Abstract
We consider a special type of sequence of arithmetic progressions, in which consecutive progressions are related by the property: ithterms ofjth, (j + 1)thprogressions of the sequence are multiplicative inverses of each other modulo(i + 1)thterm ofjthprogression. Such a sequence is uniquely defined for any pair of co-prime numbers. A computational problem, defined in the context of such a sequence and its generalization, is shown to be equivalent to the integer factoring problem. The proof is probabilistic. As an application of the equivalence result, we propose a method for how users securely agree upon secret keys, which are ensured to be random. We compare our method with factoring based public key cryptographic systems: RSA (Rivest et al., ACM 21, 120–126, 1978) and Rabin systems (Rabin 1978). We discuss the advantages of the method, and its potential use-case in the post quantum scenario.
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Notes
A worst case instance of IFP is a large number n, called RSA number, which is the product of two equally sized primes p, and q. In other words, both p and q are about \(\sqrt {n}\).
Some sensitive military applications already use RSA system with 3096-bit △
In the post-quantum scenario, g should be of length at least 256 binary bits
Probable candidates for \(\mathcal {E}\) are McEliece’s cryptosystem, NTRU, HFE, and other Lattice-based cryptosystems
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Acknowledgments
The author is thankful to anonymous referees for their valuable comments and suggestions. The author is grateful to his doctoral advisor Prof.Veni Madhavan and a senior member Kumara Swamy for their invaluable inputs on the research problem.
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This article belongs to the Topical Collection: Special Issue of Sequences and Their Applications II
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Srikanth, C. Certain sequence of arithmetic progressions and a new key sharing method. Cryptogr. Commun. 12, 597–612 (2020). https://doi.org/10.1007/s12095-019-00416-z
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DOI: https://doi.org/10.1007/s12095-019-00416-z
Keywords
- Sequence of arithmetic progressions
- Euclidean gcd algorithm
- RSA system
- Integer factoring
- Quadratic residuosity
- Key sharing method