${\sf {FAST}}$: Disk encryption and beyond

Debrup Chakraborty; Sebati Ghosh; Cuauhtemoc Mancillas López; Palash Sarkar

doi:10.3934/amc.2020108

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February 2022, 16(1): 185-230. doi: 10.3934/amc.2020108

${\sf {FAST}}$: Disk encryption and beyond

Debrup Chakraborty ^1, , Sebati Ghosh ^1,, , Cuauhtemoc Mancillas López ^2, and Palash Sarkar ^1,

1.	Indian Statistical Institute, 203, B.T. Road, Kolkata, India 700108
2.	Computer Science Department, CINVESTAV-IPN, Mexico, D.F., 07360, Mexico

^* Corresponding author: Sebati Ghosh

Received February 2020 Published February 2022 Early access September 2020

This work introduces ${\sf {FAST}}$ which is a new family of tweakable enciphering schemes. Several instantiations of ${\sf {FAST}}$ are described. These are targeted towards two goals, the specific task of disk encryption and a more general scheme suitable for a wide variety of practical applications. A major contribution of this work is to present detailed and careful software implementations of all of these instantiations. For disk encryption, the results from the implementations show that ${\sf {FAST}}$ compares very favourably to the IEEE disk encryption standards XCB and EME2 as well as the more recent proposal AEZ. ${\sf {FAST}}$ is built using a fixed input length pseudo-random function and an appropriate hash function. It uses a single-block key, is parallelisable and can be instantiated using only the encryption function of a block cipher. The hash function can be instantiated using either the Horner's rule based usual polynomial hashing or hashing based on the more efficient Bernstein-Rabin-Winograd polynomials. Security of ${\sf {FAST}}$ has been rigorously analysed using the standard provable security approach and concrete security bounds have been derived. Based on our implementation results, we put forward ${\sf {FAST}}$ as a serious candidate for standardisation and deployment.

Keywords: Disk encryption, tweakable enciphering schemes, pseudo-random function, Horner, BRW.

Mathematics Subject Classification: 11T71, 68P25, 94A60.

Citation: Debrup Chakraborty, Sebati Ghosh, Cuauhtemoc Mancillas López, Palash Sarkar. ${\sf {FAST}}$: Disk encryption and beyond. Advances in Mathematics of Communications, 2022, 16 (1) : 185-230. doi: 10.3934/amc.2020108

References:

