Group of quadratic residue over Blum integer

Sean

10/29/22, 12:53 AM

Let $x$ be a random element from $QR_n$, the quadratic residue group over Blum integer n (where $n=p*q$ and $p$ and $q$ are safe primes), and $g$ a generator of $QR_n$. Are the following computationally indistinguishable?

$$(x^2 \mod n, g^x) (r^2 \mod n, g^x)$$

The intuition is that it's hard to compute $x$ from $x^2$ and $g^x$. Could this be reduced to some standard assumptions?

0 + 4

hardness-assumptions

Ievgeni

11/1/22, 8:08 AM

Does it miss a $\mod$ ?

Sean

11/1/22, 12:29 PM

That is right. I've made the corrections.

Fractalice

11/1/22, 12:48 PM

How is $x \in QR_n$ is represented? E.g. If it is sampled uniformly from $[0; n\cdot ord(g))$, then these are indistinguishable since $x^2$ and $g^x$ are independent (because $x \mod n$ and $x \mod ord(g)$ are independent).

Sean

11/1/22, 7:55 PM

Thanks very much for the insights? What about x^2 \mod \totient(n) is given. Then I guess the argument wouldn't apply?

Elon Musk

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

EN: Group of quadratic residue over Blum integer

TH: กลุ่มของเศษส่วนกำลังสองส่วนจำนวนเต็ม Blum

RO: Grup de reziduuri pătratice peste întregul Blum

RU: Группа квадратичного вычета над целым числом Блюма

VI: Nhóm dư lượng bậc hai trên số nguyên Blum

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