Generating Graphs Packed with Paths Estimation of Linear Approximations and Differentials

  • Mathias Hall-Andersen University of Copenhagen, Copenhagen, Denmark
  • Philip S. Vejre Technical University of Denmark, Kongens Lyngby, Denmark
Keywords: Linear cryptanalysis, Differential cryptanalysis, Linear Approximations, Differentials, Trail search, Correlation distributions, Graph theory

Abstract

When designing a new symmetric-key primitive, the designer must show resistance to known attacks. Perhaps most prominent amongst these are linear and differential cryptanalysis. However, it is notoriously difficult to accurately demonstrate e.g. a block cipher’s resistance to these attacks, and thus most designers resort to deriving bounds on the linear correlations and differential probabilities of their design. On the other side of the spectrum, the cryptanalyst is interested in accurately assessing the strength of a linear or differential attack.
While several tools have been developed to search for optimal linear and differential trails, e.g. MILP and SAT based methods, only few approaches specifically try to find as many trails of a single approximation or differential as possible. This can result in an overestimate of a cipher’s resistance to linear and differential attacks, as was for example the case for PRESENT.
In this work, we present a new algorithm for linear and differential trail search. The algorithm represents the problem of estimating approximations and differentials as the problem of finding many long paths through a multistage graph. We demonstrate that this approach allows us to find a very large number of good trails for each approximation or differential. Moreover, we show how the algorithm can be used to efficiently estimate the key dependent correlation distribution of a linear approximation, facilitating advanced linear attacks. We apply the algorithm to 17 different ciphers, and present new and improved results on several of these.

References

Mohamed Ahmed Abdelraheem, Javad Alizadeh, Hoda A. AlKhzaimi, Mohammad Reza Aref, Nasour Bagheri, and Praveen Gauravaram. Improved Linear Cryptanalysis of Reduced-Round SIMON-32 and SIMON-48. In Progress in Cryptology - INDOCRYPT 2015 - 16th International Conference on Cryptology in India, Bangalore, India, December 6-9, 2015, Proceedings, pages 153–179, 2015.

Mohamed Ahmed Abdelraheem. Estimating the Probabilities of Low-Weight Differential and Linear Approximations on PRESENT-Like Ciphers. In Information Security and Cryptology - ICISC 2012 - 15th International Conference, Seoul, Korea, November 28-30, 2012, Revised Selected Papers, pages 368–382, 2012.

Martin R. Albrecht, Benedikt Driessen, Elif Bilge Kavun, Gregor Leander, Christof Paar, and Tolga Yalçin. Block Ciphers - Focus on the Linear Layer (feat. PRIDE). In Advances in Cryptology - CRYPTO 2014 - 34th Annual Cryptology Conference, Santa Barbara, CA, USA, August 17-21, 2014, Proceedings, Part I, pages 57–76, 2014.

Ralph Ankele and Stefan Kölbl. Mind the Gap - A Closer Look at the Security of Block Ciphers against Differential Cryptanalysis. In Selected Areas in Cryptography - SAC 2018, 2018.

Roberto Avanzi. The QARMA Block Cipher Family. Almost MDS Matrices Over Rings With Zero Divisors, Nearly Symmetric Even-Mansour Constructions With Non-Involutory Central Rounds, and Search Heuristics for Low-Latency S-Boxes. IACR Trans. Symmetric Cryptol., 2017(1):4–44, 2017.

Subhadeep Banik, Andrey Bogdanov, Takanori Isobe, Kyoji Shibutani Harunaga Hiwatari, Toru Akishita, and Francesco Regazzoni. Midori: A Block Cipher for Low Energy. In Advances in Cryptology - ASIACRYPT 2015 - 21st International Conference on the Theory and Application of Cryptology and Information Security, Auckland, New Zealand, November 29 - December 3, 2015, Proceedings, Part II, pages 411–436, 2015.

Julia Borghoff, Anne Canteaut, Tim Güneysu, Elif Bilge Kavun, Miroslav Knežević, Lars R. Knudsen, Gregor Leander, Ventzislav Nikov, Christof Paar, Christian Rechberger, Peter Rombouts, Søren S. Thomsen, and Tolga Yalçin. PRINCE - A Low-Latency Block Cipher for Pervasive Computing Applications - Extended Abstract. In Advances in Cryptology - ASIACRYPT 2012 - 18th International Conference on the Theory and Application of Cryptology and Information Security, Beijing, China, December 2-6, 2012. Proceedings, pages 208–225, 2012.

