Does assigning DOI numbers signal anything about the authenticity of a journal? Why do some journals not assign them? I have seen few journals which are indexed in well known databases like Zentralblatt MATH, AMS Digital Mathematics, CiteFactor but still does not assign DOI numbers to published articles, what could be the reason? Is this a common thing ? Does DOI number add any value to authenticity of journal ?
Most importantly which purpose does this DOI number serve ?

I am asking this because I have heard if journal do not assign DOI number to its published articles then it could be low profile journal or even could be fake.
The similar question has been asked in Why do some papers not have a DOI?, but my query related to quality/authenticity of journal could not be resolved.

PS. My question is related to authenticity of journal "Computational Methods for Differential Equations" published by university of Tabriz, Iran.
DOI, by itself, certainly does not add "authenticity". 

The Digital Object Identifier is, by design, meant to be a method of producing and providing a persistent actionable identifier to entities. To put it in plain English: the DOI is an effort to prevent link rot, and importantly, does so by not only providing a registry of the numbered items, but also providing a mechanism for accessing/retrieving those items. The Wikipedia Article on DOI also explains pretty well what it is for. Notice that DOIs can be had for many more than just journal articles. In science, a growing practice is to use DOI referring to datasets. 

As to why a publisher may decide not to participate in the DOI scheme: for starters, it does cost money. This may have contributed to the correlation that sham publishers typically do not make use of the DOI service.