Abstract
The Rabin scheme used in public-key cryptosystem is here revisited with a focus limited to a few specific open issues. In particular, message decryption requires one out of four roots of a quadratic equation in a residue ring to be chosen, and a longstanding problem is to identify unambiguously and deterministically the encrypted message at the decryption side by adding the minimum number of extra bits to the cipher-text. While the question has already been solved for pairs of primes of the type \(4k+3\), the general problem is here addressed. As one of the major results, an explicit solution with two extra bits is provided for pairs of primes that are congruent 5 modulo 8. The Rabin signature is also reconsidered from a deterministic point of view: a padding mechanism is proposed that avoids relying on a certain number of attempts until a suitable pad is found.
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Acknowledgments
Some of this work was done while the first author was Visiting Professor with the University of Trento, funded by CIRM, and he would like to thank the Department of Mathematics for the friendly and fruitful atmosphere offered. The third author was supported by the Swiss National Science Foundation under Grant No. 132256. We would also like to thank Steven Galbraith for his comments on a preliminary version of the paper and for pointing out some references. Finally, we gratefullyacknowledge the many suggestions and corrections offered by anonymous referees which have greatly improved the readability and quality of the paper.
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Elia, M., Piva, M. & Schipani, D. The Rabin cryptosystem revisited. AAECC 26, 251–275 (2015). https://doi.org/10.1007/s00200-014-0237-0
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DOI: https://doi.org/10.1007/s00200-014-0237-0