Score:2

Crypto

Entropy of SIM PIN code

Mohammadsadeq Borjiyan

3/14/23, 12:51 PM

Each mobile SIM card has a four-digit number ($b_1$,$b_2$,$b_3$,$b_4$) called PIN code. Each digit $0 \le b_i \le 9$ (for i = 1, 2, 3, 4) is generated using a random 16-bit sequence as follows: $b_i=(r_{4i-3} + r_{4i-2} .2 + r_{4i-1}.2^2 + r_{4i}.2^3)\pmod {10} $. How we can calculate the antropy of PIN code? I know the entropy relation but I have no view.

154

0 + 0

entropy

kelalaka

3/14/23, 3:50 PM

Clearly HW, What have you tried?

Score:1

Crypto

Ievgeni

3/14/23, 1:24 PM

I'm calling here $B_1, B_2, B_3, B_4$ the four random variables representing the four digits (I do not like to call a variable $p$). It seems a good idea to compute the entropy of only $1$ digits, and then because the four digits are independently chosen, we could multiply this number by $4$. Let $q_i= \mathbb{P}(B_1 = i)$ for any $i \in\{0, \dots, 9\}$. $$H(B_1)= -\sum_{i=0}^9 q_i\log(q_i)$$.

You can notice for $0\leq j \leq 5$, $q_j =\frac{2}{16}$, and for $6\leq j\leq9$, $q_j= \frac{1}{16}$. Then, because $\log_2(\frac{2}{16})= (1-4)$, and $\log_2(\frac{1}{16})= (-4)$. \begin{align} H(B_1)&= -\left(6\cdot \frac{2}{16}(1-4) + 4\cdot \frac{1}{16}(-4)\right) \\ &=\frac{9+4}{4}= \frac{13}{4} = 3.25 \end{align}

After multiplying by $4$, because there is $4$ digits (as I've said before), we obtain $13$ bits of entropy.

0 + 0

Paul Uszak

3/14/23, 1:26 PM

Just looking at this. Are you convinced that $p_i$ is smooth and continuous? There are no gaps? And are they IID; they come from a formula...

Ievgeni

3/14/23, 1:32 PM

I've deleted one sentence does it make more sense for you?

Paul Uszak

3/14/23, 2:18 PM

Oh I'm not criticising. I'm suspicious of the formulation of $p_i$. I am of all constructs that have implications for the security services. Four truly random digits have H = 13.3 bits ($\log_2(10) \times 4$). Seed $r$ is 16 bits. 16 > 13.3, so why not directly use $r$ and rejection sample down? It whiffs.

kelalaka

3/14/23, 3:54 PM

Nobody generates such probabilistic random pins. HW.

fgrieu

3/14/23, 4:48 PM

Since this is homework, I will let it to the OP to fix a slight error.

Ievgeni

3/14/23, 4:50 PM

I've corrected the error...

kelalaka

3/14/23, 6:31 PM

Still, there is a mistake from first-class of rational numbers.

Mohammadsadeq Borjiyan

3/15/23, 5:32 PM

Dear Paul Uszak, thank you for your consideration. Yes, this scheme in comparison to a four-digit random number is some unreasonable, but the purpose was comparison of the entropy of two designs.

Mohammadsadeq Borjiyan

3/15/23, 5:35 PM

Dear levgeni, I was very thankful you are considering my problem.

fgrieu

3/15/23, 5:59 PM

For comparison: a uniformly random 10-digit PIN has ≈13.2877 bit of entropy, and generating the PIN by reducing the 16-bit value modulo 10000 yields ≈13.2835 bit of entropy.

Paul Uszak

4/26/23, 6:39 AM

I repeat: are these values independently distributed? If not, the entropy will be (much) lower. Some Python is required.

Score:0

Crypto

Mohammadsadeq Borjiyan

3/15/23, 6:10 PM

For $0 \le b_i \le 5$, $p_i=\frac{2}{16}$ and for $6 \le b_i \le 9$, $p_i=\frac{1}{16}$. Then we calculate the logarithm values, which are respectively equal to $\log_2(\frac{2}{16})=(1-4)$ and $\log_2(\frac{1}{16})=(-4)$. Now, according to the entropy relation, \begin{align} \ H(b_1) &= -\ \sum_{i=0}^{9} p_i \log_2(p_i ), \end{align} we will have:

\begin{align} \ H(b_1) &= \ -(6 \cdot \frac{2}{16}(1-4) + 4 \cdot \frac{1}{16}(-4))=3.25, \end{align} which is the entopy of one digit. As there are four independent digits, multiply the value by 4 and we get 13 bits of entropy.

0 + 0

fgrieu

3/15/23, 7:52 PM

Watch the sign, and the exclusion of indices $j$ with zero probability. It's $H(b_1)=\displaystyle\sum_{j=0\ldots9\text{ and }p_j>0}p_j \log_2(1/p_j)$ or equivalently $H(b_1)=-\displaystyle\sum_{j=0\ldots9\text{ and }p_j>0}p_j \log_2(1/p_j)$.

kelalaka

3/15/23, 10:58 PM

@fgrieu on the right, do you mean $\log_2(p_j)$? This comment shouts: delete me if I'm correct.

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: Entropy of SIM PIN code

TH: เอนโทรปีของรหัส PIN ของซิม

RO: Entropia codului PIN SIM

RU: Энтропия PIN-кода SIM-карты

VI: Entropy của mã PIN của SIM

Post an answer

Most people don’t grasp that asking a lot of questions unlocks learning and improves interpersonal bonding. In Alison’s studies, for example, though people could accurately recall how many questions had been asked in their conversations, they didn’t intuit the link between questions and liking. Across four studies, in which participants were engaged in conversations themselves or read transcripts of others’ conversations, people tended not to realize that question asking would influence—or had influenced—the level of amity between the conversationalists.