frames are injected onto the stack and a library is returned into. The following variables can be declared: (entropy bits of stack top) (entropy bits of mmap() base) (entropy bits of main executable base) (entropy bits of heap base) (attacked bits per attempt of stack entropy) (attacked bits per attempt of mmap() base entropy) (attacked bits per attempt of main executable entropy) (attacked bits per attempt of heap base entropy) (attempts made) (total amount of entropy: ) To calculate the probability of an attacker succeeding, we have to assume a number of attempts carried out without being interrupted by a signature-based IPS, law enforcement, or other factor; in the case of brute forcing, the daemon cannot be restarted. We also have to figure out how many bits are relevant and how many are being attacked in each attempt, leaving however many bits the attacker has to defeat. The following formulas represent the probability of success for a given set of attempts on bits of entropy. (isolated guessing; address space is re-randomized after each attempt) (systematic brute forcing on copies of the program with the same address space) In many systems, can be in the thousands or millions; on 64-bit systems, these numbers typically reach the millions at least, Hector Marco-Gisbert and Ismael Ripoll showed in 2014 how to bypass the ASLR in 64-bit systems in less than one second under certain circumstances. For 32-bit systems at 2004 computer speeds which have 16 bits for address randomization, Shacham and co-workers state "... 16 bits of address randomization can be defeated by a brute force attack within minutes." The authors' statement depends on the ability to attack the same application multiple times without any delay. Proper implementations of ASLR, like that included in grsecurity, provide several methods to make such brute force attacks infeasible. One method involves preventing an executable from executing for a configurable amount of time if it has crashed a certain number of times. Android, and possibly other systems, implement Library Load Order Randomization, a form of ASLR which randomizes the order in which libraries are loaded. This supplies very little entropy. An approximation of the number of bits of entropy supplied per needed library appears below; this does not yet account for varied library sizes, so the actual entropy gained is really somewhat higher. Note that attackers usually need only one library; the math is more complex with