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August  2021, 15(3): 397-413. doi: 10.3934/amc.2020073

Ironwood meta key agreement and authentication protocol

Iris Anshel , Derek Atkins , Dorian Goldfeld and  Paul E. Gunnells

100 Beard Sawmill Rd, Suite 350, Shelton, CT 06484, USA

Received  September 2019 Published  August 2021 Early access  April 2020

Number theoretic public-key solutions, currently used in many applications worldwide, will be subject to various quantum attacks, making them less attractive for longer-term use. Certain group theoretic constructs are now showing promise in providing quantum-resistant cryptographic primitives, and may provide suitable alternatives for those looking to address known quantum attacks. In this paper, we introduce a new protocol called a Meta Key Agreement and Authentication Protocol (MKAAP) that has some characteristics of a public-key solution and some of a shared-key solution. Specifically, it has the deployment benefits of a public-key system, allowing two entities that have never met before to authenticate without requiring real-time access to a third-party, but does require secure provisioning of key material from a trusted key distribution system (similar to a symmetric system) prior to deployment. We then describe a specific MKAAP instance, the Ironwood MKAAP, discuss its security, and show how it resists certain quantum attacks such as Shor's algorithm or Grover's quantum search algorithm. We also show Ironwood implemented on several "internet of things" (IoT devices), measure its performance, and show how it performs significantly better than ECC using fewer device resources.

Keywords: Public-key cryptography, braids, Diffie-Hellman, post-quantum cryptography, internet of things, IoT.
Mathematics Subject Classification: Primary: 94A60, 20F36, 68P25; Secondary: 81P94, 20G40, 20E36.
Citation: Iris Anshel, Derek Atkins, Dorian Goldfeld, Paul E. Gunnells. Ironwood meta key agreement and authentication protocol. Advances in Mathematics of Communications, 2021, 15 (3) : 397-413. doi: 10.3934/amc.2020073
References:
[1]

I. Anshel, M. Anshel, D. Goldfeld and S. Lemieux, Key agreement, the Algebraic Eraser$^{TM}$, and lightweight cryptography, Algebraic Methods in Cryptography, Contemp. Math., Amer. Math. Soc., Providence, RI, 418 (2006), 1-34.  doi: 10.1090/conm/418/07943.

[2]

I. Anshel, D. Atkins, D. Goldfeld and P. E. Gunnells, Defeating the Ben-Zvi, Blackburn, and Tsaban Attack on the Algebraic Eraser, arXiv: 1601.04780v1 [cs.CR].

[3]

I. Anshel, D. Atkins, D. Goldfeld and P. E. Gunnells, WalnutDSA$^{TM}$: A Lightweight Quantum Resistant Digital Signature Algorithm, https://eprint.iacr.org/2017/058.pdf.

[4]

I. Anshel, M. Anshel and D. Goldfeld, An algebraic method for public-key cryptography, Math. Res. Lett., 6 (1999), 287-291.  doi: 10.4310/MRL.1999.v6.n3.a3.

[5]

E. Artin, The theory of braids, Annals of Math., 48 (1947), 101-126.  doi: 10.2307/1969218.

[6]

D. Atkins and D. Goldfeld, Addressing the algebraic eraser over the air protocol, https://eprint.iacr.org/2016/205.pdf.

[7]

A. Ben-Zvi, S. R. Blackburn and B. Tsaban, A practical cryptanalysis of the algebraic eraser, Advances in Cryptology - CRYPTO 2016, Part I, Lecture Notes in Comput. Sci., Springer, Berlin, 9814 (2016), 179–189. http://eprint.iacr.org/2015/1102. doi: 10.1007/978-3-662-53018-4_7.

[8]

S. R. Blackburn and M. J. B. Robshaw, On the security of the algebraic eraser tag authentication protocol, Applied Cryptography and Network Security, Lecture Notes in Comput. Sci., Springer, 9696 (2016), 3–17. http://eprint.iacr.org/2016/091. doi: 10.1007/978-3-319-39555-5_1.

[9]

M. Düll, B. Haase, G. Hinterwälder, M. Hutter, C. Paar, A. H. Sánchez and P. Schwabe, High-speed Curve25519 on 8-bit, 16-bit, and 32-bit microcontrollers, https://eprint.iacr.org/2015/343.pdf.

