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On the (im)possibility of obfuscating programs
 Lecture Notes in Computer Science
, 2001
"... Informally, an obfuscator O is an (efficient, probabilistic) “compiler ” that takes as input a program (or circuit) P and produces a new program O(P) that has the same functionality as P yet is “unintelligible ” in some sense. Obfuscators, if they exist, would have a wide variety of cryptographic an ..."
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Cited by 189 (10 self)
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Informally, an obfuscator O is an (efficient, probabilistic) “compiler ” that takes as input a program (or circuit) P and produces a new program O(P) that has the same functionality as P yet is “unintelligible ” in some sense. Obfuscators, if they exist, would have a wide variety of cryptographic and complexitytheoretic applications, ranging from software protection to homomorphic encryption to complexitytheoretic analogues of Rice’s theorem. Most of these applications are based on an interpretation of the “unintelligibility ” condition in obfuscation as meaning that O(P) is a “virtual black box, ” in the sense that anything one can efficiently compute given O(P), one could also efficiently compute given oracle access to P. In this work, we initiate a theoretical investigation of obfuscation. Our main result is that, even under very weak formalizations of the above intuition, obfuscation is impossible. We prove this by constructing a family of efficient programs P that are unobfuscatable in the sense that (a) given any efficient program P ′ that computes the same function as a program P ∈ P, the “source code ” P can be efficiently reconstructed, yet (b) given oracle access to a (randomly selected) program P ∈ P, no efficient algorithm can reconstruct P (or even distinguish a certain bit in the code from random) except with negligible probability. We extend our impossibility result in a number of ways, including even obfuscators that (a) are not necessarily computable in polynomial time, (b) only approximately preserve the functionality, and (c) only need to work for very restricted models of computation (TC 0). We also rule out several potential applications of obfuscators, by constructing “unobfuscatable” signature schemes, encryption schemes, and pseudorandom function families.
RiceStyle Theorems for Complexity Theory
, 2001
"... Rice's Theorem states that all nontrivial language properties of recursively enumerable sets are undecidable. Borchert and Stephan [BS00] started the search for complexitytheoretic analogs of Rice's Theorem, and proved that every nontrivial counting property of boolean circuits is UPhard. ..."
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Cited by 1 (1 self)
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Rice's Theorem states that all nontrivial language properties of recursively enumerable sets are undecidable. Borchert and Stephan [BS00] started the search for complexitytheoretic analogs of Rice's Theorem, and proved that every nontrivial counting property of boolean circuits is UPhard.
LOWER BOUNDS AND THE HARDNESS OF COUNTING PROPERTIES
"... Rice’s Theorem states that all nontrivial language properties of recursively enumerable sets are undecidable. Borchert and Stephan [BS00] started the search for complexitytheoretic analogs of Rice’s Theorem, and proved that every nontrivial counting property of boolean circuits is UPhard. Hemasp ..."
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Rice’s Theorem states that all nontrivial language properties of recursively enumerable sets are undecidable. Borchert and Stephan [BS00] started the search for complexitytheoretic analogs of Rice’s Theorem, and proved that every nontrivial counting property of boolean circuits is UPhard. Hemaspaandra and Rothe [HR00] improved the UPhardness lower bound to UPO(1)hardness. The present paper raises the lower bound for nontrivial counting properties from UPO(1)hardness to FewPhardness, i.e., from constantambiguity nondeterminism to polynomialambiguity nondeterminism. Furthermore, we prove that this lower bound is rather tight with respect to relativizable techniques, i.e., no relativizable technique can raise this lower bound to FewP ≤ p 1tthardness. We also prove a Ricestyle theorem for NP, namely that every nontrivial language property of NP sets is NPhard.