[1]	Public comments on the XTS-AES mode, http://csrc.nist.gov/groups/ST/toolkit/BCM/documents/comments/XTS/collected_XTS_comments.pdf. Google Scholar
[2]	IEEE Std 1619-2007: Standard for Cryptographic Protection of Data on Block-Oriented Storage Devices, IEEE Computer Society, 2008. Available at: http://standards.ieee.org/findstds/standard/1619-2007.html. Google Scholar
[3]	IEEE Std 1619.2-2010: IEEE Standard for Wide-block Encryption for Shared Storage Media, March 2011. Available at: http://standards.ieee.org/findstds/standard/1619.2-2010.html. Google Scholar
[4]	M. Bellare, D. Cash and S. Keelveedhi, Ciphers that securely encipher their own keys, in (eds. Y. Chen, G. Danezis and V. Shmatikov) Proceedings of the 18th ACM Conference on Computer and Communications Security, CCS 2011, Chicago, Illinois, USA, October 17-21, 2011, 2011,423–432. doi: 10.1145/2046707.2046757. Google Scholar
[5]	M. Bellare, A. Desai, E. Jokipii and P. Rogaway, A concrete security treatment of symmetric encryption, in 38th Annual Symposium on Foundations of Computer Science, FOCS '97, Miami Beach, Florida, USA, October 19-22, 1997, IEEE Computer Society, 1997,394–403. doi: 10.1109/SFCS.1997.646128. Google Scholar
[6]	D. J. Bernstein, Polynomial evaluation and message authentication, 2007. Available at: http://cr.yp.to/papers.html#pema. Google Scholar
[7]	R. Bhaumik and M. Nandi, An inverse-free single-keyed tweakable enciphering scheme, in (eds. T. Iwata and J. H. Cheon) 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, Lecture Notes in Computer Science, 9453, Springer, 2015,159–180. doi: 10.1007/978-3-662-48800-3_7. Google Scholar
[8]	D. Chakraborty, S. Ghosh and P. Sarkar, A fast single-key two-level universal hash function, IACR Trans. Symmetric Cryptol., 2017 (2017), 106–128. Google Scholar
[9]	D. Chakraborty, V. Hernandez-Jimenez and P. Sarkar, Another look at XCB, Cryptography and Communications, 7 (2015), 439-468. doi: 10.1007/s12095-015-0127-8. Google Scholar
[10]	D. Chakraborty, C. Mancillas-López, F. Rodríguez-Henríquez and P. Sarkar, Efficient hardware implementations of BRW polynomials and tweakable enciphering schemes, IEEE Trans. Computers, 62 (2013), 279–294. doi: 10.1109/TC.2011.227. Google Scholar
[11]	D. Chakraborty, C. Mancillas-López and P. Sarkar, STES: A stream cipher based low cost scheme for securing stored data, IEEE Trans. Computers, 64 (2015), 2691–2707. doi: 10.1109/TC.2014.2366739. Google Scholar
[12]	D. Chakraborty and M. Nandi, An improved security bound for HCTR, in (eds. K. Nyberg) Fast Software Encryption, 15th International Workshop, FSE 2008, Lausanne, Switzerland, February 10-13, 2008, Revised Selected Papers, Lecture Notes in Computer Science, 5086, Springer, 2008,289–302. doi: 10.1007/978-3-540-71039-4_18. Google Scholar
[13]	D. Chakraborty and P. Sarkar, A new mode of encryption providing a tweakable strong pseudo-random permutation, in (eds. M. J. B. Robshaw) FSE, Lecture Notes in Computer Science, 4047, Springer, 2006,293–309. doi: 10.1007/11799313_19. Google Scholar
[14]	D. Chakraborty and P. Sarkar, HCH: A new tweakable enciphering scheme using the hash-counter-hash approach, IEEE Transactions on Information Theory, 54 (2008), 1683-1699. doi: 10.1109/TIT.2008.917623. Google Scholar
[15]	D. Chakraborty and P. Sarkar, On modes of operations of a block cipher for authentication and authenticated encryption, Cryptography and Communications, 8 (2016), 455-511. doi: 10.1007/s12095-015-0153-6. Google Scholar
[16]	P. Crowley and E. Biggers, Adiantum: Length-preserving encryption for entry-level processors, IACR Trans. Symmetric Cryptol., 2018 (2018), 39–61. Google Scholar
[17]	M. J. Dworkin, SP 800-38E. Recommendation for Block Cipher Modes of Operation: The XTS-AES Mode for Confidentiality on Storage Devices, Technical report, Gaithersburg, MD, United States, 2010. Google Scholar
[18]	S. Gueron and M. E. Kounavis, Efficient implementation of the galois counter mode using a carry-less multiplier and a fast reduction algorithm, Inf. Process. Lett., 110 (2010), 549-553. doi: 10.1016/j.ipl.2010.04.011. Google Scholar
[19]	S. Gueron, A. Langley and Y. Lindell, AES-GCM-SIV: Specification and analysis, IACR Cryptology ePrint Archive, 168 (2017). doi: 10.1007/978-3-319-52153-4. Google Scholar
