Towards Cryptanalysis of a Variant Prime Numbers Algorithm

Bashir Kagara Yusuf; Kamil Ahmad Bin Mahmood

doi:doi:10.11648/j.mcs.20200501.13

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Towards Cryptanalysis of a Variant Prime Numbers Algorithm

Bashir Kagara Yusuf, Kamil Ahmad Bin Mahmood

Published in Mathematics and Computer Science (Volume 5, Issue 1)

Received: 10 January 2020 Accepted: 31 January 2020 Published: 13 February 2020

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Abstract

A hallmark of prime numbers (primes) that both characterizes it away from other natural numbers and makes it a challenging preoccupation, is its staunch defiance to be expressed in terms of composites or as a formula listing all its sequence of elements. A classification approach, was mapped out, that fragments a prime into two: its last digit (trailer - reduced set of residue {1, 3, 7 and 9}) and the other digits (lead) whose value is incremented by either 1, 2 or 3 thus producing a modulo-3 arithmetic equation. The algorithm tracked both Polignac’s and modified Goldbach’s coefficients in order to explore such an open and computationally hard problem. Precisely 20,064,735,430 lower primes of digits 2 to 12 were parsed through validity test with the powers of 10 primes of Sloane's A006988. Adopting at most cubic terms of predictors (as the next logical step of Euler’s quadratic formula for primes) in multiple linear regression analysis, the generated outputs were analyzed to aid in building Akaike Information Criterion (AIC) best model with forward selection strategy. The main task was fragmented into atomic units of similar instances and types (an atom is a table of length 4,493,869 integer sequences where a database contains 30 relational tables with facilities for further reprocessing). A node, that supports parallel processing, stores 30 contiguous databases, and explores 4,044,482,100 successive integers. 513,649,226,700 lower natural numbers were explored by 127 hypothetical nodes yielding primes stored in 114,300 tables spread across 3,810 databases. Veriton S6630G computer system with 7.86GB usable memory and processor Intel(R) Core(TM) i7-4790 CPU @ 3.60GHz were amongst the remarkable resources. Contrary to the apparent chaotic camouflage of primes as a bundle, the partitioned sample spaces reveal some remarkable patterns in terms of intervals of both sequence numbers and distances of separation from their immediate neighborhoods.

Published in	Mathematics and Computer Science (Volume 5, Issue 1)
DOI	10.11648/j.mcs.20200501.13
Page(s)	14-30
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

AIC Best Modeling, Algorithm Analysis, Euler’s Formula, Primes Pairs

References

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[12]	B. Gates, The Road Ahead. New York: Viking, 1995.
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APA Style

Bashir Kagara Yusuf, Kamil Ahmad Bin Mahmood. (2020). Towards Cryptanalysis of a Variant Prime Numbers Algorithm. Mathematics and Computer Science, 5(1), 14-30. https://doi.org/10.11648/j.mcs.20200501.13

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ACS Style

Bashir Kagara Yusuf; Kamil Ahmad Bin Mahmood. Towards Cryptanalysis of a Variant Prime Numbers Algorithm. Math. Comput. Sci. 2020, 5(1), 14-30. doi: 10.11648/j.mcs.20200501.13

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AMA Style

Bashir Kagara Yusuf, Kamil Ahmad Bin Mahmood. Towards Cryptanalysis of a Variant Prime Numbers Algorithm. Math Comput Sci. 2020;5(1):14-30. doi: 10.11648/j.mcs.20200501.13

