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The Bit Extraction Problem or tResilient Functions
, 1985
"... \Gamma We consider the following adversarial situation. Let n, m and t be arbitrary integers, and let f : f0; 1g n 7! f0; 1g m be a function. An adversary, knowing the function f , sets t of the n input bits, while the rest (n \Gamma t input bits) are chosen at random (independently and with un ..."
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\Gamma We consider the following adversarial situation. Let n, m and t be arbitrary integers, and let f : f0; 1g n 7! f0; 1g m be a function. An adversary, knowing the function f , sets t of the n input bits, while the rest (n \Gamma t input bits) are chosen at random (independently and with uniform probability distribution). The adversary tries to prevent the outcome of f from being uniformly distributed in f0; 1g m . The question addressed is for what values of n, m and t does the adversary necessarily fail in biasing the outcome of f : f0; 1g n 7! f0; 1g m , when being restricted to set t of the input bits of f . We present various lower and upper bounds on m's allowing an affirmative answer. These bounds are relatively close for t n=3 and for t 2n=3. Our results have applications in the fields of faulttolerance and cryptography. 1. INTRODUCTION The bit extraction problem formulated above The bit extraction problem was suggested by Brassard and Robert [BRref] and by V...
Reproduced without access to the TeX macros. Adhoc macro definitions were used instead. THE BIT EXTRACTION PROBLEM OR t–RESILIENT FUNCTIONS (Preliminary Version)
"... ABSTRACT − We consider the following adversarial situation. Let n, m and t be arbitrary integers, and let f: {0, 1} n ↦ → {0, 1} m be a function. An adversary, knowing the function f, sets t of the n input bits, while the rest (n − t input bits) are chosen at random (independently and with uniform p ..."
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ABSTRACT − We consider the following adversarial situation. Let n, m and t be arbitrary integers, and let f: {0, 1} n ↦ → {0, 1} m be a function. An adversary, knowing the function f, sets t of the n input bits, while the rest (n − t input bits) are chosen at random (independently and with uniform probability distribution). The adversary tries to prevent the outcome of f from being uniformly distributed in {0, 1} m. The question addressed is for what values of n, m and t does the adversary necessarily fail in biasing the outcome of f: {0, 1} n ↦ → {0, 1} m, when being restricted to set t of the input bits of f. We present various lower and upper bounds on m’s allowing an affirmative answer. These bounds are relatively close for t ≤ n/3 and for t ≥ 2n/3. Our results have applications in the fields of faulttolerance and cryptography. 1.