Christof Beierle, Jérémy Jean, Stefan Kölbl, Gregor Leander, Amir Moradi, Thomas Peyrin, Yu Sasaki, Pascal Sasdrich, and Siang Meng Sim. The SKINNY Family of Block Ciphers and Its Low-Latency Variant MANTIS. In Advances in Cryptology - CRYPTO 2016 - 36th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 14-18, 2016, Proceedings, Part II, pages 123–153, 2016.

Andrey Bogdanov, Lars R. Knudsen, Gregor Leander, Christof Paar, Axel Poschmann, Matthew J. B. Robshaw, Yannick Seurin, and C. Vikkelsoe. PRESENT: An Ultra-Lightweight Block Cipher. In Cryptographic Hardware and Embedded Systems - CHES 2007, 9th International Workshop, Vienna, Austria, September 10-13, 2007, Proceedings, pages 450–466, 2007.

Céline Blondeau and Kaisa Nyberg. Improved Parameter Estimates for Correlation and Capacity Deviates in Linear Cryptanalysis. IACR Trans. Symmetric Cryptol., 2016(2):162–191, 2016.

Céline Blondeau and Kaisa Nyberg. Joint Data and Key Distribution of Simple, Multiple, and Multidimensional Linear Cryptanalysis Test Statistic and Its Impact to Data Complexity. Design, Codes and Cryptography, 82(1-2):319–349, 2017.

Subhadeep Banik, Sumit Kumar Pandey, Thomas Peyrin, Yu Sasaki, Siang Meng Sim, and Yosuke Todo. GIFT: A Small Present - Towards Reaching the Limit of Lightweight Encryption. In Cryptographic Hardware and Embedded Systems - CHES 2017 - 19th International Conference, Taipei, Taiwan, September 25-28, 2017, Proceedings, pages 321–345, 2017.

Paulo S.L.M. Barreto and Vincent Rijmen. The khazad legacy-level block cipher. Primitive submitted to NESSIE, 97, 2000.

Eli Biham and Adi Shamir. Differential cryptanalysis of des-like cryptosystems. In Advances in Cryptology - CRYPTO ’90, 10th Annual International Cryptology Conference, Santa Barbara, California, USA, August 11-15, 1990, Proceedings, pages 2–21, 1990.

Andrey Bogdanov and Elmar Tischhauser. On the Wrong Key Randomisation and Key Equivalence Hypotheses in Matsui’s Algorithm 2. In Fast Software Encryption - 20th International Workshop, FSE 2013, Singapore, March 11-13, 2013. Revised Selected Papers, pages 19–38, 2013.

Andrey Bogdanov, Elmar Tischhauser, and Philip S. Vejre. Multivariate Profiling of Hulls for Linear Cryptanalysis. IACR Trans. Symmetric Cryptol., 2018(1):101–125, 2018.

Stanislav Bulygin. More on linear hulls of PRESENT-like ciphers and a cryptanalysis of full-round EPCBC-96. IACR Cryptology ePrint Archive, 2013:28, 2013.

Alex Biryukov and Vesselin Velichkov. Automatic Search for Differential Trails in ARX Ciphers. In Topics in Cryptology - CT-RSA 2014 - The Cryptographer’s Track at the RSA Conference 2014, San Francisco, CA, USA, February 25-28, 2014. Proceedings, pages 227–250, 2014.

Andrey Bogdanov and Philip S. Vejre. Linear Cryptanalysis of DES with Asymmetries. In Advances in Cryptology - ASIACRYPT 2017 - 23rd International Conference on the Theory and Applications of Cryptology and Information Security, Hong Kong, China, December 3-7, 2017, Proceedings, Part I, pages 187–216, 2017.

Anne Canteaut, Thomas Fuhr, Henri Gilbert, María Naya-Plasencia, and Jean-René Reinhard. Multiple Differential Cryptanalysis of Round-Reduced PRINCE. In Fast Software Encryption - 21st International Workshop, FSE 2014, London, UK, March 3-5, 2014. Revised Selected Papers, pages 591–610, 2014.

