What is the difference between Ring Signature and Multi User Designated Verifier Signature?

I was going through some text related to designated verifier signature (DVS). I came to know that DVS can be thought of as the two party ring signature. Can we extend this concept and say that ring signature is nothing but multi user DVS.

In the question what do you mean by “Multi User” Designated Verifier Signature is not clear. Do you have a definition in mind?

In the case that by “Multi User Designated Verifier Signature” you mean having multiple signers and one designated verifier (DV), then indeed the generalization that you mentioned is correct and it also adds anonymity to the signers. The reason is that if you consider a ring signature scheme in which one of the signers is a designated verifier, then every signer can sign anonymously and only the designated verifier can verify the validity of the signatures, since it can “simulate” signatures, by using the same signing algorithm. In this scenario the signers are anonymous to the designated verifier, which is different from DVS signatures.

In case that you consider Linkable Ring Signatures (LRS) then the generalization that you have in mind doesn’t hold, because the linking tag plays the role of pseudoidentity which links signatures of the same signer. And since the DV is also a member of the ring, the signatures of the DV are distinguishable from the signers’ signatures.

The combination of DVS and LRS is achieved by Behrouz et al. in DVLRS where the signers can sign messages anonymously, the messages of each signer are linked using the linking tag and the DV can simulate signatures with any linking tag. This was later improved in Balla et al. in UDVLRS, where they achieved unconditional anonymity.

These two schemes indeed provide a kind of MultiDVS functionality (assuming the above definition of MultiDVS). It is not clear, however, if the opposite is also true, i.e., could a MultiDVS scheme provide the ring signatures functionality (probably not, as anonymity towards the DV does not typically hold in a MultiDVS scheme).

I don't quite see the connection you speak of. Briefly, ring-signatures preserve the anonymity of the signing party within a group of parties allowed to sign. Designated-verifier signatures preserve the privacy of the signing party by only allowing a designated verifier to verify the signature. That is they provide a less strict form of the non-repudiation property of classical signature schemes.

In more detail:

In a ring signature we have a set of parties $\{P_i\}$ - each with their keys - any one of which is able to create a signature $\sigma$ of a message $m$. Verification can then be done by anyone (a rather standard assumption when it comes to signature schemes), but there is no way to learn which of the parties issued the signature. Crucially if I receive a message and valid signature, I can show the $(\sigma, m)$ tuple to anyone else, and convince them that it is a valid signature.

In a designated-verifier signature scheme the signing party $S$ designates a single (or multiple) verifying parties $\{V_i\}$, each with their own keys. $S$ then generates a set of signatures $\{\sigma_i\}$ for a fixed message $m$. Any verifier $V_i$ is then able to verify that $\sigma_i$ constitutes a valid signature for $m$ issued by $S$, but is not able to convince anyone else of such.

Now it should certainly seem possible to combine Ring and designated-verifier signatures, and a cursory search did indeed produce papers doing this, such as this one by Behrouz et al. With those you have the combined properties that:

• Given a signature you cannot learn which of the potential signing parties issued it
• Given a signature you cannot convince anyone else that it actually is a valid signature issued by one of the signing parties

Another connection is that ring signatures can seemingly serve as building blocks for designated-verifier signatures, such as in this paper by Man Ho Au and Willy Susilo.