List of missing assumptions

[jnsiemer]

List of Assumptions that I'm aware of and that I'm planning to add to the Lattice Assumption Zoo (until end of June). Until then, consider the LAZ work-in-progress project. If there are any SIS-based or LWE-based assumptions that are not in the list below and not already available in the LAZ, please open another issue or add the assumption yourself via a PR. Thanks!

I excluded further LIP-related, LPN-related and NTRU-related assumptions for now to keep the following list manageable.

SIS-based:

• k-R-ISIS / k-M-ISIS
  • Twin k-R-ISIS
  • Knowledge k-R-ISIS
• BASIS
  • BASIS_rand
  • BASIS_struct
  • BASIS_pow
  • PRISIS
  • h-PRISIS
• l-Succinct SIS
• vSIS (vanishing SIS) (prepared on v-sis branch) https://eprint.iacr.org/2025/356.pdf https://eprint.iacr.org/2025/360.pdf
  • (Strong) Hinted vSIS
• Evasive SIS https://eprint.iacr.org/2025/356.pdf
• SelfTargetMSIS (Dilithium)
• Split-SIS https://eprint.iacr.org/2015/020.pdf
• H-SIS and H-ISIS https://ia.cr/2026/187
• t-R-ISIS and t-R-SPISIS https://eprint.iacr.org/2024/1769.pdf
• Asymmetric SIS 2019/510 2026/943

LWE-based:

• LWR

• k-LWE https://eprint.iacr.org/2014/494.pdf

• HintMLWE https://eprint.iacr.org/2023/623.pdf

• Evasive LWE https://eprint.iacr.org/2024/2000.pdf https://eprint.iacr.org/2025/375.pdf https://eprint.iacr.org/2023/906.pdf https://eprint.iacr.org/2022/1140.pdf https://eprint.iacr.org/2025/390.pdf 2024/2000, 2025/374, 2025/421

• Equivocal LWE https://eprint.iacr.org/2025/1129.pdf

• Hollow LWE https://eprint.iacr.org/2025/340.pdf

• l-Succinct LWE https://eprint.iacr.org/2024/1663.pdf https://eprint.iacr.org/2025/1039.pdf

• elLWE (LWE with Error-Leakage) https://eprint.iacr.org/2023/404.pdf

• Leaky LWE unification of Hint-MLWE and elLWE https://doi.org/10.62056/ah89ksuc2

• Sparse Matrix LWE

• VLWE (Variety-LWE) https://arxiv.org/pdf/2502.07284

• CRO (Circular Security with Random Opening) https://eprint.iacr.org/2025/390.pdf

• Tensor LWE https://eprint.iacr.org/2023/941.pdf https://ia.cr/2023/906

• Augmented LWE https://eprint.iacr.org/2014/733.pdf

• Multi-Hint Extended Ring-LWE https://eprint.iacr.org/2021/046.pdf

• Large Vector Problem https://www.iacr.org/archive/asiacrypt2021/130900030/130900030.pdf#page=19.49

• {0,1}-CRT-RLWE https://eprint.iacr.org/2024/1307.pdf

• Learning with Alternating Moduli https://eprint.iacr.org/2025/968.pdf

• Extended LWE https://eprint.iacr.org/2011/352.pdf https://arxiv.org/abs/1306.0281 https://ia.cr/2015/608 https://ia.cr/2022/284

• Decomposed LWE https://eprint.iacr.org/2025/339.pdf

• MLWE-PRF-RU https://ia.cr/2024/1615

• Asymmetric LWE 2019/510 2026/943

• Circular LWE 2023/1716, 2025/001

• First-is-errorless / First-are-errorless LWE https://arxiv.org/pdf/1306.0281, 2015/608

• qLWE https://eprint.iacr.org/2025/2225.pdf

• Middle-Product LWE (MP-LWE) https://eprint.iacr.org/2017/628

• Middle-Product LWR (MP-LWR) https://eprint.iacr.org/2019/1001.pdf

• Order-LWE (OLWE) https://eprint.iacr.org/2018/494

• Integer-Polynomial LWE (IP-LWE) https://eprint.iacr.org/2021/277

• Polynomial LWE https://eprint.iacr.org/2019/878.pdf

• Algebraically Structured LWE https://eprint.iacr.org/2019/878.pdf

• Adaptive LWE https://eprint.iacr.org/2018/409.pdf

• Continuous LWE https://eprint.iacr.org/2026/650.pdf https://ia.cr/2022/093 https://ia.cr/2022/437 https://doi.org/10.1145/3406325.3451000

• Secret-Power Ring LWE https://eprint.iacr.org/2026/877.pdf#page=18.62

• LWE with low-norm samples https://eprint.iacr.org/2023/941.pdf

• Block binary LWE https://eprint.iacr.org/2023/958.pdf

[adouteau]

Here is another assumptions that might be listed in the LAZ as well. (sorry if some are already included in your previous list)

SIS-based:

• Algebraic One-More MISIS (AOM-MISIS (2025/436, 2024/496)
• Inhomogeneous Short and Sparse Decision Assumption (ISSIS) (2025/844)

LWE-based:

• Algebraic One-More AOM-MLWE (2025/436, 2024/496)
• Known-Covariance MLWE (https://cic.iacr.org/p/1/4/2)
• Known-Norm LWE (https://cic.iacr.org/p/1/4/2)
• yet another LWE (2024/1984)
• Reused-A LWE (https://cic.iacr.org/p/1/4/2)
• HNF-LWE (https://web.eecs.umich.edu/~cpeikert/pubs/kdm-learning.pdf)
• Exact-Hint HNF (2024/1695)
• others Approximate Hint assumptions (generalised by Leaky LWE)
• Truncated MLWE (2025/120)
• Sparse-LWE (distinguish between sparse secret/error (defined by cryptanalysts 2025/1002, 2025/1990, 2024/1229) ) and sparse matrix A "as in 2024/1589")
• Learning With Two Errors (2025/844)
• Learning With Short and Sparse Errors Assumption (2025/844)
• Threshold-LWE (2025/1691)
• LWE-OD (2025/2225)
• LWE with physical rounding (LWPR) (2025/2226)
• EntropicLWE (https://eprint.iacr.org/2020/119.pdf, https://eprint.iacr.org/2022/245, https://eprint.iacr.org/2025/1472)
• CosetHintMLWE (https://eprint.iacr.org/2025/1958.pdf)
• DecomposedSIS (https://eprint.iacr.org/2026/1140.pdf)
• Knaspack LWE https://eprint.iacr.org/2019/1287.pdf -> JNS: Both papers describe a converting a LWE instance into a Knapsack instance. Thus, I wouldn't consider this an LWE-Assumption as the described problem is a Knapsack assumption. However, we can think about introducing a / multiple Knapsack pages in the future.
• Oracle MLWE (presented in 2025/1655 then attacked (2026/177)

I’d also like to mention this work, which (somewhat) fits within the scope of the topic. (https://ir.cwi.nl/pub/30068/30068.pdf).