[20]	S. Halevi, EME$^{ * }$: Extending EME to handle arbitrary-length messages with associated data, in (eds. A. Canteaut and K. Viswanathan) INDOCRYPT, Lecture Notes in Computer Science, 3348, Springer 2004,315–327. doi: 10.1007/978-3-540-30556-9_25. Google Scholar
[21]	S. Halevi, Invertible universal hashing and the TET encryption mode, in (eds. A. Menezes) CRYPTO, Lecture Notes in Computer Science, 4622, Springer, (2007), pages 412–429. doi: 10.1007/978-3-540-74143-5_23. Google Scholar
[22]	S. Halevi and H. Krawczyk, Security under key-dependent inputs, in (eds. P. Ning, S. De Capitani di Vimercati, and P. F. Syverson) Proceedings of the 2007 ACM Conference on Computer and Communications Security, CCS 2007, Alexandria, Virginia, USA, October 28-31, 2007, 2007,466–475. doi: 10.1007/11935308. Google Scholar
[23]	S. Halevi and P. Rogaway, A tweakable enciphering mode, in (eds. D. Boneh) CRYPTO, Lecture Notes in Computer Science, 2729, Springer, 2003,482–499. doi: 10.1007/978-3-540-45146-4_28. Google Scholar
[24]	S. Halevi and P. Rogaway, A parallelizable enciphering mode, in (eds. T. Okamoto) CT-RSA, Lecture Notes in Computer Science, 2964, Springer, 2004,292–304. doi: 10.1007/978-3-540-24660-2_23. Google Scholar
[25]	V. T. Hoang, T. Krovetz and P. Rogaway, Robust authenticated-encryption AEZ and the problem that it solves, in (eds. E. Oswald and M. Fischlin) Advances in Cryptology - EUROCRYPT 2015 - 34th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Sofia, Bulgaria, April 26-30, 2015, Proceedings, Part I, Lecture Notes in Computer Science, 9056, Springer, 2015, 15–44. doi: 10.1007/978-3-662-46800-5_2. Google Scholar
[26]	M. Liskov, R. L. Rivest and D. Wagner, Tweakable block ciphers, in (eds. M. Yung) CRYPTO, Lecture Notes in Computer Science, 2442, Springer, 2002, 31–46. doi: 10.1007/3-540-45708-9_3. Google Scholar
[27]	D. A. McGrew and S. R. Fluhrer, The extended codebook (XCB) mode of operation, Cryptology ePrint Archive, Report 2004/278, 2004, Available at: http://eprint.iacr.org/. Google Scholar
[28]	D. A. McGrew and S. R. Fluhrer, The security of the extended codebook (xcb) mode of operation, in (eds. C. M. Adams, A. Miri, and M. J. Wiener) Selected Areas in Cryptography, Lecture Notes in Computer Science, 4876, Springer, 2007,311–327. doi: 10.1007/978-3-540-77360-3_20. Google Scholar
[29]	D. A. McGrew and J. Viega, Arbitrary block length mode, 2004. Google Scholar
[30]	K. Minematsu, Parallelizable rate-1 authenticated encryption from pseudorandom functions, in (eds. P. Q. Nguyen and E. Oswald) Advances in Cryptology - EUROCRYPT 2014 - 33rd Annual International Conference on the Theory and Applications of Cryptographic Techniques, Copenhagen, Denmark, May 11-15, 2014. Proceedings, Lecture Notes in Computer Science, 8441, Springer, 2014,275–292. doi: 10.1007/978-3-642-55220-5_16. Google Scholar
[31]	M. Naor and O. Reingold, On the construction of pseudorandom permutations: Luby-Rackoff revisited, J. Cryptology, 12 (1999), 29-66. doi: 10.1007/PL00003817. Google Scholar
[32]	M. O. Rabin and S. Winograd, Fast evaluation of polynomials by rational preparation, Comm. Pure Appl. Math., 25 (1972), 433-458. doi: 10.1002/cpa.3160250405. Google Scholar
[33]	P. Rogaway, Efficient instantiations of tweakable blockciphers and refinements to modes OCB and PMAC, in (eds. P. J. Lee) ASIACRYPT, Lecture Notes in Computer Science, 3329, Springer, 2004, 16–31. doi: 10.1007/978-3-540-30539-2_2. Google Scholar
[34]	P. Rogaway and T. Shrimpton, A provable-security treatment of the key-wrap problem, in (eds. Serge Vaudenay) EUROCRYPT, Lecture Notes in Computer Science, 4004, Springer, 2006,373–390. doi: 10.1007/11761679_23. Google Scholar
[35]	P. Sarkar, A general mixing strategy for the ECB-Mix-ECB mode of operation, Inf. Process. Lett., 109 (2008), 121-123. doi: 10.1016/j.ipl.2008.09.012. Google Scholar
[36]	P. Sarkar, Efficient tweakable enciphering schemes from (block-wise) universal hash functions, IEEE Transactions on Information Theory, 55 (2009), 4749-4759. doi: 10.1109/TIT.2009.2027487. Google Scholar
[37]	P. Sarkar, Tweakable enciphering schemes using only the encryption function of a block cipher, Inf. Process. Lett., 111 (2011), 945-955. doi: 10.1016/j.ipl.2011.06.014. Google Scholar
[38]	P. Wang, D. Feng and W. Wu, HCTR: A variable-input-length enciphering mode, in (eds. D. Feng, D. Lin, and M. Yung) CISC, Lecture Notes in Computer Science, 3822, Springer, 2005,175–188. doi: 10.1007/11599548_15. Google Scholar

