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## Classical reduction of gap SVP to LWE: A concrete security analysis

 Indian Statistical Institute, 203, BT Rd, Baranagar, Kolkata, West Bengal 700108, India

* Corresponding author: Palash Sarkar

Received  July 2020 Revised  December 2020 Early access March 2021

Regev (2005) introduced the learning with errors (LWE) problem and showed a quantum reduction from a worst case lattice problem to LWE. Building on the work of Peikert (2009), a classical reduction from the gap shortest vector problem to LWE was obtained by Brakerski et al. (2013). A concrete security analysis of Regev's reduction by Chatterjee et al. (2016) identified a huge tightness gap. The present work performs a concrete analysis of the tightness gap in the classical reduction of Brakerski et al. It turns out that the tightness gap in the Brakerski et al. classical reduction is even larger than the tightness gap in the quantum reduction of Regev. This casts doubts on the implication of the reduction to security assurance of practical cryptosystems.

Citation: Palash Sarkar, Subhadip Singha. Classical reduction of gap SVP to LWE: A concrete security analysis. Advances in Mathematics of Communications, doi: 10.3934/amc.2021004
##### References:
 [1] E. Alkim, R. Avanzi, J. Bos, L. Ducas, A. de la Piedra, T. Poppelmann, P. Schwabe, D. Stebila, M. R. Albrecht, E. Orsini, V. Osheter, K. G. Paterson, G. Peer, and N. P. Smart, NewHope: Algorithm Specifications and Supporting Documentation, 2019. Available from: https://csrc.nist.gov/projects/post-quantum-cryptography/round-2-submissions. [2] E. Alkim, J. Bos, L. Ducas, P. Longa, I. Mironov, M. Naehrig, V. Nikolaenko, C. Peikert, A. Raghunathan and D. Stebila, FrodoKEM: Learning With Errors Key Encapsulation, 2019. Available from: https://csrc.nist.gov/projects/post-quantum-cryptography/round-2-submissions. [3] R. Avanzi, J. Bos, L. Ducas, E. Kiltz, T. Lepoint, V. Lyubashevsky, J. M. Schanck, P. Schwabe, G. Seiler and D. Stehlé, CRYSTALS-Kyber: Algorithm Specifications and Supporting Documentation, 2009. Available from: https://csrc.nist.gov/projects/post-quantum-cryptography/round-2-submissions. [4] H. Baan, S. Bhattacharya, S. Fluhrer, O. Garcia-Morchon, T. Laarhoven, R. Player, R. Rietman, M.-J. O. Saarinen, L. Tolhuizen, J.-L. T.-A. and Z. Zhang, Round5: KEM and PKE based on (Ring) Learning With Rounding, 2019. Available from: https://csrc.nist.gov/projects/post-quantum-cryptography/round-2-submissions. [5] D. J. Bernstein, Comparing Proofs of Security for Lattice-Based Encryption, Cryptology ePrint Archive, Report 2019/691, 2019. Available from: https://eprint.iacr.org/2019/691. [6] Z. Brakerski, A. Langlois, C. Peikert, O. Regev and D. Stehlé, Classical hardness of learning with errors, in Dan Boneh, Tim Roughgarden and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, (2013), 575–584. doi: 10.1145/2488608.2488680. [7] S. Chatterjee, N. Koblitz, A. Menezes and P. Sarkar, Another look at tightness II: practical issues in cryptography, in Raphael C.-W. Phan and Moti Yung, editors, Paradigms in Cryptology - Mycrypt 2016. Malicious and Exploratory Cryptology - Second International Conference, Mycrypt 2016, Kuala Lumpur, Malaysia, December 1-2, 2016, Revised Selected Papers, Lecture Notes in Computer Science, Springer, 10311 (2016), 21–55. doi: 10.1007/978-3-319-61273-7_3. [8] J.-P. DÁnvers, A. Karmakar, S. S. Roy and F. Vercauteren, SABER: Mod-LWR Based KEM (Round 2 Submission), 2019. Available from: https://csrc.nist.gov/projects/post-quantum-cryptography/round-2-submissions. doi: 10.1007/978-3-319-89339-6_16. [9] C. Gentry, C. Peikert and V. Vaikuntanathan, Trapdoors for hard lattices and new cryptographic constructions, in Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, ACM, (2008), 197–206. doi: 10.1145/1374376.1374407. [10] O. Goldreich and S. Goldwasser, On the limits of nonapproximability of lattice problems, J. Comput. Syst. Sci., 60 (2000), 540-563.  doi: 10.1006/jcss.1999.1686. [11] W. Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association, 58 (1963), 13-30.  doi: 10.1080/01621459.1963.10500830. [12] Y.-K. Liu, V. Lyubashevsky and D. Micciancio, On bounded distance decoding for general lattices, in Josep Díaz, Klaus Jansen, José D. P. Rolim and Uri Zwick, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006, Barcelona, Spain, August 28-30 2006, Proceedings, Lecture Notes in Computer Science, Springer, 4110 (2006), 450–461. [13] X. Lu, Y. Liu, D. Jia, H. Xue, J. He, Z.i Zhang, Z. Liu, H. Yang, B. Li and K. Wang, LAC: Lattice-Based Cryptosystems, 2019. Available from: https://csrc.nist.gov/projects/post-quantum-cryptography/round-2-submissions. [14] D. Micciancio and C. Peikert, Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller, Cryptology ePrint Archive, Report 2011/501, 2011. Available from: https://eprint.iacr.org/2011/501. doi: 10.1007/978-3-642-29011-4_41. [15] C. Peikert, Public-key cryptosystems from the worst-case shortest vector problem: Extended abstract, in Michael Mitzenmacher, editor, Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, (2009), 333–342. [16] O. Regev, On lattices, learning with errors, random linear codes, and cryptography, J. ACM, 56 (2009), 40 pp. doi: 10.1145/1568318.1568324. [17] P. Sarkar and S. Singha, Verifying solutions to LWE with implications for concrete security, Adv. Math. Commun., 15 (2020), 257-266.  doi: 10.3934/amc.2020057.

