Blum Blum Shub

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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

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

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):

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

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

Security[edit]

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.

The performance of the BBS random-number generator depends on the size of the modulus M and the number of bits per iteration j. While lowering M or increasing j makes the algorithm faster, doing so also reduces the security. A 2005 paper gives concrete, as opposed to asymptotic, security proof of BBS, for a given M and j. The result can also be used to guide choices of the two numbers by balancing expected security against computational cost.^[2]

Example[edit]

Let ${\displaystyle p=11}$, ${\displaystyle q=23}$ and ${\displaystyle s=3}$ (where ${\displaystyle s}$ is the seed). We can expect to get a large cycle length for those small numbers, because ${\displaystyle {\rm {gcd}}((p-3)/2,(q-3)/2)=2}$. The generator starts to evaluate ${\displaystyle x_{0}}$ by using ${\displaystyle x_{-1}=s}$ and creates the sequence ${\displaystyle x_{0}}$, ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$, ${\displaystyle \ldots }$ ${\displaystyle x_{5}}$ = 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.

The following is a Python implementation that does check for primality.

import sympy def blum_blum_shub(p1, p2, seed, iterations):   assert p1 % 4 == 3   assert p2 % 4 == 3   assert sympy.isprime(p1//2)   assert sympy.isprime(p2//2)   n = p1 * p2   numbers = []   for _ in range(iterations):     seed = (seed ** 2) % n     if seed in numbers:       print(f"The RNG has fallen into a loop at {len(numbers)} steps")       return numbers     numbers.append(seed)   return numbers  print(blum_blum_shub(11, 23, 3, 100)) 

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)   "Returns the number of 1-valued bits in the integer-encoded BITS."   (declare (type (integer 0 *) bits))   (the (integer 0 *) (logcount bits)))  (defun get-even-parity-bit (bits)   "Returns the even parity bit of the integer-encoded BITS."   (declare (type (integer 0 *) bits))   (the bit (mod (get-number-of-1-bits bits) 2)))  (defun get-least-significant-bit (bits)   "Returns the least significant bit of the integer-encoded BITS."   (declare (type (integer 0 *) bits))   (the bit (ldb (byte 1 0) bits)))  (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 number x[n+1],    (2) the even parity bit of the number,    (3) the least significant bit of the number.    ---    Please note that the parameters P, Q, and S are not checked in    accordance to the conditions described in the article."   (declare (type (integer 0 *) p q s))   (let ((M    (* p q))       ;; M  = p * q         (x[n] s))            ;; x0 = seed     (declare (type (integer 0 *) M x[n]))     #'(lambda ()         ;; x[n+1] = x[n]^2 mod M         (let ((x[n+1] (mod (* x[n] x[n]) M)))           (declare (type (integer 0 *) 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])))             (declare (type bit even-parity-bit))             (declare (type bit least-significant-bit))             ;; Update the state such that x[n+1] becomes the new x[n].             (setf x[n] x[n+1])             (values x[n+1]                     even-parity-bit                     least-significant-bit))))))  ;; Print the exemplary outputs. (let ((bbs (make-blum-blum-shub :p 11 :q 23 :s 3)))   (declare (type (function () (values (integer 0 *) bit bit)) bbs))   (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 (x[n+1] even-parity-bit least-significant-bit)         (funcall bbs)       (declare (type (integer 0 *) x[n+1]))       (declare (type bit           even-parity-bit))       (declare (type bit           least-significant-bit))       (format T "~&~6d | ~d | ~d"                 x[n+1] even-parity-bit least-significant-bit)))) 

References[edit]

Citations[edit]

Sources[edit]

External links[edit]

• GMPBBS, a C-language implementation by Mark Rossmiller
• BlumBlumShub, a Java-language implementation by Mark Rossmiller
• An implementation in Java
• Randomness tests