# Questions tagged [blum-blum-shub]

The Blum Blum Shub generator is a deterministic Pseudo-Random Bit Generator with security demonstrably reducible to that of integer factorization.

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### Generating arbitrary state/bit using Blum Blum Shub generaor. Simplification of Carmichael function term?

From wikipedia: 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)} ...
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### Period of Blum-Blum-Shub

Reading about Blum-Blum-Shub, I have found that everyone has stressed the importance of $gcd(p-1, q-1)$ being a small number, as this leads to a large period. I found it really difficult to find a ...
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### Chosen plaintext attack on Blum–Goldwasser cryptosystem

As I was reading on some encryption schemes the other day, I stumbled upon the Blum–Goldwasser cryptosystem and read that it is vulnerable to adaptive chosen ciphertext attack. I searched for some ...
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### Significance of having remainder $3$ when divided by $4$ for both $p$ and $q$ in BBS

In the Blum Blum Shub random number generator, we take two random prime numbers $p$ and $q$ such that both have a remainder of $3$ when divided by $4$. My question is why can't we just take any $2$ ...
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### How big does M need to be in Blum Blum Shub?

I've read that Blum Blum Shub is a CSPRNG, defined by $x_{n+1} = x_n^2 \bmod M$. I didn't understand that, and couldn't find any sources on how big $M$ should be. Are 32 bits enough? 64 bits? Or are ...
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### using non prime numbers in Blum Blum Shub or Blum Micali

I would like to use Blum Blum Shub, or Blum Micali methods as random number generators only. I am not interested in their encryption capabilities. I was wondering, if I replaced the prime numbers ...
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### Are factorization algorithms parallelizable?

I was reading about the Blum-Blum-Shub random number generator, and its security depends on the hardness of factoring very large numbers (like many things in crypto do). I'm just wondering, if I have ...
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### If Blum Blum Shub is modified to use a prime modulus, is it still secure?

The definition of the Blum Blum Shub cryptographically secure pseudorandom number generator is $x=x^2 \mod N$ where $N=p \times q$, $p \in \mathbb P$, and $q \in \mathbb P$. Supposedly, the security ...
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### Why is knowing M not enough to break Blum Blum Shub?

In Blum Blum Shub, the generator is $x_{n+1}={x_n}^2 \mod M$ where $M=p \cdot q$, $p \in \mathbb P$, and $q \in \mathbb P$. Supposedly, knowing $p$ and $q$ is enough to break the system. But if I know ...
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### Exploration of Blum Micali Security By Seed Size

I'm new to cryptography and am most intrigued by mathematically based pseudo random number generators. With reference to the Blum Micali algorithm: $X_{i+1} = G^{X_i} \bmod P$ can security be ...
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### Is BBS used for generation of keys for any modern cryptosystem?

Is there any (current) cryptosystem or digital signature or cryptographic primitive that uses the BBS PRG in order to generate its secret keys?
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### Advantages to knowing $p$ and $q$ in Blum Blum Shub?

Do you gain any advantage by knowing the factorization of $M$ (over just knowing $M$ itself) in the Blum Blum Shub generator? The only advantage I see is being able to calculate the $i$-th number ...
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### Why is $-1$ an illegal message in the Goldwasser-Micali Encryption Scheme

I'm not sure why in the Goldwasser-Micali encryption scheme with a Blum integer $N$, the message $-1$ is always a illegal message. Can you give me some direction for starting? What is illegal ...
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### Blum Blum Shub vs. AES-CTR or other CSPRNGs

Following on from D.W.'s comments on a previous question, what properties does Blum Blum Shub have that make it better / worse than other PRNGs? Are there significant implementation difficulties or ...
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### How many bits can be safely extracted from the BBS generator at each step?

The Blum-Blum-Shub generator is a deterministic Pseudo-Random Bit Generator with security reducible to that of integer factorization. Setup: Secretly chose random primes $P$, $Q$, with \$P\equiv Q\...
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