# Is $X^{E} \equiv X \;(\text{mod}\; n)$ a potential problem in the RSA cryptosystem?

This is Exercises 11 from Section 7.3 of the book "Abstract Algebra: Theory and Applications" (aata-20180801) by Thomas W. Judson.

Find integers $$n$$, $$E$$, $$X$$ such that $$X^{E} \equiv X \;(\text{mod}\; n).$$ Is this a potential problem in the RSA cryptosystem?

What does the "potential problem" mean in this problem? Is it related to the Iterated Encryption Attack described in this article?

• Imagine that the secret message you wanted to encrypt happened to be equal to $X$, what would it get encrypted to? – Matthew Towers May 6 at 9:26

Assume that you find integers $$n$$, $$E$$, $$X$$ such that $$X^{E} \equiv X \;(\text{mod}\; n).$$ Then;

$$X^{E} -X = k \cdot n \quad\text{ for some } k\in \mathbb{Z}.$$

Take out $$X$$

$$X ( X^{E-1} -1) = k \cdot n \quad (= k \cdot p \cdot q)$$

Now, calculate $$\gcd(X,n)$$ and $$\gcd(X^{E-1} -1,n)$$.

If you are lucky, you might find $$p$$ and $$q$$.

• You might just find factors of $k$ instead... – Henno Brandsma May 9 at 21:51
• @HennoBrandsma thanks, I think, in the worst case $X$ is equal to a divisor $t$ of $k$ and $X^{E-1}-1,n$ is equal to $tn$. – kelalaka May 10 at 13:13