gistfile1.cs

//CustomMembershipProvider.cs
using System;
using System.Collections.Generic;
using System.Linq;
using System.Web;
using CMS.CMSHelper;
using CMS.GlobalHelper;

public class CustomMembershipProvider : CMS.MembershipProvider.CMSMembershipProvider
{
    public override bool ValidateUser(string username, string password)
    {
        if (ValidationHelper.IsEmail(username))
        {
            var customUsername = base.GetUserNameByEmail(username);
            if (!string.IsNullOrEmpty(customUsername))
                username = customUsername;
        }
        

        return base.ValidateUser(username, password);
    }
}


//CustomUserInfoProvider.cs
using System;
using System.Collections.Generic;
using System.Linq;
using System.Web;
using CMS.SiteProvider;
using CMS.GlobalHelper;
using CMS.MembershipProvider;

public class CustomUserInfoProvider : UserInfoProvider
{
    protected override UserInfo GetUserInfoInternal(string userName)
    {
        if (ValidationHelper.IsEmail(userName))
        {
            var membershipProvider = new CMSMembershipProvider();
            var customUserName = membershipProvider.GetUserNameByEmail(userName);
            if (!string.IsNullOrEmpty(customUserName))
                userName = customUserName;
        }

        return base.GetUserInfoInternal(userName);
    }
}

//CMSAppBase.cs
/// <summary>
/// Application start event handler.
/// </summary>
public static void CMSApplicationStart()
{
#if DEBUG
    // Set debug mode
    SystemHelper.IsWebProjectDebug = true;
#endif

    UserInfoProvider.ProviderObject = new CustomUserInfoProvider();
}

//Web.config
<!--<add name="CMSProvider" type="CMS.MembershipProvider.CMSMembershipProvider" connectionStringName="CMSConnectionString" enablePasswordRetrieval="false" enablePasswordReset="true" requiresQuestionAndAnswer="false" requiresUniqueEmail="true" passwordFormat="Hashed"/>-->
<add name="CMSProvider" type="CustomMembershipProvider" connectionStringName="CMSConnectionString" enablePasswordRetrieval="false" enablePasswordReset="true" requiresQuestionAndAnswer="false" requiresUniqueEmail="true" passwordFormat="Hashed"/>