator, configured to use the
   generator parameters P, Q, and S (seed), and returning three values:
   (1) the even parity bit of the number,
   (2) the least significant bit of the number,
   (3) the number x[n+1].
   ---
   Please note that the parameters P, Q, and S are not checked in
   accordance to the conditions described in the article."
  (let ((M    (* p q))       ;; M  = p * q
        (x[n] s))            ;; x0 = seed
    (declare (integer p q M x[n]))
    #'(lambda ()
        ;; x[n+1] = x[n]^2 mod M
        (let ((x[n+1] (mod (* x[n] x[n]) M)))
          (declare (integer x[n+1]))
          ;; Compute the random bit(s) based on x[n+1].
          (let ((even-parity-bit       (get-even-parity-bit       x[n+1]))
                (least-significant-bit (get-least-significant-bit x[n+1])))
            ;; Update the state such that x[n+1] becomes the new x[n].
            (setf x[n] x[n+1])
            (values even-parity-bit
                    least-significant-bit
                    x[n+1]))))))

;; Print the exemplary outputs.
(let ((bbs (make-blum-blum-shub :p 11 :q 23 :s 3)))
  (format T "~&Keys: E = even parity, ~
                     L = least significant")
  (format T "~2%")
  (format T "~&x[n+1] | E | L")
  (format T "~&------------------")
  (loop repeat 6 do
    (multiple-value-bind (even-parity-bit odd-parity-bit
                          least-significant-bit x[n+1])
        (funcall bbs)
      (format T "~&~6d | ~d | ~d"
        x[n+1] even-parity-bit least-significant-bit))))

References

Blum, Lenore; Blum, Manuel; Shub, Mike (1 May 1986). "A Simple Unpredictable Pseudo-Random Number Generator". SIAM Journal on Computing. 15 (2): 364–383. doi:10.1137/0215025.

General

Blum, Lenore; Blum, Manuel; Shub, Mike (1982). "Comparison of Two Pseudo-Random Number Generators". Advances in Cryptology: Proceedings of CRYPTO '82. Plenum: 61–78. Cite journal requires |journal= (help)
Geisler, Martin; Krøigård, Mikkel; Danielsen, Andreas (December 2004). "About Random Bits". CiteSeerX 10.1.1.90.3779. Cite journal requires |journal= (help) available as PDF and gzipped Postscript

External links

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[blum1986-1] Blum, Lenore; Blum, Manuel; Shub, Mike (1 May 1986). "A Simple Unpredictable Pseudo-Random Number Generator". SIAM Journal on Computing. 15 (2): 364–383. doi:10.1137/0215025.