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Hard-Core Bit

  • Reference work entry
Encyclopedia of Cryptography and Security

  • 143 Accesses

Related Concepts

One-Way Function; Pseudorandom Number Generator

Definition

A hard-core bit of a one-way function y = f(x) is any bit or other binary function of the input x that is hard to compute significantly better than guessing given the output y alone.

Theory

Let f be a one-way function. According to the definition of such a function, it is difficult, given y = f(x), where x is random, to recover x. However, it may be easy to determine certain information about x. For instance, the RSA function \(f(x) = {x}^{e}\) mod n (RSA public-key encryption) is believed to be one-way, yet it is easy to compute the Jacobi symbol of x, given f(x):

$$ \left (\frac{{x}^{e}\mathrm{mod}\ n} {n} \right ) ={ \left (\frac{x} {n}\right )}^{e} = \left (\frac{x} {n}\right ). $$

Another example is found in the discrete exponentiation function \(f(x) = {g}^{x}\) mod p (discrete logarithm problem), where the least-significant bit of x is revealed from the Legendre symbol of f(x), i.e., \(f{(x)}^{(p-1)/2}\)...

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Recommended Reading

  1. Alexi WB, Chor B, Goldreich O, Schnorr C-P (1988) RSA and Rabin functions: certain parts are as hard as the whole. SIAM J Comput 17(2):194–209

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  2. Blum M, Micali S (1984) How to generate cryptographically strong sequences of pseudo-random bits. SIAM J Comput 13(4):850–863

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  3. Goldreich O, Levin L (1989) A hard-core predicate for all one-way functions. In: Proceedings of the 21st annual ACM symposium on theory of computing. ACM, New York, pp 25–32

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  4. Håstad J, Näslund M (2004) The security of all RSA and discrete log bits. J ACM 51(2):187–230

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  5. Håstad J, Schrift AW, Shamir A (1993) The discrete logarithm modulo a composite hides O(n) bits. J Comput Syst Sci 47(3):376–404

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  6. Yao A (1982) Theory and applications of trapdoor functions. In: Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society Press, Los Alamitos, pp 80–91

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Author information

Authors and Affiliations

  1. Office of the CTO, EMC Corporation, 176 South Street, 01748, Hopkinton, MA, USA

    Burt Kaliski Jr. Dr.

Authors
  1. Burt Kaliski Jr. Dr.
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Computing Science, Eindhoven University of Technology, 5600 MB, Eindhoven, The Netherlands

    Henk C. A. van Tilborg

  2. Center for Secure Information Systems, George Mason University, Fairfax, VA, 22030-4422, USA

    Sushil Jajodia

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Kaliski, B. (2011). Hard-Core Bit. In: van Tilborg, H.C.A., Jajodia, S. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5906-5_412

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