show all references

References:

[1]	Public comments on the XTS-AES mode, http://csrc.nist.gov/groups/ST/toolkit/BCM/documents/comments/XTS/collected_XTS_comments.pdf. Google Scholar
[2]	IEEE Std 1619-2007: Standard for Cryptographic Protection of Data on Block-Oriented Storage Devices, IEEE Computer Society, 2008. Available at: http://standards.ieee.org/findstds/standard/1619-2007.html. Google Scholar
[3]	IEEE Std 1619.2-2010: IEEE Standard for Wide-block Encryption for Shared Storage Media, March 2011. Available at: http://standards.ieee.org/findstds/standard/1619.2-2010.html. Google Scholar
[4]	M. Bellare, D. Cash and S. Keelveedhi, Ciphers that securely encipher their own keys, in (eds. Y. Chen, G. Danezis and V. Shmatikov) Proceedings of the 18th ACM Conference on Computer and Communications Security, CCS 2011, Chicago, Illinois, USA, October 17-21, 2011, 2011,423–432. doi: 10.1145/2046707.2046757. Google Scholar
[5]	M. Bellare, A. Desai, E. Jokipii and P. Rogaway, A concrete security treatment of symmetric encryption, in 38th Annual Symposium on Foundations of Computer Science, FOCS '97, Miami Beach, Florida, USA, October 19-22, 1997, IEEE Computer Society, 1997,394–403. doi: 10.1109/SFCS.1997.646128. Google Scholar
[6]	D. J. Bernstein, Polynomial evaluation and message authentication, 2007. Available at: http://cr.yp.to/papers.html#pema. Google Scholar
[7]	R. Bhaumik and M. Nandi, An inverse-free single-keyed tweakable enciphering scheme, in (eds. T. Iwata and J. H. Cheon) 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, Lecture Notes in Computer Science, 9453, Springer, 2015,159–180. doi: 10.1007/978-3-662-48800-3_7. Google Scholar
[8]	D. Chakraborty, S. Ghosh and P. Sarkar, A fast single-key two-level universal hash function, IACR Trans. Symmetric Cryptol., 2017 (2017), 106–128. Google Scholar
[9]	D. Chakraborty, V. Hernandez-Jimenez and P. Sarkar, Another look at XCB, Cryptography and Communications, 7 (2015), 439-468. doi: 10.1007/s12095-015-0127-8. Google Scholar
[10]	D. Chakraborty, C. Mancillas-López, F. Rodríguez-Henríquez and P. Sarkar, Efficient hardware implementations of BRW polynomials and tweakable enciphering schemes, IEEE Trans. Computers, 62 (2013), 279–294. doi: 10.1109/TC.2011.227. Google Scholar
[11]	D. Chakraborty, C. Mancillas-López and P. Sarkar, STES: A stream cipher based low cost scheme for securing stored data, IEEE Trans. Computers, 64 (2015), 2691–2707. doi: 10.1109/TC.2014.2366739. Google Scholar
[12]	D. Chakraborty and M. Nandi, An improved security bound for HCTR, in (eds. K. Nyberg) Fast Software Encryption, 15th International Workshop, FSE 2008, Lausanne, Switzerland, February 10-13, 2008, Revised Selected Papers, Lecture Notes in Computer Science, 5086, Springer, 2008,289–302. doi: 10.1007/978-3-540-71039-4_18. Google Scholar
