How difficult would it be to find valid answers for this hash arrangement?

Question

If $A$ is a 160-bit number, and $X \& Y$ are two SHA-1 hashes, to be generated such that the 320-bit number $X\mathbin\|A$ hashed to $Y$, and the 320-bit number $A \mathbin\| Y$ hashed to $X$?

How hard is it to find pairs of values of $X\&Y$ that would fulfill the above criteria? How many valid pairs of $X \& Y$ would one expect to find for an arbitrary value of $A$?

Answer 1

If you are interested in attack for SHA-1, the best-known attack on SHA-1 is SHAttered.

SHAttered paper is an identical-prefix collision attack, where a given prefix $p$ is extended with two distinct near-collision block pairs such that they collide for any suffix S:

Specifically, the attack takes a prefix $p$ and finds two pairs of message blocks $(m_0, m_1) \ne (m'_0, m'_1)$ such that $$\operatorname{SHA-1}(p \mathbin\Vert m_0 \mathbin\Vert m_1 \mathbin\Vert s) = \operatorname{SHA-1}(p \mathbin\Vert m'_0 \mathbin\Vert m'_1 \mathbin\Vert s)$$ for any suffix $s$.

The computational effort spent on our attack is estimated to be equivalent to $2^{63.1}$ SHA-1 calls

Also, a recent article might interest you;

• From Collisions to Chosen-Prefix Collisions Application to Full SHA-1

We might close to signature forgeries.