show all references

##### References:
 [1] E. Alkim, R. Avanzi, J. Bos, L. Ducas, A. de la Piedra, T. Poppelmann, P. Schwabe, D. Stebila, M. R. Albrecht, E. Orsini, V. Osheter, K. G. Paterson, G. Peer, and N. P. Smart, NewHope: Algorithm Specifications and Supporting Documentation, 2019. Available from: https://csrc.nist.gov/projects/post-quantum-cryptography/round-2-submissions. [2] E. Alkim, J. Bos, L. Ducas, P. Longa, I. Mironov, M. Naehrig, V. Nikolaenko, C. Peikert, A. Raghunathan and D. Stebila, FrodoKEM: Learning With Errors Key Encapsulation, 2019. Available from: https://csrc.nist.gov/projects/post-quantum-cryptography/round-2-submissions. [3] R. Avanzi, J. Bos, L. Ducas, E. Kiltz, T. Lepoint, V. Lyubashevsky, J. M. Schanck, P. Schwabe, G. Seiler and D. Stehlé, CRYSTALS-Kyber: Algorithm Specifications and Supporting Documentation, 2009. Available from: https://csrc.nist.gov/projects/post-quantum-cryptography/round-2-submissions. [4] H. Baan, S. Bhattacharya, S. Fluhrer, O. Garcia-Morchon, T. Laarhoven, R. Player, R. Rietman, M.-J. O. Saarinen, L. Tolhuizen, J.-L. T.-A. and Z. Zhang, Round5: KEM and PKE based on (Ring) Learning With Rounding, 2019. Available from: https://csrc.nist.gov/projects/post-quantum-cryptography/round-2-submissions. [5] D. J. Bernstein, Comparing Proofs of Security for Lattice-Based Encryption, Cryptology ePrint Archive, Report 2019/691, 2019. Available from: https://eprint.iacr.org/2019/691. [6] Z. Brakerski, A. Langlois, C. Peikert, O. Regev and D. Stehlé, Classical hardness of learning with errors, in Dan Boneh, Tim Roughgarden and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, (2013), 575–584. doi: 10.1145/2488608.2488680. [7] S. Chatterjee, N. Koblitz, A. Menezes and P. Sarkar, Another look at tightness II: practical issues in cryptography, in Raphael C.-W. Phan and Moti Yung, editors, Paradigms in Cryptology - Mycrypt 2016. Malicious and Exploratory Cryptology - Second International Conference, Mycrypt 2016, Kuala Lumpur, Malaysia, December 1-2, 2016, Revised Selected Papers, Lecture Notes in Computer Science, Springer, 10311 (2016), 21–55. doi: 10.1007/978-3-319-61273-7_3. [8] J.-P. DÁnvers, A. Karmakar, S. S. Roy and F. Vercauteren, SABER: Mod-LWR Based KEM (Round 2 Submission), 2019. Available from: https://csrc.nist.gov/projects/post-quantum-cryptography/round-2-submissions. doi: 10.1007/978-3-319-89339-6_16. [9] C. Gentry, C. Peikert and V. Vaikuntanathan, Trapdoors for hard lattices and new cryptographic constructions, in Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, ACM, (2008), 197–206. doi: 10.1145/1374376.1374407. [10] O. Goldreich and S. Goldwasser, On the limits of nonapproximability of lattice problems, J. Comput. Syst. Sci., 60 (2000), 540-563.  doi: 10.1006/jcss.1999.1686. [11] W. Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association, 58 (1963), 13-30.  doi: 10.1080/01621459.1963.10500830. [12] Y.