# 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

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

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

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 xn 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

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.

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

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

### Citations

1. ^ a b Blum, Blum & Shub 1986, pp. 364–383.
2. ^ Sidorenko, Andrey; Schoenmakers, Berry (2005). "Concrete Security of the Blum-Blum-Shub Pseudorandom Generator". Cryptography and Coding. 3796: 355–375. doi:10.1007/11586821_24.