# SIS vs LWE Problem

The Ajtai one way function is defined by

$$f_A(x)= Ax \; mod\; q$$ where the x $$\in \{0,1\}^m$$ and A $$\in \mathbb{Z_q}^{n \times m}$$. $$f_A(x)$$ is one way function ( Ajtai 96)

While the Regev One way function(Regev 05) is defined over x $$\in \mathbb{Z_q}^k$$ and $$e \in \mathscr{E}^m$$ and A $$\in \mathbb{Z_q}^{m \times k}$$ .The one way function is defined as

$$g_A(x,e) =Ax +e \; mod\; q \; (LWE)$$

$$g_A(x,e)$$ is a One-way function. My question is does Regev's One way function provide advantage over Ajtai One way function in terms of designing new schemes of encryption or are they equivalent with respect to their use cases? Also are they equivalent with respect to hardness?

Regarding Hardness, solving SIS over $$A^t$$ quite directly allows to solve LWE over $$A$$. In the other direction there is also a reduction which is quantum. So, at least to quantum computers, the problems are equivalent.