El Gamal signature scheme variant
First, you should not use $p-1$ as modulus. Instead , use a big prime factor $q$ of $p-1$, as it is done in the DSA algorithm. 

The DSA variant  you describe exists as an elliptic curve version. It is called EC-GDSA ( Elliptic Curve German Digital Signature Algorithm) . It is described here: https://www.bsi.bund.de/SharedDocs/Downloads/EN/BSI/Publications/TechGuidelines/TR03111/BSI-TR-03111_pdf.pdf?__blob=publicationFile