[10]

D. Garber, S. Kaplan, M. Teicher, B. Tsaban and U. Vishne, Length-based conjugacy search in the braid group, Algebraic Methods in Cryptography, Contemp. Math., Amer. Math. Soc., Providence, RI, 418 (2006), 75-87.  doi: 10.1090/conm/418/07947.

[11]

V. Gebhardt, A new approach to the conjugacy problem in Garside groups, J. Algebra, 292 (2005), 282-302.  doi: 10.1016/j.jalgebra.2005.02.002.

[12]

D. Goldfeld and P. E. Gunnells, Defeating the Kalka-Teicher-Tsaban linear algebra attack on the Algebraic Eraser, (2012), arXiv: 1202.0598.

[13] M. I. González Vasco, S. Magliveras and R. Steinwandt, Group-Theoretic Cryptography, Chapman & Hall/CRC Cryptography and Network Security, CRC Press, Boca Raton, FL, 2015. 
[14]

L. K. Grover, A fast quantum mechanical algorithm for database search, Proceedings, 28th Annual ACM, Symposium on the Theory of Computing, ACM, New York, (1996), 212-219.  doi: 10.1145/237814.237866.

[15]

P. E. Gunnells, On the cryptanalysis of the generalized simultaneous conjugacy search problem and the security of the Algebraic Eraser, arXiv: 1105.1141v1 [cs.CR].

[16]

D. Hofheinz and R. Steinwandt, A practical attack on some braid group based cryptographic primitives, Public Key Cryptography—PKC 2003, Lecture Notes in Comput. Sci., Springer, Berlin, 2567 (2002), 187-198.  doi: 10.1007/3-540-36288-6_14.

[17]

A. Kalka, M. Teicher and B. Tsaban, Short expressions of permutations as products and cryptanalysis of the Algebraic Eraser, Advances in Applied Mathematics, 49 (2012), 57-76.  doi: 10.1016/j.aam.2012.03.001.

[18]

K. H. Ko, S. J. Lee, J. H. Cheon, J. W. Han, J. Kang and C. Park, New public-key cryptosystem using braid groups, Advances in Cryptology - CRYPTO 2000, Lecture Notes in Computer Science, Springer, Berlin, 1880 (2000), 166-183.  doi: 10.1007/3-540-44598-6_10.

[19]

C. Lomont, The hidden subgroup problem - review and open problems, (2004), arXiv: quant-ph/0411037.

[20]

H. R. Morton, The multivariable Alexander polynomial for a closed braid, Low-dimensional topology, Contemp. Math., Amer. Math. Soc., Providence, RI, 233 (1999), 167-172.  doi: 10.1090/conm/233/03427.

[21]

C. Mulland and B. Tsaban, $ \rm SL _2$ homomorphic hash functions: Worst case to average case reduction and short collision search, Des. Codes Cryptogr., 81 (2016), 83–107, arXiv: 1306.5646v3 [cs.CR]. doi: 10.1007/s10623-015-0129-8.

[22]

A. D. Myasnikov and A. Ushakov, Cryptanalysis of the Anshel-Anshel-Goldfeld-Lemieux key agreement protocol, Groups Complex. Cryptol., 1 (2009), 63-75.  doi: 10.1515/GCC.2009.63.

[23]

A. Myasnikov, V. Shpilrain and A. Ushakov, Group-Based Cryptography, Advanced Courses in Mathematics, CRM Barcelona Birkhäuser, Basel, 2008.

[24]

P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Computing, 26 (1997), 1484-1509.  doi: 10.1137/S0097539795293172.

[25]

G. Zémor, Hash functions and graphs with large girths, Eurocrypt '91, Lecture Notes in Computer Science, 547 (1991), 508-511. 

show all references

References:
[1]

I. Anshel, M. Anshel, D. Goldfeld and S. Lemieux, Key agreement, the Algebraic Eraser$^{TM}$, and lightweight cryptography, Algebraic Methods in Cryptography, Contemp. Math., Amer. Math. Soc., Providence, RI, 418 (2006), 1-34.  doi: 10.1090/conm/418/07943.