[13]	D. Chakraborty and P. Sarkar, A new mode of encryption providing a tweakable strong pseudo-random permutation, in (eds. M. J. B. Robshaw) FSE, Lecture Notes in Computer Science, 4047, Springer, 2006,293–309. doi: 10.1007/11799313_19. Google Scholar
[14]	D. Chakraborty and P. Sarkar, HCH: A new tweakable enciphering scheme using the hash-counter-hash approach, IEEE Transactions on Information Theory, 54 (2008), 1683-1699. doi: 10.1109/TIT.2008.917623. Google Scholar
[15]	D. Chakraborty and P. Sarkar, On modes of operations of a block cipher for authentication and authenticated encryption, Cryptography and Communications, 8 (2016), 455-511. doi: 10.1007/s12095-015-0153-6. Google Scholar
[16]	P. Crowley and E. Biggers, Adiantum: Length-preserving encryption for entry-level processors, IACR Trans. Symmetric Cryptol., 2018 (2018), 39–61. Google Scholar
[17]	M. J. Dworkin, SP 800-38E. Recommendation for Block Cipher Modes of Operation: The XTS-AES Mode for Confidentiality on Storage Devices, Technical report, Gaithersburg, MD, United States, 2010. Google Scholar
[18]	S. Gueron and M. E. Kounavis, Efficient implementation of the galois counter mode using a carry-less multiplier and a fast reduction algorithm, Inf. Process. Lett., 110 (2010), 549-553. doi: 10.1016/j.ipl.2010.04.011. Google Scholar
[19]	S. Gueron, A. Langley and Y. Lindell, AES-GCM-SIV: Specification and analysis, IACR Cryptology ePrint Archive, 168 (2017). doi: 10.1007/978-3-319-52153-4. Google Scholar
[20]	S. Halevi, EME$^{ * }$: Extending EME to handle arbitrary-length messages with associated data, in (eds. A. Canteaut and K. Viswanathan) INDOCRYPT, Lecture Notes in Computer Science, 3348, Springer 2004,315–327. doi: 10.1007/978-3-540-30556-9_25. Google Scholar
[21]	S. Halevi, Invertible universal hashing and the TET encryption mode, in (eds. A. Menezes) CRYPTO, Lecture Notes in Computer Science, 4622, Springer, (2007), pages 412–429. doi: 10.1007/978-3-540-74143-5_23. Google Scholar
[22]	S. Halevi and H. Krawczyk, Security under key-dependent inputs, in (eds. P. Ning, S. De Capitani di Vimercati, and P. F. Syverson) Proceedings of the 2007 ACM Conference on Computer and Communications Security, CCS 2007, Alexandria, Virginia, USA, October 28-31, 2007, 2007,466–475. doi: 10.1007/11935308. Google Scholar
[23]	S. Halevi and P. Rogaway, A tweakable enciphering mode, in (eds. D. Boneh) CRYPTO, Lecture Notes in Computer Science, 2729, Springer, 2003,482–499. doi: 10.1007/978-3-540-45146-4_28. Google Scholar
[24]	S. Halevi and P. Rogaway, A parallelizable enciphering mode, in (eds. T. Okamoto) CT-RSA, Lecture Notes in Computer Science, 2964, Springer, 2004,292–304. doi: 10.1007/978-3-540-24660-2_23. Google Scholar
[25]	V. T. Hoang, T. Krovetz and P. Rogaway, Robust authenticated-encryption AEZ and the problem that it solves, in (eds. E. Oswald and M. Fischlin) Advances in Cryptology - EUROCRYPT 2015 - 34th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Sofia, Bulgaria, April 26-30, 2015, Proceedings, Part I, Lecture Notes in Computer Science, 9056, Springer, 2015, 15–44. doi: 10.1007/978-3-662-46800-5_2. Google Scholar
