Non-Black-Box Techniques In Crytpography Thesis for the Ph.D degree

f(x) def = x + 1 for any integer x. Given a program Π and a value x, it is possible to compute f(x), where f(·) is the function associated with the program Π, by executing the program Π on a computer. We sometimes call the string Π the representation or code of the function f(·). Since a program is a string, it sometimes makes sense to use it as input to a different program. Indeed, algorithms that take other programs as input are very common in Computer Science. In most cases, these algorithms use their input program as a subroutine. By this we mean that, on input a program Π, the algorithm’s operation does not depend on the particular representation of the program Π, but rather, the algorithm only uses its input program Π to evaluate the function that the program Π computes. That is, the algorithm only uses Π to obtain a “black box” for computing f(·), such that the algorithm can feed inputs to and receive outputs from this box. We call such an algorithm (that only uses black-box access to the program it gets as input) a black-box algorithm. As mentioned above, black-box algorithms are very popular in computer science. Many times, when trying to solve a particular task, one would write an algorithm that solves the task if it is given black-box access to programs that solve some simpler tasks, and then write programs that solve these simpler tasks. This approach is a basic paradigm of software engineering, and almost all programming languages implement mechanisms (such as function calls) to facilitate it. Black-box algorithms are also very popular in the more theoretical aspects of Computer Science. For example, when proving that a decision problem L is NP-complete, one needs to show the existence of a black-box algorithm B (where B is usually called a reduction), such that if B is given black-box access to an algorithm A that solves the problem L, then B can solve the Satisfiability problem. Such reductions also appear in cryptography. For example, consider the constructions of public key encryption schemes whose security can be reduced to the factoring problem [Rab79, BG84]. This is shown by providing a black-box algorithm B, such that if B is given black-box access to an algorithm A that compromises the security of the encryption scheme, then B can solve the factoring problem.