Huiju Cheng, Howard M. Heys, and Cheng Wang. PUFFIN: A Novel Compact Block Cipher Targeted to Embedded Digital Systems. In 11th Euromicro Conference on Digital System Design: Architectures, Methods and Tools, DSD 2008, Parma, Italy, September 3-5, 2008, pages 383–390, 2008.

Jiageng Chen, Atsuko Miyaji, Chunhua Su, and Jesen Teh. Accurate Estimation of the Full Differential Distribution for General Feistel Structures. In Information Security and Cryptology - 11th International Conference, Inscrypt 2015, Beijing, China, November 1-3, 2015, Revised Selected Papers, pages 108–124, 2015.

Jiageng Chen, Atsuko Miyaji, Chunhua Su, and Jesen Teh. Improved Differential Characteristic Searching Methods. In IEEE 2nd International Conference on Cyber Security and Cloud Computing, CSCloud 2015, New York, NY, USA, November 3-5, 2015, pages 500–508, 2015.

Christoph Dobraunig, Maria Eichlseder, and Florian Mendel. Heuristic Tool for Linear Cryptanalysis with Applications to CAESAR Candidates. In Advances in Cryptology - ASIACRYPT 2015 - 21st International Conference on the Theory and Application of Cryptology and Information Security, Auckland, New Zealand, November 29 - December 3, 2015, Proceedings, Part II, pages 490–509, 2015.

Joan Daemen and Vincent Rijmen. The Design of Rijndael: AES - The Advanced Encryption Standard. Information Security and Cryptography. Springer, 2002.

Joan Daemen and Vincent Rijmen. Probability distributions of correlation and differentials in block ciphers. J. Mathematical Cryptology, 1(3):221–242, 2007.

Pierre-Alain Fouque, Gaëtan Leurent, and Phong Q. Nguyen. Automatic Search of Differential Path in MD4. IACR Cryptology ePrint Archive, 2007:206, 2007.

Kai Fu, Meiqin Wang, Yinghua Guo, Siwei Sun, and Lei Hu. MILP-Based Automatic Search Algorithms for Differential and Linear Trails for Speck. In Fast Software Encryption - 23rd International Conference, FSE 2016, Bochum, Germany, March 20-23, 2016, Revised Selected Papers, pages 268–288, 2016.

Zheng Gong, Svetla Nikova, and Yee Wei Law. KLEIN: A New Family of Lightweight Block Ciphers. In RFID. Security and Privacy - 7th International Workshop, RFIDSec 2011, Amherst, USA, June 26-28, 2011, Revised Selected Papers, pages 1–18, 2011.

Jian Guo, Thomas Peyrin, Axel Poschmann, and Matthew J. B. Robshaw. The LED Block Cipher. In Cryptographic Hardware and Embedded Systems - CHES 2011 - 13th International Workshop, Nara, Japan, September 28 - October 1, 2011. Proceedings, pages 326–341, 2011.

Jialin Huang, Serge Vaudenay, Xuejia Lai, and Kaisa Nyberg. Capacity and Data Complexity in Multidimensional Linear Attack. In Advances in Cryptology - CRYPTO 2015 - 35th Annual Cryptology Conference, Santa Barbara, CA, USA, August 16-20, 2015, Proceedings, Part I, pages 141–160, 2015.

Pierre Karpman and Benjamin Grégoire. The LITTLUN S-box and the FLY block cipher. In Lightweight Cryptography Workshop, pages 17–18, 2016.

Stefan Kölbl, Gregor Leander, and Tyge Tiessen. Observations on the SIMON Block Cipher Family. In Advances in Cryptology - CRYPTO 2015 - 35th Annual Cryptology Conference, Santa Barbara, CA, USA, August 16-20, 2015, Proceedings, Part I, pages 161–185, 2015.

Xuejia Lai, James L. Massey, and Sean Murphy. Markov ciphers and differential cryptanalysis. In Advances in Cryptology - EUROCRYPT ’91, Workshop on the Theory and Application of of Cryptographic Techniques, Brighton, UK, April 8-11, 1991, Proceedings, pages 17–38, 1991.

Mitsuru Matsui. Linear cryptanalysis method for DES cipher. In Advances in Cryptology - EUROCRYPT ’93, Workshop on the Theory and Application of Cryptographic Techniques, Lofthus, Norway, May 23-27, 1993, Proceedings, pages 386–397, 1993.

