mod_rewrite and htaccess password protection together - .htaccess

Question

I am currently trying to combine the two - htaccess password protection and mod_rewrite. My problem is, that the code below is resulting an error: "No input file specified."
<IfModule mod_rewrite.c>
Options +FollowSymLinks -MultiViews
DirectoryIndex index.php
AuthType Basic
AuthName "Password Protected Area"
AuthUserFile .htpasswd
Require valid-user
RewriteEngine on
RewriteCond %{HTTP_HOST} !^www\.
RewriteRule ^(.*)$ http://www.%{HTTP_HOST}/$1 [R=301,L]
RewriteRule ^$ public/ [L]
RewriteRule (.*) public/$1 [L]
</IfModule>
Can anyone see, what I am doing wrong?
EDIT #1:
<IfModule mod_rewrite.c>
Options +FollowSymLinks -MultiViews
DirectoryIndex index.php
AuthType Basic
AuthName "Password Protected Area"
AuthUserFile /.htpasswd
Require valid-user
RewriteEngine on
RewriteCond %{HTTP_HOST} !^www\. [NC]
RewriteRule ^ http://www.%{HTTP_HOST}%{REQUEST_URI} [R=301,L,NE]
RewriteRule ^((?!public/).*)$ public/$1 [L,NC]
</IfModule>

Answer 1

Make sure AuthUserFile has full path to a valid encrypted passwords file.
Rest of the code, have it like this:
<IfModule mod_rewrite.c>
Options +FollowSymLinks -MultiViews
DirectoryIndex index.php
AuthType Basic
AuthName "Password Protected Area"
AuthUserFile /full/path/to/.htpasswd
Require valid-user
RewriteEngine on
RewriteCond %{HTTP_HOST} !^www\. [NC]
RewriteRule ^ http://www.%{HTTP_HOST}%{REQUEST_URI} [R=301,L,NE]
RewriteRule ^((?!public/).*)$ /public/$1 [L,NC]
</IfModule>