The DOI matches its Crossref record; 15 of the 15 references it cites were found in a registry or an index.
RSA and Rabin Functions: Certain Parts are as Hard as the Whole
Abstract
The RSA and Rabin encryption functions $E_N ( \cdot )$ are respectively defined by raising $x \in Z_N $ to the power e (where e is relatively prime to $\varphi (N)$) and squaring modulo N (i.e., $E_N (x) = x^e (\bmod N)$, $E_N (x) = x^2 (\bmod N)$, respectively). We prove that for both functions, the following problems are computationally equivalent (each is probabilistic polynomial-time reducible to the other): (1) Given $E_N (x)$, find x. (2) Given $E_N (x)$, guess the least-significant bit of x with success probability $\tfrac{1}{2} + {1 {{\operatorname{poly}}(n)}}$ (where n is the length of the modulus N). This equivalence implies that an adversary, given the RSA/Rabin ciphertext, cannot have a non-negligible advantage (over a random coin flip) in guessing the least-significant bit of the plaintext, unless he can invert RSA/factor N. The proof techniques also yield the simultaneous security of the $\log n$ least-significant bits. Our results improve the efficiency of pseudorandom number generation and probabilistic encryption schemes based on the intractability of factoring.
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Citations
Cites
New directions in cryptography
Diffie, W., & Hellman, M. (1976) · IEEE Transactions on Information Theory · doi:10.1109/tit.1976.1055638 · Cited by 14,713
A method for obtaining digital signatures and public-key cryptosystems
Rivest, R. L., Shamir, A., & Adleman, L. (1978) · Communications of the ACM · doi:10.1145/359340.359342 · Cited by 13,462
A method for obtaining digital signatures and public-key cryptosystems
Rivest, R. L., Shamir, A., & Adleman, L. (1983) · Communications of the ACM · doi:10.1145/357980.358017 · Cited by 13,012
An Introduction to Probability Theory and its Applications.
Kingman, J. F. C., & Feller, W. (1972) · Journal of the Royal Statistical Society. Series A (General) · doi:10.2307/2344620 · Cited by 4,938
An introduction to the theory of numbers
(1960) · Journal of the Franklin Institute · doi:10.1016/0016-0032(60)90676-1 · Cited by 3,574
Goldwasser, S., & Micali, S. (1984) · Journal of Computer and System Sciences · doi:10.1016/0022-0000(84)90070-9 · Cited by 2,603
An Introduction to the Theory of Numbers.
LeVeque, W. J., Niven, I., & Zuckerman, H. S. (1961) · The American Mathematical Monthly · doi:10.2307/2311171 · Cited by 1,907
How to Generate Cryptographically Strong Sequences of Pseudorandom Bits
Blum, M., & Micali, S. (1984) · SIAM Journal on Computing · doi:10.1137/0213053 · Cited by 1,322
A Simple Unpredictable Pseudo-Random Number Generator
Blum, L., Blum, M., & Shub, M. (1986) · SIAM Journal on Computing · doi:10.1137/0215025 · Cited by 1,036
Theory and application of trapdoor functions
Yao, A. C. (1982) · 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982) · doi:10.1109/sfcs.1982.45 · Cited by 968
Cited by
Software protection and simulation on oblivious RAMs
Goldreich, O., & Ostrovsky, R. (1996) · Journal of the ACM · doi:10.1145/233551.233553 · Cited by 1,617
Self-citationDolev, D., Dwork, C., & Naor, M. (1991) · Proceedings of the twenty-third annual ACM symposium on Theory of computing - STOC '91 · doi:10.1145/103418.103474 · Cited by 990
How to Construct Pseudorandom Permutations from Pseudorandom Functions
Luby, M., & Rackoff, C. (1988) · SIAM Journal on Computing · doi:10.1137/0217022 · Cited by 938
A concrete security treatment of symmetric encryption
Bellare, M., Desai, A., Jokipii, E., & Rogaway, P. (2002) · Proceedings 38th Annual Symposium on Foundations of Computer Science · doi:10.1109/sfcs.1997.646128 · Cited by 931
Dolev, D., Dwork, C., & Naor, M. (2000) · SIAM Journal on Computing · doi:10.1137/s0097539795291562 · Cited by 746
Metadata
- Authors
- Alexi, W., Chor, B., Goldreich, O., & Schnorr, C. P.
- Venue
- SIAM Journal on Computing
- Published
- 1988-04
- Volume
- 17
- Issue
- 2
- Pages
- 194-209
- ISSN
- 0097-5397 (ISSN-L)
- 1095-7111
- OpenAlex
- W2108104892
- Cited by
- 339
- References
- 15
- FWCI
- 11.60
- Type
- journal article
- Language
- en
- Open access
- closed
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