[2]

I. Anshel, D. Atkins, D. Goldfeld and P. E. Gunnells, Defeating the Ben-Zvi, Blackburn, and Tsaban Attack on the Algebraic Eraser, arXiv: 1601.04780v1 [cs.CR].

[3]

I. Anshel, D. Atkins, D. Goldfeld and P. E. Gunnells, WalnutDSA$^{TM}$: A Lightweight Quantum Resistant Digital Signature Algorithm, https://eprint.iacr.org/2017/058.pdf.

[4]

I. Anshel, M. Anshel and D. Goldfeld, An algebraic method for public-key cryptography, Math. Res. Lett., 6 (1999), 287-291.  doi: 10.4310/MRL.1999.v6.n3.a3.

[5]

E. Artin, The theory of braids, Annals of Math., 48 (1947), 101-126.  doi: 10.2307/1969218.

[6]

D. Atkins and D. Goldfeld, Addressing the algebraic eraser over the air protocol, https://eprint.iacr.org/2016/205.pdf.

[7]

A. Ben-Zvi, S. R. Blackburn and B. Tsaban, A practical cryptanalysis of the algebraic eraser, Advances in Cryptology - CRYPTO 2016, Part I, Lecture Notes in Comput. Sci., Springer, Berlin, 9814 (2016), 179–189. http://eprint.iacr.org/2015/1102. doi: 10.1007/978-3-662-53018-4_7.

[8]

S. R. Blackburn and M. J. B. Robshaw, On the security of the algebraic eraser tag authentication protocol, Applied Cryptography and Network Security, Lecture Notes in Comput. Sci., Springer, 9696 (2016), 3–17. http://eprint.iacr.org/2016/091. doi: 10.1007/978-3-319-39555-5_1.

[9]

M. Düll, B. Haase, G. Hinterwälder, M. Hutter, C. Paar, A. H. Sánchez and P. Schwabe, High-speed Curve25519 on 8-bit, 16-bit, and 32-bit microcontrollers, https://eprint.iacr.org/2015/343.pdf.

[10]

D. Garber, S. Kaplan, M. Teicher, B. Tsaban and U. Vishne, Length-based conjugacy search in the braid group, Algebraic Methods in Cryptography, Contemp. Math., Amer. Math. Soc., Providence, RI, 418 (2006), 75-87.  doi: 10.1090/conm/418/07947.

[11]

V. Gebhardt, A new approach to the conjugacy problem in Garside groups, J. Algebra, 292 (2005), 282-302.  doi: 10.1016/j.jalgebra.2005.02.002.

[12]

D. Goldfeld and P. E. Gunnells, Defeating the Kalka-Teicher-Tsaban linear algebra attack on the Algebraic Eraser, (2012), arXiv: 1202.0598.

[13] M. I. González Vasco, S. Magliveras and R. Steinwandt, Group-Theoretic Cryptography, Chapman & Hall/CRC Cryptography and Network Security, CRC Press, Boca Raton, FL, 2015. 
[14]

L. K. Grover, A fast quantum mechanical algorithm for database search, Proceedings, 28th Annual ACM, Symposium on the Theory of Computing, ACM, New York, (1996), 212-219.  doi: 10.1145/237814.237866.

[15]

P. E. Gunnells, On the cryptanalysis of the generalized simultaneous conjugacy search problem and the security of the Algebraic Eraser, arXiv: 1105.1141v1 [cs.CR].

[16]

D. Hofheinz and R. Steinwandt, A practical attack on some braid group based cryptographic primitives, Public Key Cryptography—PKC 2003, Lecture Notes in Comput. Sci., Springer, Berlin, 2567 (2002), 187-198.  doi: 10.1007/3-540-36288-6_14.

[17]

A. Kalka, M. Teicher and B. Tsaban, Short expressions of permutations as products and cryptanalysis of the Algebraic Eraser, Advances in Applied Mathematics, 49 (2012), 57-76.  doi: 10.1016/j.aam.2012.03.001.

[18]

K. H. Ko, S. J. Lee, J. H. Cheon, J. W. Han, J. Kang and C. Park, New public-key cryptosystem using braid groups, Advances in Cryptology - CRYPTO 2000, Lecture Notes in Computer Science, Springer, Berlin, 1880 (2000), 166-183.  doi: 10.1007/3-540-44598-6_10.

