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Nov 01, 2023 •216 likes •1.59k views

Algorithms for hard lattice problems Thijs Laarhoven ts ttts Tenerife PQCrypto Conference (January 31, 2018) Lattices What is a lattice? O Lattices What

  1. v 9 v 9 Sieving v 12 v 12 2. Split L into centers C and reducible vectors R v 5 v 5 v 13 v 13 v 13 v 16 v 7 v 7 v 15 v 1 v 1 v 3 v 3 v 8 v 8 v 2 v 2 O v 6 v 6 v 11 v 11 v 10 v 10 v 14 v 4 v 4

  2. v 9 v 9 Sieving v 12 v 12 2. Split L into centers C and reducible vectors R v 5 v 5 v 13 v 13 v 16 v 7 v 7 v 15 v 1 v 1 v 3 v 3 v 8 v 8 v 2 v 2 O v 6 v 6 v 11 v 11 v 10 v 10 v 14 v 14 v 14 v 4 v 4

  3. v 9 v 9 Sieving v 12 v 12 2. Split L into centers C and reducible vectors R v 5 v 5 v 13 v 13 v 16 v 7 v 7 v 15 v 15 v 15 v 1 v 1 v 3 v 3 v 8 v 8 v 2 v 2 O v 6 v 6 v 11 v 11 v 10 v 10 v 14 v 14 v 4 v 4

  4. v 9 v 9 Sieving v 12 v 12 2. Split L into centers C and reducible vectors R v 5 v 5 v 13 v 13 v 16 v 16 v 16 v 7 v 7 v 15 v 15 v 1 v 1 v 3 v 3 v 8 v 8 v 2 v 2 O v 6 v 6 v 11 v 11 v 10 v 10 v 14 v 14 v 4 v 4

  5. v 9 v 9 Sieving v 12 v 12 2. Split L into centers C and reducible vectors R v 5 v 5 v 13 v 13 v 16 v 16 v 7 v 7 v 15 v 15 v 1 v 1 v 3 v 3 v 8 v 8 v 2 v 2 O v 6 v 6 v 11 v 11 v 10 v 10 v 14 v 14 v 4 v 4

  6. v 9 v 9 Sieving v 12 v 12 3. Repeat with reduced vectors until we find a shortest vector v 5 v 5 v 13 v 13 v 16 v 16 v 7 v 7 v 15 v 15 v 1 v 1 v 3 v 3 v 8 v 8 v 2 v 2 O v 6 v 6 v 11 v 11 v 10 v 10 v 14 v 14 v 4 v 4

  7. v 9 v 9 Sieving v 12 v 12 3. Repeat with reduced vectors until we find a shortest vector v 5 v 5 v 13 v 13 v 16 v 16 v 7 v 7 v 15 v 15 v 1 v 1 v 3 v 3 v 8 v 8 v 2 v 2 O v 6 v 6 v 11 v 11 v 10 v 10 v 14 v 14 v 4 v 4

  8. v 9 v 9 Sieving v 12 v 12 3. Repeat with reduced vectors until we find a shortest vector v 5 v 5 v 13 v 13 v 16 v 16 v 7 v 7 v 15 v 15 v 1 v 1 v 3 v 3 v 8 v 8 v 2 v 2 O v 6 v 6 v 11 v 11 v 10 v 10 v 14 v 14 v 4 v 4

  9. v 9 Sieving v 12 3. Repeat with reduced vectors until we find a shortest vector v 5 v 13 v 16 v 7 v 15 v 1 v 5 v 1 v 3 v 4 v 2 v 7 O v 3 v 2 v 6 v 6 v 11 v 10 v 14 v 4

  10. v 9 Sieving v 12 Overview v 5 v 13 v 16 v 7 v 15 v 1 v 5 v 1 v 3 v 4 v 2 v 7 O v 3 v 2 v 6 v 6 v 11 v 10 v 14 v 4

  11. v 9 Sieving v 12 Overview v 5 v 13 v 16 v 7 v 15 Heuristic (Nguyen–Vidick, J. Math. Crypt. ’08) Sieving solves SVP in time ( 4 / 3 ) n + o ( n ) and space ( 4 / 3 ) n / 2 + o ( n ) . v 1 v 5 v 1 v 3 v 4 v 2 v 7 O v 3 v 2 v 6 v 6 v 11 v 10 v 14 v 4

