First Open Firefox and click "Tools" top of Firefox and click Option you see a window, Click "General" . Type home page url you want in "Home Page" box and click "Ok" you are done.
When using a Google Custom Search Engine and a Google Account, the respective privacy policies of Google apply. The Google Custom Search Engine Definiton - that is, the definition which sites to search and how to rank pages - can be hosted by Google itself (so called "hosted CSE") or by the third party website quicksear.ch (so called "linked CSE") which provides this as an anonymous service. For a Google hosted CSE, a personal Google Account is required.
For linked Google Custom Search Engine Definitions hosted by quicksear.ch, no registration is required at all. For this reason, quicksear.ch is used by default in this latest version of the Custom Google Search extension.It is technically impossible to provide a linked custom google search engine without making the associated search engine definition available publicly on the internet. So everyone who knows your search engine link can actually retrieve your search engine definition.
Technically, there is no way around this. If you want to keep its definition private, you should choose a Google hosted CSE. Quicksear.ch is an anonymous service it does not require you to provide information which allows to identify you, such as your name.
Quicksear.ch stores only the information required to provide the service, and will use it only to provide the service - the stored information will never be sold to third parties or used for personalized marketing purposes or alike. The Google Custom Search results are provided directly by Google - quicksear.ch stores only the customization information you provide. Search requests going to a quicksear.ch hosted Custom Search Engine are routed through a quicksear.ch server.
This is done in order to ensure that the latest version of the custom search engine definition file gets used, so that updates to the Custom Search Engine can be seen immediately. Without this, changes would only become active once google dismisses the cached version of the linked CSE. Quicksear.ch does not store search terms that have been searched for, nor any other personally identifiable information about any searches you make.
If you do not like to use quicksear.ch, you are still free to use Google hosted Custom Search Engines.