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@article{10.11648/j.mcs.20200501.13,
  author = {Bashir Kagara Yusuf and Kamil Ahmad Bin Mahmood},
  title = {Towards Cryptanalysis of a Variant Prime Numbers Algorithm},
  journal = {Mathematics and Computer Science},
  volume = {5},
  number = {1},
  pages = {14-30},
  doi = {10.11648/j.mcs.20200501.13},
  url = {https://doi.org/10.11648/j.mcs.20200501.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20200501.13},
  abstract = {A hallmark of prime numbers (primes) that both characterizes it away from other natural numbers and makes it a challenging preoccupation, is its staunch defiance to be expressed in terms of composites or as a formula listing all its sequence of elements. A classification approach, was mapped out, that fragments a prime into two: its last digit (trailer - reduced set of residue {1, 3, 7 and 9}) and the other digits (lead) whose value is incremented by either 1, 2 or 3 thus producing a modulo-3 arithmetic equation. The algorithm tracked both Polignac’s and modified Goldbach’s coefficients in order to explore such an open and computationally hard problem. Precisely 20,064,735,430 lower primes of digits 2 to 12 were parsed through validity test with the powers of 10 primes of Sloane's A006988. Adopting at most cubic terms of predictors (as the next logical step of Euler’s quadratic formula for primes) in multiple linear regression analysis, the generated outputs were analyzed to aid in building Akaike Information Criterion (AIC) best model with forward selection strategy. The main task was fragmented into atomic units of similar instances and types (an atom is a table of length 4,493,869 integer sequences where a database contains 30 relational tables with facilities for further reprocessing). A node, that supports parallel processing, stores 30 contiguous databases, and explores 4,044,482,100 successive integers. 513,649,226,700 lower natural numbers were explored by 127 hypothetical nodes yielding primes stored in 114,300 tables spread across 3,810 databases. Veriton S6630G computer system with 7.86GB usable memory and processor Intel(R) Core(TM) i7-4790 CPU @ 3.60GHz were amongst the remarkable resources. Contrary to the apparent chaotic camouflage of primes as a bundle, the partitioned sample spaces reveal some remarkable patterns in terms of intervals of both sequence numbers and distances of separation from their immediate neighborhoods.},
 year = {2020}
}

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TY  - JOUR
T1  - Towards Cryptanalysis of a Variant Prime Numbers Algorithm
AU  - Bashir Kagara Yusuf
AU  - Kamil Ahmad Bin Mahmood
Y1  - 2020/02/13
PY  - 2020
N1  - https://doi.org/10.11648/j.mcs.20200501.13
DO  - 10.11648/j.mcs.20200501.13
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 14
EP  - 30
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20200501.13
AB  - A hallmark of prime numbers (primes) that both characterizes it away from other natural numbers and makes it a challenging preoccupation, is its staunch defiance to be expressed in terms of composites or as a formula listing all its sequence of elements. A classification approach, was mapped out, that fragments a prime into two: its last digit (trailer - reduced set of residue {1, 3, 7 and 9}) and the other digits (lead) whose value is incremented by either 1, 2 or 3 thus producing a modulo-3 arithmetic equation. The algorithm tracked both Polignac’s and modified Goldbach’s coefficients in order to explore such an open and computationally hard problem. Precisely 20,064,735,430 lower primes of digits 2 to 12 were parsed through validity test with the powers of 10 primes of Sloane's A006988. Adopting at most cubic terms of predictors (as the next logical step of Euler’s quadratic formula for primes) in multiple linear regression analysis, the generated outputs were analyzed to aid in building Akaike Information Criterion (AIC) best model with forward selection strategy. The main task was fragmented into atomic units of similar instances and types (an atom is a table of length 4,493,869 integer sequences where a database contains 30 relational tables with facilities for further reprocessing). A node, that supports parallel processing, stores 30 contiguous databases, and explores 4,044,482,100 successive integers. 513,649,226,700 lower natural numbers were explored by 127 hypothetical nodes yielding primes stored in 114,300 tables spread across 3,810 databases. Veriton S6630G computer system with 7.86GB usable memory and processor Intel(R) Core(TM) i7-4790 CPU @ 3.60GHz were amongst the remarkable resources. Contrary to the apparent chaotic camouflage of primes as a bundle, the partitioned sample spaces reveal some remarkable patterns in terms of intervals of both sequence numbers and distances of separation from their immediate neighborhoods.
VL  - 5
IS  - 1
ER  -

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Author Information

Bashir Kagara Yusuf

Department of Computer, Ibrahim Badamasi Babangida University, Lapai, Nigeria
Kamil Ahmad Bin Mahmood

Department of Computer and Information Sciences, Universiti Teknologi Petronas (UTP), Bandar Seri Iskandar, Malaysia

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APA Style

Bashir Kagara Yusuf, Kamil Ahmad Bin Mahmood. (2020). Towards Cryptanalysis of a Variant Prime Numbers Algorithm. Mathematics and Computer Science, 5(1), 14-30. https://doi.org/10.11648/j.mcs.20200501.13

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ACS Style

Bashir Kagara Yusuf; Kamil Ahmad Bin Mahmood. Towards Cryptanalysis of a Variant Prime Numbers Algorithm. Math. Comput. Sci. 2020, 5(1), 14-30. doi: 10.11648/j.mcs.20200501.13

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AMA Style

Bashir Kagara Yusuf, Kamil Ahmad Bin Mahmood. Towards Cryptanalysis of a Variant Prime Numbers Algorithm. Math Comput Sci. 2020;5(1):14-30. doi: 10.11648/j.mcs.20200501.13