[19]

C. Lomont, The hidden subgroup problem - review and open problems, (2004), arXiv: quant-ph/0411037.

[20]

H. R. Morton, The multivariable Alexander polynomial for a closed braid, Low-dimensional topology, Contemp. Math., Amer. Math. Soc., Providence, RI, 233 (1999), 167-172.  doi: 10.1090/conm/233/03427.

[21]

C. Mulland and B. Tsaban, $ \rm SL _2$ homomorphic hash functions: Worst case to average case reduction and short collision search, Des. Codes Cryptogr., 81 (2016), 83–107, arXiv: 1306.5646v3 [cs.CR]. doi: 10.1007/s10623-015-0129-8.

[22]

A. D. Myasnikov and A. Ushakov, Cryptanalysis of the Anshel-Anshel-Goldfeld-Lemieux key agreement protocol, Groups Complex. Cryptol., 1 (2009), 63-75.  doi: 10.1515/GCC.2009.63.

[23]

A. Myasnikov, V. Shpilrain and A. Ushakov, Group-Based Cryptography, Advanced Courses in Mathematics, CRM Barcelona Birkhäuser, Basel, 2008.

[24]

P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Computing, 26 (1997), 1484-1509.  doi: 10.1137/S0097539795293172.

[25]

G. Zémor, Hash functions and graphs with large girths, Eurocrypt '91, Lecture Notes in Computer Science, 547 (1991), 508-511. 

Figure 1.  Ironwood Data Flows
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Figure 2.  Ironwood Protocol Flow
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Table 1.  Performance on MSP430, LPC1768 (in Cycles)
Artin Length MSP430 LPC1768
$ |\beta| $ $ |\beta'| $
2626 5272 6002668 2026216
2332 3580 4532480 1538472
2414 3944 4862464 1648742
3172 4266 5661952 1914009
2168 4514 5101824 1728545
3092 4698 5922048 2000312
2978 3968 5297664 1792959
2744 4420 5459456 1845502
2430 4762 5479424 1854446
2636 3600 4771840 1617670
2659.2 4302.4 5309182 1796687
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Download as excel

Artin Length MSP430 LPC1768
$ |\beta| $ $ |\beta'| $
2626 5272 6002668 2026216
2332 3580 4532480 1538472
2414 3944 4862464 1648742
3172 4266 5661952 1914009
2168 4514 5101824 1728545
3092 4698 5922048 2000312
2978 3968 5297664 1792959
2744 4420 5459456 1845502
2430 4762 5479424 1854446
2636 3600 4771840 1617670
2659.2 4302.4 5309182 1796687
Table 2.  Performance on MSP430, LPC1768, CC2650 (in ms)
Artin Length MSP430 LPC1768 CC2650 48Mhz
$ |\beta| $ $ |\beta'| $ 25MHz 48MHz (SST 5) (SST 2)
2626 5272 240.1 42.2 42 42
2332 3580 181.3 32.2 32 32
2414 3944 194.5 34.3 34 35
3172 4266 226.5 39.9 40 40
2168 4514 204.1 36.0 36 36
3092 4698 236.9 41.2 42 42
2978 3968 211.9 37.4 37 37
2744 4420 218.4 38.4 38 39
2430 4762 219.2 38.6 39 39
2636 3600 190.9 33.7 34 34
2659.2 4302.4 212.4 37.4 37.4 37.6
Table Options

Download as excel

Artin Length MSP430 LPC1768 CC2650 48Mhz
$ |\beta| $ $ |\beta'| $ 25MHz 48MHz (SST 5) (SST 2)
2626 5272 240.1 42.2 42 42
2332 3580 181.3 32.2 32 32
2414 3944 194.5 34.3 34 35
3172 4266 226.5 39.9 40 40
2168 4514 204.1 36.0 36 36
3092 4698 236.9 41.2 42 42
2978 3968 211.9 37.4 37 37
2744 4420 218.4 38.4 38 39
2430 4762 219.2 38.6 39 39
2636 3600 190.9 33.7 34 34
2659.2 4302.4 212.4 37.4 37.4 37.6
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