-K. Liu, V. Lyubashevsky and D. Micciancio, On bounded distance decoding for general lattices, in Josep Díaz, Klaus Jansen, José D. P. Rolim and Uri Zwick, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006, Barcelona, Spain, August 28-30 2006, Proceedings, Lecture Notes in Computer Science, Springer, 4110 (2006), 450–461. [13] X. Lu, Y. Liu, D. Jia, H. Xue, J. He, Z.i Zhang, Z. Liu, H. Yang, B. Li and K. Wang, LAC: Lattice-Based Cryptosystems, 2019. Available from: https://csrc.nist.gov/projects/post-quantum-cryptography/round-2-submissions. [14] D. Micciancio and C. Peikert, Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller, Cryptology ePrint Archive, Report 2011/501, 2011. Available from: https://eprint.iacr.org/2011/501. doi: 10.1007/978-3-642-29011-4_41. [15] C. Peikert, Public-key cryptosystems from the worst-case shortest vector problem: Extended abstract, in Michael Mitzenmacher, editor, Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, (2009), 333–342. [16] O. Regev, On lattices, learning with errors, random linear codes, and cryptography, J. ACM, 56 (2009), 40 pp. doi: 10.1145/1568318.1568324. [17] P. Sarkar and S. Singha, Verifying solutions to LWE with implications for concrete security, Adv. Math. Commun., 15 (2020), 257-266.  doi: 10.3934/amc.2020057.
 Algorithm 1 Reducing GapSVP$_{\zeta,\gamma}$ to LWE$_{q,\Psi_{\alpha}}$, where $\gamma=\gamma(n)\geq n/(\alpha\sqrt{\log n})$ and $q=q(n)\geq \zeta(n)\cdot \omega(\sqrt{\log n/n})$. 1: function ${\sf {solveGapSVP}}_{\zeta,\gamma}$ $\mathbf{B},d$ 2:        Let $\mathbf{D}$ be the reverse dual basis of $\mathbf{B}$; 3:        $d^{\prime} =d{\cdot}{\sqrt{n/(4\ln{n})}}$; $r = q\sqrt{2n} / (\gamma d)$; 4:        for $i\leftarrow 1$ to $N$ do 5:              $\mathbf{w} \xleftarrow{＄} {d^{\prime}}{\cdot}{\mathcal{B}}_n$; $\mathbf{x} = \mathbf{w} \bmod \mathbf{B}$; 6:              $\mathcal{L} \leftarrow \{\}$; 7:              for $j\leftarrow 1$ to $n^c$ do 8:                    $\mathcal{L}\leftarrow \mathcal{L}\cup {\sf {sample}}(D,r)$; 9:              end for 10:              $\mathbf{v} \leftarrow {\sf {solveCVP}}(\mathbf{B},\mathcal{L},\mathbf{x})$ 11:              if $\mathbf{v} \neq \mathbf{x} - \mathbf{w}$ then 12:                    return ${\sf {accept}}$; 13:              end if 14:        end for 15:        return ${\sf {reject}}$; 16: end function
 Algorithm 1 Reducing GapSVP$_{\zeta,\gamma}$ to LWE$_{q,\Psi_{\alpha}}$, where $\gamma=\gamma(n)\geq n/(\alpha\sqrt{\log n})$ and $q=q(n)\geq \zeta(n)\cdot \omega(\sqrt{\log n/n})$. 1: function ${\sf {solveGapSVP}}_{\zeta,\gamma}$ $\mathbf{B},d$ 2:        Let $\mathbf{D}$ be the reverse dual basis of $\mathbf{B}$; 3:        $d^{\prime} =d{\cdot}{\sqrt{n/(4\ln{n})}}$; $r = q\sqrt{2n} / (\gamma d)$; 4:        for $i\leftarrow 1$ to $N$ do 5:              $\mathbf{w} \xleftarrow{＄} {d^{\prime}}{\cdot}{\mathcal{B}}_n$; $\mathbf{x} = \mathbf{w} \bmod \mathbf{B}$; 6:              $\mathcal{L} \leftarrow \{\}$; 7:              for $j\leftarrow 1$ to $n^c$ do 8:                    $\mathcal{L}\leftarrow \mathcal{L}\cup {\sf {sample}}(D,r)$; 9:              end for 10:              $\mathbf{v} \leftarrow {\sf {solveCVP}}(\mathbf{B},\mathcal{L},\mathbf{x})$ 11:              if $\mathbf{v} \neq \mathbf{x} - \mathbf{w}$ then 12:                    return ${\sf {accept}}$; 13:              end if 14:        end for 15:        return ${\sf {reject}}$; 16: end function
