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iliyas bagwaniliyas bagwan 

Suggest the api to register contact and save with password and login using same credentials

User should able to register himself/herself and able to login using the credentials like email and password provided in the registration.

Please,
Suggest API
VinayVinay (Salesforce Developers) 
Why API is required you can use salesforce communities.

https://help.salesforce.com/articleView?id=networks_customize_login.htm&type=5

Thanks,