Let \(1 < \tau \in \mathbb{Z}\) be a security parameter. Let \(N=pq\) be a product of two random τ-bit primes and let s be an element of the group \(\mathbb{Z}_N^*\) (see also modular arithmetic). The strong-RSA problem is defined as follows:
given \((N,s)\) as input, output a pair \(a,b \in \mathbb{Z}\)} such that \(a^b = s \bmod N\) and \(b \neq \pm 1\).
Loosely speaking, the Strong-RSA assumption states that for a sufficiently large τ the strong RSA problem is intractable.
The Strong-RSA assumption was introduced by Baric and Pfitzman [2]. The assumption is used to construct efficient signature schemes that are existentially unforgeable under a chosen message attack without the random oracle model. One such system is described in [4] and another in [3]. The Strong-RSA assumption is also the basis of several efficient group signature schemes [1].
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References
Ateniese, Giuseppe, Jan Camenisch, Marc Joye, and Gene Tsudik (2000). “A practical and provably secure coalition-resistant group signature scheme.” Advances in Cryptology—CRYPTO 2000, August, Lecture Notes in Computer Science, vol. 1880, ed. M. Bellare. Springer-Verlag, Berlin, 255–70.
Baric, N. and B. Pfitzman (1997). “Collision-free accumulators and fail-stop signature schemes without trees.” Proceedings of Eurocrypt, Lecture Notes in Computer Science, vol. 1233, ed. W. Fumy. Springer-Verlag, Berlin, 480–494.
Cramer, Ronald and Victor Shoup (2000). “Signature schemes based on the strong RSA assumption.” ACM Transactions on Information and System Security (ACM TISSEC), 3 (3), 161–185, extended abstract in Proc. 6th ACM Conf. on Computer and Communications Security, 1999.
Gennaro, Rosario, Shai Halevi, and Tal Rabin (1999). “Secure hash-and-sign signatures without the random oracle.” Advances in Cryptology—EUROCRYPT'99, Lecture Notes in Computer Science, vol. 1592, ed. J. Stern. Springer-Verlag, Berlin, 123–139.
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Boneh, D. (2005). Strong RSA Assumption. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_414
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