permissions to work with fopen

Dear Sirs, I am planning to make a log file with fopen command. I am succesfully read and write the file with fopen. But as I see fopen is requiring a file with read and write permission for public. I was using same method with perl in cgi-bin directory; but I was giving read and write permission for only owner. If I give write and read permission for only owner; fopen command can not read the file. Do you have any idea, how can I make log file with protecting from public? Best Regards