IC (Index Calculus) algorithm is the most effective probability algorithm for solving discrete logarithm of finite prime fields, and IICA (improved Index Calculus algorithm) is an improved algorithm based on IC in the third stage. The essence of IICA is to convert the number required to solve the discrete logarithm into the product of the power of prime factors, and then multiply every prime factor larger than the smooth bound by a smooth number approximating a large prime p from the right end, that is, to perform congruence transformation for every prime factor larger than the smooth bound. If all prime factors larger than the smooth bound fall within the smooth bound, the number required to solve the discrete logarithm is successfully solved. Unfortunately, given a large prime number p, some prime factors do not have congruent transformations of smooth numbers. For this, this paper analyzes the features of the IICA algorithm, based on the characteristics of IICA algorithm, is given when the IICA algorithm cannot undertake congruence transformation termination conditions, namely when ⌈ p/p_{i} ⌉ not smooth algorithm is terminated, where p_{i} is greater than a smooth boundary element factor. According to the ⌈ p/p_{i} ⌉ not smooth algorithm was terminated when judging conditions, optimized the IICA algorithm, and the correctness of the optimization algorithm is verified by an example.
Published in | Mathematics and Computer Science (Volume 8, Issue 2) |
DOI | 10.11648/j.mcs.20230802.14 |
Page(s) | 57-61 |
Creative Commons |
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Copyright © The Author(s), 2023. Published by Science Publishing Group |
Discrete Logarithm, Decomposition Base, Smooth Boundary, Factorization, Prime Finite Fields
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APA Style
Hu Jianjun. (2023). Analysis and Optimization of Improved Index Calculus Algorithm. Mathematics and Computer Science, 8(2), 57-61. https://doi.org/10.11648/j.mcs.20230802.14
ACS Style
Hu Jianjun. Analysis and Optimization of Improved Index Calculus Algorithm. Math. Comput. Sci. 2023, 8(2), 57-61. doi: 10.11648/j.mcs.20230802.14
AMA Style
Hu Jianjun. Analysis and Optimization of Improved Index Calculus Algorithm. Math Comput Sci. 2023;8(2):57-61. doi: 10.11648/j.mcs.20230802.14
@article{10.11648/j.mcs.20230802.14, author = {Hu Jianjun}, title = {Analysis and Optimization of Improved Index Calculus Algorithm}, journal = {Mathematics and Computer Science}, volume = {8}, number = {2}, pages = {57-61}, doi = {10.11648/j.mcs.20230802.14}, url = {https://doi.org/10.11648/j.mcs.20230802.14}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20230802.14}, abstract = {IC (Index Calculus) algorithm is the most effective probability algorithm for solving discrete logarithm of finite prime fields, and IICA (improved Index Calculus algorithm) is an improved algorithm based on IC in the third stage. The essence of IICA is to convert the number required to solve the discrete logarithm into the product of the power of prime factors, and then multiply every prime factor larger than the smooth bound by a smooth number approximating a large prime p from the right end, that is, to perform congruence transformation for every prime factor larger than the smooth bound. If all prime factors larger than the smooth bound fall within the smooth bound, the number required to solve the discrete logarithm is successfully solved. Unfortunately, given a large prime number p, some prime factors do not have congruent transformations of smooth numbers. For this, this paper analyzes the features of the IICA algorithm, based on the characteristics of IICA algorithm, is given when the IICA algorithm cannot undertake congruence transformation termination conditions, namely when ⌈ p/pi ⌉ not smooth algorithm is terminated, where pi is greater than a smooth boundary element factor. According to the ⌈ p/pi ⌉ not smooth algorithm was terminated when judging conditions, optimized the IICA algorithm, and the correctness of the optimization algorithm is verified by an example.}, year = {2023} }
TY - JOUR T1 - Analysis and Optimization of Improved Index Calculus Algorithm AU - Hu Jianjun Y1 - 2023/04/13 PY - 2023 N1 - https://doi.org/10.11648/j.mcs.20230802.14 DO - 10.11648/j.mcs.20230802.14 T2 - Mathematics and Computer Science JF - Mathematics and Computer Science JO - Mathematics and Computer Science SP - 57 EP - 61 PB - Science Publishing Group SN - 2575-6028 UR - https://doi.org/10.11648/j.mcs.20230802.14 AB - IC (Index Calculus) algorithm is the most effective probability algorithm for solving discrete logarithm of finite prime fields, and IICA (improved Index Calculus algorithm) is an improved algorithm based on IC in the third stage. The essence of IICA is to convert the number required to solve the discrete logarithm into the product of the power of prime factors, and then multiply every prime factor larger than the smooth bound by a smooth number approximating a large prime p from the right end, that is, to perform congruence transformation for every prime factor larger than the smooth bound. If all prime factors larger than the smooth bound fall within the smooth bound, the number required to solve the discrete logarithm is successfully solved. Unfortunately, given a large prime number p, some prime factors do not have congruent transformations of smooth numbers. For this, this paper analyzes the features of the IICA algorithm, based on the characteristics of IICA algorithm, is given when the IICA algorithm cannot undertake congruence transformation termination conditions, namely when ⌈ p/pi ⌉ not smooth algorithm is terminated, where pi is greater than a smooth boundary element factor. According to the ⌈ p/pi ⌉ not smooth algorithm was terminated when judging conditions, optimized the IICA algorithm, and the correctness of the optimization algorithm is verified by an example. VL - 8 IS - 2 ER -