Enforcement First Immigration Reform Act of 2005 - Amends the Immigration and Nationality Act and other immigration-related provisions with respect to: (1) enforcement and detention of, and criminal penalties for, illegal aliens; (2) the institutional removal (IRP) and criminal alien assistance (SCAAP) programs; (3) alien smuggling, document fraud, gang violence, and drug trafficking; (4) border security, personnel increases, and border-related assignment of armed forces members; (5) provision of social security numbers; (6) work authorization and enforcement; (7) secure identification standards, including birth certificates; (8) reform of legal immigration, including elimination of the diversity lottery visa category and certain limitations on entrants from Mexico; (9) citizenship reform, including limitations on citizenship by birth; and (10) wages paid to unauthorized aliens.