A solution to using the CRTP with private members.

crtp.cpp

/*
Copyright (C) 2019 Sergey Tomilin

Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the
"Software"), to deal in the Software without restriction, including
without limitation the rights to use, copy, modify, merge, publish,
distribute, sublicense, and/or sell copies of the Software, and to
permit persons to whom the Software is furnished to do so, subject to
the following conditions:

The above copyright notice and this permission notice shall be
included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
*/

namespace CRTP
{
	template <template <typename, typename...> class _Base, typename _Derived, typename... _BaseArgs>
	struct Friend { using Base = _Base<_Derived, _BaseArgs...>; };

	template <template <typename, typename...> class _Base, typename ..._BaseArgs>
	struct Base
	{
		template <template <typename...> class _Derived, typename... _DerivedArgs>
		struct Derived : public _Base<_Derived<_DerivedArgs...>, _BaseArgs...> {};
	};
}

//Templated variadic base
template <typename Derived, typename...>
struct Interface
{
	using CRTP = Interface; //Magic!
	void f() { static_cast<Derived*>(this)->f(); }
};

//Simple usage of the base with extra types
//This can only be used when the derived is NON templated
class A : public Interface<A, void>
{
	friend CRTP;
	void f() {}
};

//Templated derived with extra, non-dependant types, passed to the base
//The arguments passed to CRTP::Base::Derived<, ARGS> must exactly match
//	the template
template <typename T, typename... Args>
class B : public CRTP::Base<Interface, void>::Derived<B, T, Args...>
{
	friend typename CRTP::Friend<Interface, B, void>::Base;
	void f() {}
};

//Templated derived with extra dependant types passed to the base
//Notice the addition of the "template" keyword
template <typename... Args>
class C : public CRTP::Base<Interface, Args...>::template Derived<C, Args...>
{
	friend typename CRTP::Friend<Interface, C, Args...>::Base;
	void f() {}
};

template <typename Fake = void>
class D : public CRTP::Base<Interface>::Derived<D, Fake>
{
	friend typename CRTP::Friend<Interface, D>::Base;
	void f() {}
};


template <typename... Args>
void invoke(Interface<Args...> & base)
{
	base.f();
}

int main(int, char *[])
{
	{
		A derived;

		//Direct invocation through cast to base (derived.f() is private)
		static_cast<A::CRTP &>(derived).f();

		//Invocation through template function accepting the base
		invoke(derived);
	}

	{
		B<int> derived;
		static_cast<B<int>::CRTP &>(derived).f();
		invoke(derived);
	}

	{
		C<void> derived;
		static_cast<C<void>::CRTP &>(derived).f();
		invoke(derived);
	}

	{
		D<void> derived;
		static_cast<D<>::CRTP &>(derived).f();
		invoke(derived);
	}

	return 0;
}