How to specify last $t$ bits are only sent when a signature is sent? I have read that in cases where there is a limit on the amount of data sent over the network, one idea is to sent only $t$ bits of data rather than sending all the bits. For example, in the Schnorr scheme this can be employed.

My question is, whether this can be specified by sending the $t$ bits along with the other information?
In general, this depends entirely on the signature scheme in question; usually however, the answer is that truncation is generally not possible.

For very short signatures, depending on your risk appetite for other kinds of somewhat novel public-key cryptography, you may find success with pairing-based BLS signatures or multivariate cryptography (GeMSS comes to mind).



For Schnorr signatures in particular, you generally either send:


the result of applying the group operation with the nonce $r$, $x = \alpha^r$ (or $[r]G$ in the elliptic curve case) and the scalar $y$, or
the tuple of the hash value $e=h(\overline{x}, m)$ and the scalar $y$.


EdDSA, a relative of Schnorr signatures, sends $([r]G, y)$ and thus doesn't lend itself to shortening unless you change your implementation to send $(e, y)$ instead.

If you send $(e, y)$, you can essentially truncate $e$ to an arbitrary extent in your scheme, but the security of the scheme also sinks by that much. Schnorr's original paper proposed using a $t$-bit hash for a $2t$-bit group order. The academic community still isn't sure whether this is a good idea, however, see e.&thinsp;g. Jeremiah Blocki and Seunghoon Lee. On the Multi-User Security of Short Schnorr Signatures, 2019 and Daniel R.&nbsp;L.&nbsp;Brown. Short Schnorr signatures require a hash function with more than just random-prefix resistance, 2015 and their references.