Saber: Module-LWR Based Key Exchange, CPA-Secure Encryption and CCA-Secure KEM

  • Jan-Pieter D’Anvers
  • Angshuman KarmakarEmail author
  • Sujoy Sinha Roy
  • Frederik Vercauteren
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10831)


In this paper, we introduce Saber, a package of cryptographic primitives whose security relies on the hardness of the Module Learning With Rounding problem (Mod-LWR). We first describe a secure Diffie-Hellman type key exchange protocol, which is then transformed into an IND-CPA encryption scheme and finally into an IND-CCA secure key encapsulation mechanism using a post-quantum version of the Fujisaki-Okamoto transform. The design goals of this package were simplicity, efficiency and flexibility resulting in the following choices: all integer moduli are powers of 2 avoiding modular reduction and rejection sampling entirely; the use of LWR halves the amount of randomness required compared to LWE-based schemes and reduces bandwidth; the module structure provides flexibility by reusing one core component for multiple security levels. A constant-time AVX2 optimized software implementation of the KEM with parameters providing more than 128 bits of post-quantum security, requires only 101K, 125K and 129K cycles for key generation, encapsulation and decapsulation respectively on a Dell laptop with an Intel i7-Haswell processor.



This work was supported in part by the Research Council KU Leuven: C16/15/058. In addition, this work was supported by the European Commission through the Horizon 2020 research and innovation programme under grant agreement No H2020-ICT-2014-645622 PQCRYPTO, H2020-ICT-2014-644209 HEAT, Cathedral ERC Advanced Grant 695305 and in part by Flemish Government, by the Hercules Foundation AKUL/11/19.

