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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?

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There are important constraint in the parameters for Ajtai's function, that makes it highly surjective (each image has many preimages). We do not know how to get an encryption scheme from that.

On the contrary, Regev's one is typically used in an injective regime. And we do know how to build and encryption scheme from it.

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.

This should be taken with a grain of salt: there are huge losses in this reduction. In particular, LWE is with some huge parameters provably quantumly harder than SIS with certain more reasonable parameter.

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