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@article{10.11648/j.mcs.20200501.13,
  author = {Bashir Kagara Yusuf and Kamil Ahmad Bin Mahmood},
  title = {Towards Cryptanalysis of a Variant Prime Numbers Algorithm},
  journal = {Mathematics and Computer Science},
  volume = {5},
  number = {1},
  pages = {14-30},
  doi = {10.11648/j.mcs.20200501.13},
  url = {https://doi.org/10.11648/j.mcs.20200501.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20200501.13},
  abstract = {A hallmark of prime numbers (primes) that both characterizes it away from other natural numbers and makes it a challenging preoccupation, is its staunch defiance to be expressed in terms of composites or as a formula listing all its sequence of elements. A classification approach, was mapped out, that fragments a prime into two: its last digit (trailer - reduced set of residue {1, 3, 7 and 9}) and the other digits (lead) whose value is incremented by either 1, 2 or 3 thus producing a modulo-3 arithmetic equation. The algorithm tracked both Polignac’s and modified Goldbach’s coefficients in order to explore such an open and computationally hard problem. Precisely 20,064,735,430 lower primes of digits 2 to 12 were parsed through validity test with the powers of 10 primes of Sloane's A006988. Adopting at most cubic terms of predictors (as the next logical step of Euler’s quadratic formula for primes) in multiple linear regression analysis, the generated outputs were analyzed to aid in building Akaike Information Criterion (AIC) best model with forward selection strategy. The main task was fragmented into atomic units of similar instances and types (an atom is a table of length 4,493,869 integer sequences where a database contains 30 relational tables with facilities for further reprocessing). A node, that supports parallel processing, stores 30 contiguous databases, and explores 4,044,482,100 successive integers. 513,649,226,700 lower natural numbers were explored by 127 hypothetical nodes yielding primes stored in 114,300 tables spread across 3,810 databases. Veriton S6630G computer system with 7.86GB usable memory and processor Intel(R) Core(TM) i7-4790 CPU @ 3.60GHz were amongst the remarkable resources. Contrary to the apparent chaotic camouflage of primes as a bundle, the partitioned sample spaces reveal some remarkable patterns in terms of intervals of both sequence numbers and distances of separation from their immediate neighborhoods.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Towards Cryptanalysis of a Variant Prime Numbers Algorithm
AU  - Bashir Kagara Yusuf
AU  - Kamil Ahmad Bin Mahmood
Y1  - 2020/02/13
PY  - 2020
N1  - https://doi.org/10.11648/j.mcs.20200501.13
DO  - 10.11648/j.mcs.20200501.13
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 14
EP  - 30
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20200501.13
AB  - A hallmark of prime numbers (primes) that both characterizes it away from other natural numbers and makes it a challenging preoccupation, is its staunch defiance to be expressed in terms of composites or as a formula listing all its sequence of elements. A classification approach, was mapped out, that fragments a prime into two: its last digit (trailer - reduced set of residue {1, 3, 7 and 9}) and the other digits (lead) whose value is incremented by either 1, 2 or 3 thus producing a modulo-3 arithmetic equation. The algorithm tracked both Polignac’s and modified Goldbach’s coefficients in order to explore such an open and computationally hard problem. Precisely 20,064,735,430 lower primes of digits 2 to 12 were parsed through validity test with the powers of 10 primes of Sloane's A006988. Adopting at most cubic terms of predictors (as the next logical step of Euler’s quadratic formula for primes) in multiple linear regression analysis, the generated outputs were analyzed to aid in building Akaike Information Criterion (AIC) best model with forward selection strategy. The main task was fragmented into atomic units of similar instances and types (an atom is a table of length 4,493,869 integer sequences where a database contains 30 relational tables with facilities for further reprocessing). A node, that supports parallel processing, stores 30 contiguous databases, and explores 4,044,482,100 successive integers. 513,649,226,700 lower natural numbers were explored by 127 hypothetical nodes yielding primes stored in 114,300 tables spread across 3,810 databases. Veriton S6630G computer system with 7.86GB usable memory and processor Intel(R) Core(TM) i7-4790 CPU @ 3.60GHz were amongst the remarkable resources. Contrary to the apparent chaotic camouflage of primes as a bundle, the partitioned sample spaces reveal some remarkable patterns in terms of intervals of both sequence numbers and distances of separation from their immediate neighborhoods.
VL  - 5
IS  - 1
ER  -

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