Supplementary material


  1. 1.
    National Institute of Standards and Technology: SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions. FIPS PUB 202 (2015)Google Scholar
  2. 2.
    Albrecht, M.R.: On dual lattice attacks against small-secret LWE and parameter choices in HElib and SEAL. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10211, pp. 103–129. Springer, Cham (2017). Scholar
  3. 3.
    Alkim, E., Ducas, L., Pöppelmann, T., Schwabe, P.: NewHope without reconciliation (2016).
  4. 4.
    Alkim, E., Ducas, L., Pöppelmann, T., Schwabe, P.: Post-quantum key exchange - a new hope. In: USENIX Security 2016 (2016)Google Scholar
  5. 5.
    Alperin-Sheriff, J., Apon, D.: Dimension-preserving reductions from LWE to LWR. Cryptology ePrint Archive, Report 2016/589 (2016)Google Scholar
  6. 6.
    Alwen, J., Krenn, S., Pietrzak, K., Wichs, D.: Learning with rounding, revisited. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 57–74. Springer, Heidelberg (2013). Scholar
  7. 7.
    Arora, S., Ge, R.: New algorithms for learning in presence of errors. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 403–415. Springer, Heidelberg (2011). Scholar
  8. 8.
    Baan, H., Bhattacharaya, S., Garcia-Morchon, O., Rietman, R., Tolhuizen, L., Torre-Arce, J.L., Zhang, Z.: Round2: KEM and PKE based on GLWR. Cryptology ePrint Archive, Report 2017/1183 (2017).
  9. 9.
    Bai, S., Galbraith, S.D.: Lattice decoding attacks on binary LWE. In: Susilo, W., Mu, Y. (eds.) ACISP 2014. LNCS, vol. 8544, pp. 322–337. Springer, Cham (2014). Scholar
  10. 10.
    Banerjee, A., Peikert, C., Rosen, A.: Pseudorandom functions and lattices. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 719–737. Springer, Heidelberg (2012). Scholar
  11. 11.
    Bernstein, D.J., Chuengsatiansup, C., Lange, T., van Vredendaal, C.: NTRU prime: reducing attack surface at low cost. Cryptology ePrint Archive, Report 2016/461 (2016).
  12. 12.
    Bhattacharya, S., Garcia-Morchon, O., Rietman, R., Tolhuizen, L.: spKEX: an optimized lattice-based key exchange. Cryptology ePrint Archive, Report 2017/709 (2017).
  13. 13.
    Birkett, J., Dent, A.W.: Relations among notions of plaintext awareness. In: Cramer, R. (ed.) PKC 2008. LNCS, vol. 4939, pp. 47–64. Springer, Heidelberg (2008). Scholar
  14. 14.
    Bodrato, M., Zanoni, A.: Integer and polynomial multiplication: towards optimal Toom-Cook matrices. In: ISSAC 2007, pp. 17–24. ACM (2007).
  15. 15.
    Bogdanov, A., Guo, S., Masny, D., Richelson, S., Rosen, A.: On the hardness of learning with rounding over small modulus. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016. LNCS, vol. 9562, pp. 209–224. Springer, Heidelberg (2016). Scholar
  16. 16.
    Bos, J., Ducas, L., Kiltz, E., Lepoint, T., Lyubashevsky, V., Schanck, J.M., Schwabe, P., Stehlé, D.: CRYSTALS - kyber: a CCA-secure module-lattice-based KEM. Cryptology ePrint Archive, Report 2017/634 (2017).
  17. 17.
    Bos, J.W., Costello, C., Ducas, L., Mironov, I., Naehrig, M., Nikolaenko, V., Raghunathan, A., Stebila, D.: Frodo: take off the ring! practical, quantum-secure key exchange from LWE. In: CCS 2016, pp. 1006–1018. ACM (2016).
  18. 18.
    Chen, L., Jordan, S.P., Liu, Y.K., Moody, D., Peralta, R.C., Perlner, R.A., Smith-Tone, D.C.: Report on post-quantum cryptography. In: NIST Internal Report (NISTIR) - 8105 (2016).
  19. 19.
    Chen, Y., Nguyen, P.Q.: BKZ 2.0: better lattice security estimates. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 1–20. Springer, Heidelberg (2011). Scholar
  20. 20.
    Cheon, J.H., Han, K., Kim, J., Lee, C., Son, Y.: A practical post-quantum public-key cryptosystem based on spLWE. In: Hong, S., Park, J.H. (eds.) ICISC 2016. LNCS, vol. 10157, pp. 51–74. Springer, Cham (2017). Scholar
  21. 21.
    Cheon, J.H., Kim, D., Lee, J., Song, Y.: Lizard: cut off the tail! practical post-quantum public-key encryption from LWE and LWR. Cryptology ePrint Archive, Report 2016/1126 (2016).
  22. 22.
    Diffie, W., Hellman, M.: New directions in cryptography. IEEE Trans. Inf. Theory 22(6), 644–654 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ding, J.: New cryptographic constructions using generalized learning with errors problem. Cryptology ePrint Archive, Report 2012/387 (2012).
  24. 24.
    Hoffstein, J., Pipher, J., Schanck, J.M., Silverman, J.H., Whyte, W., Zhang, Z.: Choosing parameters for NTRUEncrypt. In: Handschuh, H. (ed.) CT-RSA 2017. LNCS, vol. 10159, pp. 3–18. Springer, Cham (2017). Scholar
  25. 25.
    Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: a ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998). Scholar
  26. 26.
    Hofheinz, D., Hövelmanns, K., Kiltz, E.: A modular analysis of the Fujisaki-Okamoto transformation. Cryptology ePrint Archive, Report 2017/604 (2017).
  27. 27.
    Hulsing, A., Rijneveld, J., Schanck, J.M., Schwabe, P.: High-speed key encapsulation from NTRU. Cryptology ePrint Archive, Report 2017/667 (2017).
  28. 28.
    Jiang, H., Zhang, Z., Chen, L., Wang, H., Ma, Z.: Post-quantum IND-CCA-secure KEM without additional hash. Cryptology ePrint Archive, Report 2017/1096 (2017).
  29. 29.
    Jin, Z., Zhao, Y.: Optimal key consensus in presence of noise. Cryptology ePrint Archive, Report 2017/1058 (2017).
  30. 30.
    Kirchner, P., Fouque, P.-A.: An improved BKW algorithm for LWE with applications to cryptography and lattices. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 43–62. Springer, Heidelberg (2015). Scholar
  31. 31.
    Langlois, A., Stehlé, D.: Worst-case to average-case reductions for module lattices. Des. Codes Crypt. 75(3), 565–599 (2015). Scholar
  32. 32.
    Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010). Scholar
  33. 33.
    Peikert, C.: Lattice cryptography for the internet. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 197–219. Springer, Cham (2014). Scholar
  34. 34.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: STOC 2005, pp. 84–93. ACM (2005).
  35. 35.
    Saito, T., Xagawa, K., Yamakawa, T.: Tightly-secure key-encapsulation mechanism in the quantum random oracle model. Cryptology ePrint Archive, Report 2017/1005 (2017).
  36. 36.
    Schnorr, C.P., Euchner, M.: Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math. Program. 66(1–3), 181–199 (1994). Scholar
  37. 37.
    Stehlé, D., Steinfeld, R.: Making NTRU as secure as worst-case problems over ideal lattices. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 27–47. Springer, Heidelberg (2011). Scholar
  38. 38.
    Targhi, E.E., Unruh, D.: Post-quantum security of the Fujisaki-Okamoto and OAEP transforms. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 192–216. Springer, Heidelberg (2016). Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Jan-Pieter D’Anvers
    • 1
  • Angshuman Karmakar
    • 1
    Email author
  • Sujoy Sinha Roy
    • 1
  • Frederik Vercauteren
    • 1
  1. 1.imec-COSICKU LeuvenLeuven-HeverleeBelgium

Personalised recommendations