Alessandro Raffa

In some cases you want your wordpress users to login with their email address instead of the username. To do this, you can use a good plugin like ‘WP Email Login’ or you can activate a ‘Register Plus Redux’ plugin’s option, but when the users will ask for lost password, the old / original username will be emailed to them, and this can cause confusion. To avoid this situation, just install and activate Login Email Sync, and it will sync the username ( user_login database field ) with the email ( user_email database field ) every time a user updates her/his profile.
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Login Email Sync

In some cases you want your wordpress users to login with their email address instead of the username. To do this, you can use a good plugin like &#03 …

curlybracket

This plugin will allow you to automatically rehash the passwords of users you have beforehand imported from a Prestashop database to your WordPress database.
This is useful if you are migrating a website from Prestashop to WordPress.
It allows users to authenticate via their email address, like on a Prestashop installation.
Upon login, the plugin will check the password entered by the user, against the value stored in the database. Using the COOKIE_KEY of the former Prestashop installation, it’ll verify if the password is correct and, only in that case, it’ll rehash the password using WordPress’ default hashing algorithm and insert the new password to the database. Then it’ll try logging in.
In every other case, login will function normally, so if there is an error, it’ll be returned.
In a while, every former Prestashop user you have imported to your WP DB will have their passwords securely rehashed.
First tested successfully with Prestashop 1.3.5 and WordPress 3.6. 
Lately tested successfully with Prestashop 1.5.6.1 and WordPress 3.9.

Prestashop User Compatibility

This plugin will automatically rehash the passwords of users you have beforehand imported from a Prestashop database to your WP database.

Login Email Sync vs Prestashop User Compatibility