  We define a pseudorandom function (PRF) $F: \mathcal{K} \times \mathcal{X}
\rightarrow \mathcal{Y}$ to be bi-homomorphic when it is fully Key homomorphic
and partially Input Homomorphic (KIH), i.e., given $F(k_1, x_1)$ and $F(k_2,
x_2)$, there is an efficient algorithm to compute $F(k_1 \oplus k_2, x_1
\ominus x_2)$, where $\oplus$ and $\ominus$ are (binary) group operations. The
homomorphism on the input is restricted to a fixed subset of the input bits,
i.e., $\ominus$ operates on some pre-decided $m$-out-of-$n$ bits, where $|x_1|
= |x_2| = n$, and the remaining $n-m$ bits are identical in both inputs. In
addition, the output length, $\ell$, of the operator $\ominus$ is not fixed and
is defined as $n \leq \ell \leq 2n$, hence leading to Homomorphically induced
Variable input Length (HVL) as $n \leq |x_1 \ominus x_2| \leq 2n$. We present a
learning with errors (LWE) based construction for a HVL-KIH-PRF family. Our
construction is inspired by the key homomorphic PRF construction due to
Banerjee and Peikert (Crypto 2014).
  An updatable encryption scheme allows rotations of the encryption key, i.e.,
moving existing ciphertexts from old to new key. These updates are carried out
via \emph{update tokens}, which can be used by an untrusted party since the
update procedure does not involve decryption of the ciphertext. We use our
novel PRF family to construct an updatable encryption scheme, named QPC-UE-UU,
which is quantum-safe, post-compromise secure and supports unidirectional
ciphertext updates, i.e., the update tokens can be used to perform ciphertext
updates but they cannot be used to undo already completed updates. Our PRF
family also leads to the first left/right key homomorphic constrained-PRF
family with HVL.