[26]	M. Liskov, R. L. Rivest and D. Wagner, Tweakable block ciphers, in (eds. M. Yung) CRYPTO, Lecture Notes in Computer Science, 2442, Springer, 2002, 31–46. doi: 10.1007/3-540-45708-9_3. Google Scholar
[27]	D. A. McGrew and S. R. Fluhrer, The extended codebook (XCB) mode of operation, Cryptology ePrint Archive, Report 2004/278, 2004, Available at: http://eprint.iacr.org/. Google Scholar
[28]	D. A. McGrew and S. R. Fluhrer, The security of the extended codebook (xcb) mode of operation, in (eds. C. M. Adams, A. Miri, and M. J. Wiener) Selected Areas in Cryptography, Lecture Notes in Computer Science, 4876, Springer, 2007,311–327. doi: 10.1007/978-3-540-77360-3_20. Google Scholar
[29]	D. A. McGrew and J. Viega, Arbitrary block length mode, 2004. Google Scholar
[30]	K. Minematsu, Parallelizable rate-1 authenticated encryption from pseudorandom functions, in (eds. P. Q. Nguyen and E. Oswald) Advances in Cryptology - EUROCRYPT 2014 - 33rd Annual International Conference on the Theory and Applications of Cryptographic Techniques, Copenhagen, Denmark, May 11-15, 2014. Proceedings, Lecture Notes in Computer Science, 8441, Springer, 2014,275–292. doi: 10.1007/978-3-642-55220-5_16. Google Scholar
[31]	M. Naor and O. Reingold, On the construction of pseudorandom permutations: Luby-Rackoff revisited, J. Cryptology, 12 (1999), 29-66. doi: 10.1007/PL00003817. Google Scholar
[32]	M. O. Rabin and S. Winograd, Fast evaluation of polynomials by rational preparation, Comm. Pure Appl. Math., 25 (1972), 433-458. doi: 10.1002/cpa.3160250405. Google Scholar
[33]	P. Rogaway, Efficient instantiations of tweakable blockciphers and refinements to modes OCB and PMAC, in (eds. P. J. Lee) ASIACRYPT, Lecture Notes in Computer Science, 3329, Springer, 2004, 16–31. doi: 10.1007/978-3-540-30539-2_2. Google Scholar
[34]	P. Rogaway and T. Shrimpton, A provable-security treatment of the key-wrap problem, in (eds. Serge Vaudenay) EUROCRYPT, Lecture Notes in Computer Science, 4004, Springer, 2006,373–390. doi: 10.1007/11761679_23. Google Scholar
[35]	P. Sarkar, A general mixing strategy for the ECB-Mix-ECB mode of operation, Inf. Process. Lett., 109 (2008), 121-123. doi: 10.1016/j.ipl.2008.09.012. Google Scholar
[36]	P. Sarkar, Efficient tweakable enciphering schemes from (block-wise) universal hash functions, IEEE Transactions on Information Theory, 55 (2009), 4749-4759. doi: 10.1109/TIT.2009.2027487. Google Scholar
[37]	P. Sarkar, Tweakable enciphering schemes using only the encryption function of a block cipher, Inf. Process. Lett., 111 (2011), 945-955. doi: 10.1016/j.ipl.2011.06.014. Google Scholar
[38]	P. Wang, D. Feng and W. Wu, HCTR: A variable-input-length enciphering mode, in (eds. D. Feng, D. Lin, and M. Yung) CISC, Lecture Notes in Computer Science, 3822, Springer, 2005,175–188. doi: 10.1007/11599548_15. Google Scholar

