doi: 10.3934/amc.2021029
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On the number of factorizations of $ t $ mod $ N $ and the probability distribution of Diffie-Hellman secret keys for many users

Institute of Cybernetics, Tallinn University of Technology, Estonia

Received  December 2020 Revised  June 2021 Early access August 2021

We study the number
$ R_n(t,N) $
of tuplets
$ (x_1,\ldots, x_n) $
of congruence classes modulo
$ N $
such that
$ \begin{equation*} x_1\cdots x_n \equiv t \pmod{N}. \end{equation*} $
As a result, we derive a recurrence for
$ R_n(t,N) $
and prove some multiplicative properties of
$ R_n(t,N) $
. Furthermore, we apply the result to study the probability distribution of Diffie-Hellman keys used in multiparty communication. We show that this probability distribution is not uniform.
Citation: Alar Leibak. On the number of factorizations of $ t $ mod $ N $ and the probability distribution of Diffie-Hellman secret keys for many users. Advances in Mathematics of Communications, doi: 10.3934/amc.2021029
References:
[1]

J. B. Friedlander, Uniform distribution, exponential sums, and cryptography, Equidistributions in Number Theory, An Introduction, Nato Science Series Ⅱ. Mathematics, Physics and Chemistry, Vol. 237, Springer, Dordrecht, 2007, 29–57. doi: 10.1007/978-1-4020-5404-4_3.  Google Scholar

[2]

P. H. van der Kamp, On the Fourier transform of the greatest common divisor, INTEGERS, 13 (2013), 1-16.   Google Scholar

[3]

D. Neuenschwander, Probabilistic and Statistical Methods in Cryptology: An Introduction by Selected Topics, Lecture Notes in Computer Science, Vol. 3028, Springer-Verlag, Berlin, 2004. doi: 10.1007/b97045.  Google Scholar

[4]

R. W. K. OdoniV. Varadharajan and P. W. Sanders, Public key distribution in matrix rings, Electronic Letters, 20 (1984), 386-387.  doi: 10.1049/el:19840267.  Google Scholar

[5] H. E. Rose, A Course in Number Theory, 2 edition, Oxford University Press, New York, 1994.   Google Scholar
[6]

J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., 6 (1962), 64-94.   Google Scholar

[7]

I. Shparlinski, Cryptographic Applications of Analytic Number Theory: Complexity Lower Bounds and Pseudorandomness, Progress in Computer Science and Applied Logic, 2003. doi: 10.1007/978-3-0348-8037-4.  Google Scholar

[8]

C. P. Waldvogel and J. L. Massey, The probability distribution of the Diffie-Hellmann key, Advances in Cryptology-AUSCRYPT '92, Lecture Notes in Comp. Sci., Vol. 718, Springer, Berlin, 1993,492–504. doi: 10.1007/3-540-57220-1_87.  Google Scholar

show all references

References:
[1]

J. B. Friedlander, Uniform distribution, exponential sums, and cryptography, Equidistributions in Number Theory, An Introduction, Nato Science Series Ⅱ. Mathematics, Physics and Chemistry, Vol. 237, Springer, Dordrecht, 2007, 29–57. doi: 10.1007/978-1-4020-5404-4_3.  Google Scholar

[2]

P. H. van der Kamp, On the Fourier transform of the greatest common divisor, INTEGERS, 13 (2013), 1-16.   Google Scholar

[3]

D. Neuenschwander, Probabilistic and Statistical Methods in Cryptology: An Introduction by Selected Topics, Lecture Notes in Computer Science, Vol. 3028, Springer-Verlag, Berlin, 2004. doi: 10.1007/b97045.  Google Scholar

[4]

R. W. K. OdoniV. Varadharajan and P. W. Sanders, Public key distribution in matrix rings, Electronic Letters, 20 (1984), 386-387.  doi: 10.1049/el:19840267.  Google Scholar

[5] H. E. Rose, A Course in Number Theory, 2 edition, Oxford University Press, New York, 1994.   Google Scholar
[6]

J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., 6 (1962), 64-94.   Google Scholar

[7]

I. Shparlinski, Cryptographic Applications of Analytic Number Theory: Complexity Lower Bounds and Pseudorandomness, Progress in Computer Science and Applied Logic, 2003. doi: 10.1007/978-3-0348-8037-4.  Google Scholar

[8]

C. P. Waldvogel and J. L. Massey, The probability distribution of the Diffie-Hellmann key, Advances in Cryptology-AUSCRYPT '92, Lecture Notes in Comp. Sci., Vol. 718, Springer, Berlin, 1993,492–504. doi: 10.1007/3-540-57220-1_87.  Google Scholar

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