  12. v 9 Sieving v 12 Overview v 5 v 13 v 16 v 7 v 15 Heuristic (Nguyen–Vidick, J. Math. Crypt. ’08) Sieving solves SVP in time ( 4 / 3 ) n + o ( n ) and space ( 4 / 3 ) n / 2 + o ( n ) . v 1 v 5 v 1 v 3 v 4 v 2 v 7 O The list size comes from heuristic packing / saturation arguments, v 3 the time complexity is quadratic in the list size. v 2 v 6 v 6 v 11 v 10 v 14 v 4

  13. Sieving Near neighbor techniques O

  14. v 9 Sieving v 12 Near neighbor techniques v 5 v 13 v 16 v 7 v 15 v 1 v 3 v 8 v 2 O v 6 v 11 v 10 v 14 v 4

  15. v 9 Sieving v 12 Near neighbor techniques v 5 v 13 v 16 v 7 v 15 v 1 v 3 v 8 v 2 O v 6 v 11 v 10 v 14 v 4

  16. v 9 Sieving v 12 Near neighbor techniques v 5 v 13 v 16 v 7 v 15 v 1 v 3 v 8 v 2 O v 6 v 11 v 10 v 14 v 4

  17. v 9 Sieving v 12 Near neighbor techniques v 5 v 13 v 16 v 7 v 15 v 1 v 3 v 8 v 2 O O v 6 v 11 v 10 v 14 v 4

  18. v 9 Sieving v 12 Near neighbor techniques v 5 v 13 v 16 v 7 v 15 v 1 v 1 v 1 v 3 v 8 v 2 O v 6 v 11 v 10 v 14 v 4

  19. v 9 Sieving v 12 Near neighbor techniques v 5 v 13 v 16 v 7 v 15 v 1 v 1 v 3 v 8 v 2 v 2 v 2 O v 6 v 11 v 10 v 14 v 4

  20. v 9 Sieving v 12 Near neighbor techniques v 5 v 13 v 16 v 7 v 15 v 1 v 1 v 3 v 3 v 3 v 8 v 2 v 2 O v 6 v 11 v 10 v 14 v 4

  21. v 9 Sieving v 12 Near neighbor techniques v 5 v 13 v 16 v 7 v 15 v 1 v 1 v 3 v 3 v 8 v 2 v 2 O v 6 v 11 v 10 v 14 v 4 v 4 v 4

  22. v 9 Sieving v 12 Near neighbor techniques v 5 v 5 v 5 v 13 v 16 v 7 v 15 v 1 v 1 v 3 v 3 v 8 v 2 v 2 O v 6 v 11 v 10 v 14 v 4 v 4

  23. v 9 Sieving v 12 Near neighbor techniques v 5 v 5 v 13 v 16 v 7 v 15 v 1 v 1 v 3 v 3 v 8 v 2 v 2 O v 6 v 6 v 6 v 11 v 10 v 14 v 4 v 4

  24. v 9 Sieving v 12 Near neighbor techniques v 5 v 5 v 13 v 16 v 7 v 7 v 7 v 15 v 1 v 1 v 3 v 3 v 8 v 2 v 2 O v 6 v 6 v 11 v 10 v 14 v 4 v 4

  25. v 9 Sieving v 12 Near neighbor techniques v 5 v 5 v 13 v 16 v 7 v 7 v 15 v 1 v 1 v 3 v 3 v 8 v 8 v 8 v 2 v 2 O v 6 v 6 v 11 v 10 v 14 v 4 v 4

  26. v 9 v 9 v 9 Sieving v 12 Near neighbor techniques v 5 v 5 v 13 v 16 v 7 v 7 v 15 v 1 v 1 v 3 v 3 v 8 v 8 v 2 v 2 O v 6 v 6 v 11 v 10 v 14 v 4 v 4

  27. v 9 v 9 Sieving v 12 Near neighbor techniques v 5 v 5 v 13 v 16 v 7 v 7 v 15 v 1 v 1 v 3 v 3 v 8 v 8 v 2 v 2 O v 6 v 6 v 11 v 10 v 10 v 10 v 14 v 4 v 4

  28. v 9 v 9 Sieving v 12 Near neighbor techniques v 5 v 5 v 13 v 16 v 7 v 7 v 15 v 1 v 1 v 3 v 3 v 8 v 8 v 2 v 2 O v 6 v 6 v 11 v 11 v 11 v 10 v 10 v 14 v 4 v 4

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