Lattice reduction algorithms application in cryptanalysis

BY KEYHUNTER 19.11.2024

Lattice reduction algorithms such as the LLL algorithm and the large number theorem (LNT) play an important role in cryptanalysis. These methods allow for efficient data processing and analysis, which is especially important in the context of cryptographic systems where the security of information depends on the complexity of mathematical problems.

Lattice reduction algorithm

The LLL algorithm is a method developed by Arjen Lenstra, Hendrik Lenstra, and Laszle Lovasz in 1982 that allows lattice bases to be reduced in polynomial time. The algorithm has applications in a variety of fields, including cryptography, where it is used to attack systems based on complex mathematical problems such as factorization and discrete logarithm.[2]

Application in cryptanalysis

In cryptanalysis, the LLL algorithm can be used to:

  • Finding Short Vectors : Basis reduction allows finding short vectors in a lattice that can be used to attack cryptographic schemes.
  • Attacks on lattice-based systems : Many modern cryptographic systems rely on the difficulty of lattice-based problems. The LLL algorithm can help find vulnerabilities in such systems.

Theorem of Large Numbers

The law of large numbers (LLN) is a principle of probability theory that states that as the number of trials increases, the mean of a sample will tend toward the expected value.[4] This property can be used to estimate the probability of a successful attack on a cryptographic system over multiple trials.

Relationship with cryptanalysis

The ZBC can be applied in cryptanalysis as follows:

  • Probability assessment : When analyzing random processes such as key generation or data encryption, the BLK allows one to estimate the probability of a successful attack.
  • Statistical Analysis : Using the law of large numbers, cryptanalysts can perform statistical analysis of the results of their attacks to identify patterns and vulnerabilities.

There are several scientific articles and tutorials in Russian that deal with cryptanalysis using the large number theorem. Here are some of them:

  1. Mathematical Foundations of Cryptology  – A textbook that covers the basic mathematical concepts used in cryptography and cryptanalysis. It discusses theoretical aspects that may be related to the application of the large number theorem to the analysis of cryptographic algorithms 1 .
  2. Applied Cryptology: Encryption Techniques  – This tutorial introduces encryption techniques and cryptographic analysis. It covers various aspects of cryptography, including cipher analysis and the application of statistical methods, which may be related to probability theory and the large number theorem 2 .
  3. Cryptanalysis: Current Status and Future Trends  – An article covering modern cryptanalysis techniques, including linear and differential analysis of block ciphers. It also discusses approaches that can use statistical methods to assess the security of cryptographic systems 3 .
  4. A Genetic Algorithm for Cryptanalysis of the Vigenere Cipher  – This paper describes the application of genetic algorithms to cryptanalysis problems. Although the focus is on genetic algorithms, the paper also discusses statistical properties of texts, which may be related to the application of the theorem of large numbers in estimating symbol frequencies 4 .

These works can serve as a basis for further study of the application of the large number theorem in the field of cryptanalysis.

There are several cryptanalysis techniques using the large number theorem that have proven effective in various scenarios. The main ones include:

  1. Statistical frequency analysis :
    • This method is based on the assumption that in any sufficiently large text, the frequency of occurrence of characters will follow a certain distribution, which corresponds to the theorem of large numbers. For example, in Russian texts, the average frequencies of letters are known. Cryptanalysts can use this data to analyze ciphertexts and identify ciphers, such as the Vigenère cipher, by comparing the actual frequencies with the expected ones 2 .
  2. Methods based on genetic algorithms :
    • Genetic algorithms are used to optimize key search in cryptanalysis. They use the properties of the symbol frequency distribution and the theorem of large numbers to find the most probable keys, which allows for efficient solution of combinatorial optimization problems 2 .
  3. Known Plaintext Attacks :
    • In this method, the cryptanalyst uses known plaintext and ciphertext pairs to estimate probabilities and find keys. The LBN helps estimate how often different combinations can occur in the ciphertext, which greatly speeds up the cryptanalysis process 1 .
  4. Side-channel attacks :
    • These attacks use statistics of the execution time of encryption operations or other physical parameters to extract information about the keys. The theorem of large numbers can help in estimating the probabilities of various outcomes and their impact on the security of the system 1 .