 Algorithm 2: Construction of a distinguisher $\mathcal{B}$ for binLWE$_{n,m_1,q,\leq \alpha}$ using a distinguisher $\mathcal{A}$ for binLWE$_{n,m_2,q,\alpha}$. In the algorithm, $\theta$ is a known lower bound on the advantage of $\mathcal{A}$. 1: function $\mathcal{B}$($\mathcal{J}$) 2:        let $\mathcal{S}$ be a collection of $m_1$ samples drawn independently and uniformly from $\mathbb{Z}^n_p{\times}\mathbb{T}$; 3:        partition $\mathcal{S}$ as $\mathcal{S}=\cup_{i=1}^{\mathfrak{k}}\mathcal{S}_i$, such that $\#\mathcal{S}_i=m_2$, $i=1,\ldots,\mathfrak{k}$; 4:        let $\hat{p}_{＄} = (\mathcal{A}(\mathcal{S}_1)+\cdots+\mathcal{A}(\mathcal{S}_{\mathfrak{k}}))/\mathfrak{k}$; 5:        $m_3\leftarrow 6 (m_2/\theta)^{1/2}$; 6:        let $Z$ be the set of all integer multiples of $m_3^{-2}\alpha^2$ in the range $(0,\alpha^2]$; 7:        for $\gamma$ in $Z$ do 8:                $\mathcal{J}^{\prime} \leftarrow \emptyset$; 9:                for $(\mathbf{a},e) \in \mathcal{J}$ do 10:                   sample $\varepsilon$ from $\Psi_{\sqrt{\gamma}}$; 11:                    $\mathcal{J}^{\prime} \leftarrow \mathcal{J}^{\prime} \cup \{(\mathbf{a},e+\varepsilon)\}$; 12:                end for 13:                partition $\mathcal{J}^{\prime}$ as $\mathcal{J}^{\prime}=\cup_{i=1}^{k}\mathcal{J}_i$, such that $\#\mathcal{J}_i=m_2$, $i=1,\ldots,k$; 14:                let $p = (\mathcal{A}(\mathcal{J}_1)+\cdots+\mathcal{A}(\mathcal{J}_{\mathfrak{k}}))/\mathfrak{k}$; 15:                if $|p-\hat{p}_{＄}| > \theta/2$ then 16:                    return 1; 17:                end if 18:        end for 19:        return 0; 20: end function.
 Algorithm 2: Construction of a distinguisher $\mathcal{B}$ for binLWE$_{n,m_1,q,\leq \alpha}$ using a distinguisher $\mathcal{A}$ for binLWE$_{n,m_2,q,\alpha}$. In the algorithm, $\theta$ is a known lower bound on the advantage of $\mathcal{A}$. 1: function $\mathcal{B}$($\mathcal{J}$) 2:        let $\mathcal{S}$ be a collection of $m_1$ samples drawn independently and uniformly from $\mathbb{Z}^n_p{\times}\mathbb{T}$; 3:        partition $\mathcal{S}$ as $\mathcal{S}=\cup_{i=1}^{\mathfrak{k}}\mathcal{S}_i$, such that $\#\mathcal{S}_i=m_2$, $i=1,\ldots,\mathfrak{k}$; 4:        let $\hat{p}_{＄} = (\mathcal{A}(\mathcal{S}_1)+\cdots+\mathcal{A}(\mathcal{S}_{\mathfrak{k}}))/\mathfrak{k}$; 5:        $m_3\leftarrow 6 (m_2/\theta)^{1/2}$; 6:        let $Z$ be the set of all integer multiples of $m_3^{-2}\alpha^2$ in the range $(0,\alpha^2]$; 7:        for $\gamma$ in $Z$ do 8:                $\mathcal{J}^{\prime} \leftarrow \emptyset$; 9:                for $(\mathbf{a},e) \in \mathcal{J}$ do 10:                   sample $\varepsilon$ from $\Psi_{\sqrt{\gamma}}$; 11:                    $\mathcal{J}^{\prime} \leftarrow \mathcal{J}^{\prime} \cup \{(\mathbf{a},e+\varepsilon)\}$; 12:                end for 13:                partition $\mathcal{J}^{\prime}$ as $\mathcal{J}^{\prime}=\cup_{i=1}^{k}\mathcal{J}_i$, such that $\#\mathcal{J}_i=m_2$, $i=1,\ldots,k$; 14:                let $p = (\mathcal{A}(\mathcal{J}_1)+\cdots+\mathcal{A}(\mathcal{J}_{\mathfrak{k}}))/\mathfrak{k}$; 15:                if $|p-\hat{p}_{＄}| > \theta/2$ then 16:                    return 1; 17:                end if 18:        end for 19:        return 0; 20: end function.
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