Mitsuru Matsui. On Correlation Between the Order of S-boxes and the Strength of DES. In Advances in Cryptology - EUROCRYPT ’94, Workshop on the Theory and Application of Cryptographic Techniques, Perugia, Italy, May 9-12, 1994, Proceedings, pages 366–375, 1994.

Nicky Mouha and Bart Preneel. Towards Finding Optimal Differential Characteristics for ARX: Application to Salsa20. IACR Cryptology ePrint Archive, 2013:328, 2013.

Nicky Mouha, Qingju Wang, Dawu Gu, and Bart Preneel. Differential and Linear Cryptanalysis Using Mixed-Integer Linear Programming. In Information Security and Cryptology - 7th International Conference, Inscrypt 2011, Beijing, China, November 30 - December 3, 2011. Revised Selected Papers, pages 57–76, 2011.

Kaisa Nyberg. Linear approximation of block ciphers. In Advances in Cryptology - EUROCRYPT ’94, Workshop on the Theory and Application of Cryptographic Techniques, Perugia, Italy, May 9-12, 1994, Proceedings, pages 439–444, 1994.

Kenji Ohkuma. Weak Keys of Reduced-Round PRESENT for Linear Cryptanalysis. In Selected Areas in Cryptography, 16th Annual International Workshop, SAC 2009, Calgary, Alberta, Canada, August 13-14, 2009, Revised Selected Papers, pages 249–265, 2009.

National Institute of Standards and Technology. 197: Advanced encryption standard (AES). Federal information processing standards publication, 197(441):0311, 2001.

Siwei Sun, Lei Hu, Peng Wang, Kexin Qiao, Xiaoshuang Ma, and Ling Song. Automatic Security Evaluation and (Related-key) Differential Characteristic Search: Application to SIMON, PRESENT, LBlock, DES(L) and Other Bit-Oriented Block Ciphers. In Advances in Cryptology - ASIACRYPT 2014 - 20th International Conference on the Theory and Application of Cryptology and Information Security, Kaoshiung, Taiwan, R.O.C., December 7-11, 2014. Proceedings, Part I, pages 158–178, 2014.

Marc Stevens. New Collision Attacks on SHA-1 Based on Optimal Joint Local-Collision Analysis. In Advances in Cryptology - EUROCRYPT 2013, 32nd Annual International Conference on the Theory and Applications of Cryptographic Techniques, Athens, Greece, May 26-30, 2013. Proceedings, pages 245–261, 2013.

S. H. Yen, David Hung-Chang Du, and Subbarao Ghanta. Efficient Algorithms for Extracting the K most Critical Paths in Timing Analysis. In Proceedings of the 26th ACM/IEEE Design Automation Conference, Las Vegas, Nevada, USA, June 25-29, 1989., pages 649–654, 1989. Huihui Yap, Khoongming Khoo, Axel Poschmann, and Matt Henricksen. EPCBC - A Block Cipher Suitable for Electronic Product Code Encryption. In Cryptology and Network Security - 10th International Conference, CANS 2011, Sanya, China, December 10-12, 2011. Proceedings, pages 76–97, 2011.

Jun Yin, Chuyan Ma, Lijun Lyu, Jian Song, Guang Zeng, Chuangui Ma, and Fushan Wei. Improved Cryptanalysis of an ISO Standard Lightweight Block Cipher with Refined MILP Modelling. In Information Security and Cryptology - 13th International Conference, Inscrypt 2017, Xi’an, China, November 3-5, 2017, Revised Selected Papers, pages 404–426, 2017.

Wentao Zhang, Zhenzhen Bao, Dongdai Lin, Vincent Rijmen, Bohan Yang, and Ingrid Verbauwhede. RECTANGLE: A Bit-slice Ultra-Lightweight Block Cipher Suitable for Multiple Platforms. IACR Cryptology ePrint Archive, 2014:84, 2014.

Baoyu Zhu, Xiaoyang Dong, and Hongbo Yu. MILP-based Differential Attack on Round-reduced GIFT. Cryptology ePrint Archive, Report 2018/390, 2018. https://eprint.iacr.org/2018/390.

Published
2018-09-04
How to Cite
Hall-Andersen, M., & Vejre, P. S. (2018). Generating Graphs Packed with Paths Estimation of Linear Approximations and Differentials. IACR Transactions on Symmetric Cryptology, 2018(3), 265-289. https://doi.org/10.13154/tosc.v2018.i3.265-289
Section
Articles