Figure 1. The hash functions $ \mathbf{H} $ and $ \mathbf{G} $

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Table 1. Encryption and decryption algorithms for ${\sf FAST}$

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Table 2. A two-round Feistel construction required in Table 1

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Table 3. Computations of ${\sf {vecHorner}}$ and ${\sf {vecHash2L}}$. The string $ 1^n $ denotes the element of $ { \mathbb{F} } $ whose binary representation consists of the all-one string. Here $ \eta $ is a positive integer $ \geq 3 $ and $ \mathfrak{d}(\eta) $ denote the degree of $ {\sf {BRW}}_{\tau}(X_1, \ldots, X_\eta) $, where $ X_1, \ldots, X_\eta \in { \mathbb{F} } $

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Table 4. Game $ G_{{\sf {real}}} $

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Table 5. Game $ G_{{\sf {int}}} $

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Table 6. Game $ G_{{\sf {rnd}}} $

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Table 7. Comparison of different tweakable enciphering schemes according to computational efficiency. [BC] denotes the number of block cipher calls; [M] denotes the number of field multiplications; [D] denotes the number of doubling ('multiplication by $ \alpha $') operations;

type	scheme	[BC]	[M]	[D]
enc-mix-enc	CMC [23]	$ 2m+1 $	–	–
	EME2$ ^* $ [20]	$ 2m+1+m/n $	–	2
	AEZ [25]	$ (5m+4)/2 $	–	$ \frac{m-2}{4} $
	FMix [7]	$ 2m+1 $	–	–
hash-enc-hash	XCB [27]	$ m+1 $	$ 2(m+3) $	–
	HCTR [38]	$ m $	$ 2(m+1) $	–
	HCHfp [14]	$ m+2 $	$ 2(m-1) $	–
	TET [21]	$ m+1 $	$ 2m $	$ 2(m-1) $
	HEH-BRW[36]	$ m+1 $	$ 2+2\lfloor(m-1)/2\rfloor $	$ 2(m-1) $
	$ {\sf {TESX}} $ with BRW [37]	$ m+1 $	$ 4+2\lfloor(m-1)/2\rfloor $	$ 2(m-1) $
	$ {\sf {FAST}}[{\sf {Fx}}_m, {\sf Horner}] $	$ m+1 $	$ 2m+1 $	–
	$ {\sf {FAST}}[{\sf {Fx}}_m, {\sf {BRW}}] $	$ m+1 $	$ 2+2\lfloor(m-1)/2\rfloor $	–

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type	scheme	[BC]	[M]	[D]
enc-mix-enc	CMC [23]	$ 2m+1 $	–	–
	EME2$ ^* $ [20]	$ 2m+1+m/n $	–	2
	AEZ [25]	$ (5m+4)/2 $	–	$ \frac{m-2}{4} $
	FMix [7]	$ 2m+1 $	–	–
hash-enc-hash	XCB [27]	$ m+1 $	$ 2(m+3) $	–
	HCTR [38]	$ m $	$ 2(m+1) $	–
	HCHfp [14]	$ m+2 $	$ 2(m-1) $	–
	TET [21]	$ m+1 $	$ 2m $	$ 2(m-1) $
	HEH-BRW[36]	$ m+1 $	$ 2+2\lfloor(m-1)/2\rfloor $	$ 2(m-1) $
	$ {\sf {TESX}} $ with BRW [37]	$ m+1 $	$ 4+2\lfloor(m-1)/2\rfloor $	$ 2(m-1) $
	$ {\sf {FAST}}[{\sf {Fx}}_m, {\sf Horner}] $	$ m+1 $	$ 2m+1 $	–
	$ {\sf {FAST}}[{\sf {Fx}}_m, {\sf {BRW}}] $	$ m+1 $	$ 2+2\lfloor(m-1)/2\rfloor $	–

Table 8. Comparison of different tweakable enciphering schemes according to practical and implementation simplicity. [BCK] denotes the number of block cipher keys; and [HK] denotes the number of blocks in the hash key

type	scheme	[BCK]	[HK]	dec module	parallel
enc-mix-enc	CMC [23]	1	-	reqd	no
	EME2$ ^* $ [20]	1	2	reqd	yes
	AEZ [25]	1	2	not reqd	yes
	FMix [7]	1	-	not reqd	no
hash-enc-hash	XCB [27]	3	2	reqd	yes
	HCTR [38]	1	1	reqd	yes
	HCHfp [14]	1	1	reqd	yes
	TET [21]	2	3	reqd	yes
	HEH-BRW[36]	1	1	reqd	yes
	$ {\sf {TESX}} $ with BRW [37]	1	2	not reqd	yes
	$ {\sf {FAST}}[{\sf {Fx}}_m, {\sf Horner}] $	1	-	not reqd	yes
	$ {\sf {FAST}}[{\sf {Fx}}_m, {\sf {BRW}}] $	1	-	not reqd	yes
Note: for both Tables 7 and 8, the block size is $n$ bits, the tweak is a single $n$-bit block and the number of blocks $m\geq 3$ in the message is fixed.

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type	scheme	[BCK]	[HK]	dec module	parallel
enc-mix-enc	CMC [23]	1	-	reqd	no
	EME2$ ^* $ [20]	1	2	reqd	yes
	AEZ [25]	1	2	not reqd	yes
	FMix [7]	1	-	not reqd	no
hash-enc-hash	XCB [27]	3	2	reqd	yes
	HCTR [38]	1	1	reqd	yes
	HCHfp [14]	1	1	reqd	yes
	TET [21]	2	3	reqd	yes
	HEH-BRW[36]	1	1	reqd	yes
	$ {\sf {TESX}} $ with BRW [37]	1	2	not reqd	yes
	$ {\sf {FAST}}[{\sf {Fx}}_m, {\sf Horner}] $	1	-	not reqd	yes
	$ {\sf {FAST}}[{\sf {Fx}}_m, {\sf {BRW}}] $	1	-	not reqd	yes
Note: for both Tables 7 and 8, the block size is $n$ bits, the tweak is a single $n$-bit block and the number of blocks $m\geq 3$ in the message is fixed.