These methods demonstrate how theoretical concepts from statistics can be effectively integrated into practical approaches to cryptanalysis, improving the chances of successfully breaking ciphers.

There are several exotic cryptanalysis techniques based on neural networks and genetic algorithms that can be used in combination with the large number theorem. Here are some of them:

  1. Genetic algorithms for cryptanalysis :
    • Genetic algorithms (GA) can be used to find secret keys in ciphers such as the Vigenere cipher. This approach uses an evolutionary mechanism, where “individuals” represent potential keys and a “fitness function” estimates their efficiency. GA allows for efficient exploration of the space of possible keys using statistical properties of texts and symbol frequencies, which is consistent with the principles of the large number theorem 1 .
  2. Neural networks for cipher analysis :
    • Neural networks can be used to classify ciphertexts and identify patterns that may indicate the structure of the cipher. Training a neural network on a large amount of data allows it to identify statistical relationships that can be interpreted using the theorem of large numbers. This can help in determining the probabilities of certain symbols or combinations, which is useful in attacking ciphers 2 .
  3. Combined methods :
    • Using neural networks in combination with genetic algorithms to optimize model parameters can significantly improve the efficiency of cryptanalysis. Genetic algorithms can optimize the architecture of a neural network or its parameters, which allows finding more accurate models for analyzing ciphertexts. This combination can use statistical properties of the data to improve the results of the analysis 3 .
  4. Parametric optimization methods :
    • The use of parametric optimization methods using genetic algorithms to tune neural networks allows for effective solving of cryptanalysis problems. These methods may include the use of the theorem of large numbers to assess the stability and reliability of models when analyzing large amounts of data 3 .

These methods show how modern approaches to machine learning and evolutionary computation can be integrated into cryptanalysis, using mathematical principles to improve the effectiveness of attacks on cryptographic systems.

Conclusion

The use of lattice reduction algorithms and the large number theorem in cryptanalysis opens up new possibilities for increasing the effectiveness of attacks on modern cryptographic systems. The LLL algorithm provides a powerful tool for working with lattices, and the LN allows one to make reasonable conclusions about the success probabilities of various attack methods. These approaches emphasize the importance of mathematical foundations in the field of information security.

Citations:
[1] https://ru.wikipedia.org/wiki/%D0%A0%D0%B5%D1%88%D0%B5%D1%82%D0%BE_%D0%AD%D1%80% D0%B0%D1%82%D0%BE%D1%81%D1%84%D0%B5%D0%BD%D0%B0
[2] https://cyclowiki.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC_%D0%B1%D0%B0 %D0%B7%D0%B8%D1%81%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1 %80%D0%B0%D0%B7% D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F_%D0%BA%D1%80%D0%B8%D0%BF%D1%82%D0%BE%D0% B3%D1%80%D0%B0%D1%84%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B9_%D1%80%D0%B5% D1%88%D1%91%D1%82%D0%BA%D0%B8
[3] https://portal.tpu.ru/SHARED/v/VIR/Met/Tab1/OPTIMISATION.pdf
[4] https://ru.wikipedia.org/wiki/%D0%97%D0%B0% D0%BA%D0%BE%D0%BD_%D0%B1%D0%BE%D0%BB%D1%8C%D1%88%D0%B8%D1%85_%D1%87%D0%B8%D1% 81%D0%B5%D0%BB
[5] https://cyberleninka.ru/article/n/effektivnoe-masshtabirovanie-v-sisteme-ostatochnyh-klassov-s-ispolzovaniem-intervalnyh-otsenok
[6] https:// homepage.mi-ras.ru/~podolskii/files/integer.pdf
[7] http://www.ict.nsc.ru/matmod/files/textbooks/SharyNuMeth.pdf
[8] https://teach-in.ru/file/synopsis/pdf/numerical-methods-in-physics-M. pdf