Authentication of data using XOR and a secure function
It can be.

Suppose $y_i = F_K(1 \mathbin\| i \mathbin\| x_i)$ and $t = \alpha \oplus F_K(0 \mathbin\| n)$ is the authentication tag, for some PRF $F_K$ and per-message nonce $n$.  The security of this structure is called XMACC and analyzed by Bellare, Guérin, and Rogaway, ‘XOR MACs: New Methods for Message Authentication Using Finite Pseudorandom Functions’, Proceedings of CRYPTO '95, Springer LNCS 963, 1995.

Suppose $y_i = x_i {K_0}^{i + 1}$ and $t = \alpha + K_1$ with arithmetic in $\operatorname{GF}(2^{128})$, where $K_0, K_1 \in \operatorname{GF}(2^{128})$ are uniform random, and $K_1$ is only ever used once.  This is the structure of GHASH, which is used by AES-GCM in Carter–Wegman mode with $K_0 = \operatorname{AES}_K(0)$ and $K_1 = \operatorname{AES}_K(n)$ for a per-message nonce $n$.