Table 9. Comparison of the cycles per byte measure of ${\sf {FAST}}$ with those of XCB, EME2 and AEZ in the setting of $ {\sf {Fx}}_{256} $

scheme	Skylake	Kabylake
XCB	1.92	1.85
EME2	2.07	1.99
AEZ	1.74	1.70
$ {\sf {FAST}}[{\sf {Fx}}_{256}, {\sf {Horner}}] $	1.63	1.56
$ {\sf {FAST}}[{\sf {Fx}}_{256}, {\sf {BRW}}] $	1.24	1.19

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scheme	Skylake	Kabylake
XCB	1.92	1.85
EME2	2.07	1.99
AEZ	1.74	1.70
$ {\sf {FAST}}[{\sf {Fx}}_{256}, {\sf {Horner}}] $	1.63	1.56
$ {\sf {FAST}}[{\sf {Fx}}_{256}, {\sf {BRW}}] $	1.24	1.19

Table 10. Report of cycles per byte measure for the setting of ${\sf {Gn}}$ for $ {\sf {FAST}}[{\sf {Gn}}, \mathfrak{k}, {\sf {vecHorner}}] $ and $ {\sf {FAST}}[{\sf {Gn}}, \mathfrak{k}, 31, {\sf {vecHash2L}}] $

		Skylake			Kabylake
msg len	$ \mathfrak{k} $	$ {\sf {vecHorner}} $	$ {\sf {vecHash2L}} $	$ {\sf {vecHash2L}} $	$ {\sf {vecHorner}} $	$ {\sf {vecHash2L}} $	$ {\sf {vecHash2L}} $
(bytes)			(delayed)	(normal)		(delayed)	(normal)
	2	1.51	1.38	1.59	1.42	1.32	1.56
512	3	1.40	1.38	1.39	1.32	1.31	1.35
	4	1.34	1.37	1.36	1.26	1.31	1.33
	2	1.53	1.34	1.48	1.42	1.27	1.42
1024	3	1.45	1.34	1.34	1.35	1.27	1.30
	4	1.40	1.33	1.32	1.29	1.27	1.30
	2	1.57	1.30	1.35	1.45	1.24	1.30
4096	3	1.54	1.29	1.31	1.43	1.24	1.27
	4	1.51	1.29	1.30	1.40	1.24	1.26
	2	1.57	1.27	1.32	1.45	1.22	1.27
8192	3	1.56	1.27	1.30	1.44	1.22	1.25
	4	1.54	1.27	1.30	1.43	1.22	1.25

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		Skylake			Kabylake
msg len	$ \mathfrak{k} $	$ {\sf {vecHorner}} $	$ {\sf {vecHash2L}} $	$ {\sf {vecHash2L}} $	$ {\sf {vecHorner}} $	$ {\sf {vecHash2L}} $	$ {\sf {vecHash2L}} $
(bytes)			(delayed)	(normal)		(delayed)	(normal)
	2	1.51	1.38	1.59	1.42	1.32	1.56
512	3	1.40	1.38	1.39	1.32	1.31	1.35
	4	1.34	1.37	1.36	1.26	1.31	1.33
	2	1.53	1.34	1.48	1.42	1.27	1.42
1024	3	1.45	1.34	1.34	1.35	1.27	1.30
	4	1.40	1.33	1.32	1.29	1.27	1.30
	2	1.57	1.30	1.35	1.45	1.24	1.30
4096	3	1.54	1.29	1.31	1.43	1.24	1.27
	4	1.51	1.29	1.30	1.40	1.24	1.26
	2	1.57	1.27	1.32	1.45	1.22	1.27
8192	3	1.56	1.27	1.30	1.44	1.22	1.25
	4	1.54	1.27	1.30	1.43	1.22	1.25

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American Institute of Mathematical Sciences

${\sf {FAST}}$: Disk encryption and beyond

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