1. Introduction -- 1.1 Our results -- 1.2 Brief overview -- 1.3 Organization --

2. Survey of applications -- 2.1 How flexible can we make access to encrypted data? -- 2.2 Program obfuscation -- 2.3 Other applications --

3. Multilinear maps and graded encoding systems -- 3.1 Cryptographic multilinear maps -- 3.1.1 Efficient procedures -- 3.1.2 Hardness assumptions -- 3.2 Graded encoding schemes -- 3.2.1 Efficient procedures, the dream version -- 3.2.2 Efficient procedures, the real-life version -- 3.2.3 Hardness assumptions --

4. Preliminaries I: lattices -- 4.1 Lattices -- 4.2 Gaussians on lattices -- 4.3 Sampling from discrete Gaussian --

5. Preliminaries II: algebraic number theory background -- 5.1 Number fields and rings of integers -- 5.2 Embeddings and geometry -- 5.3 Ideals in the ring of integers -- 5.4 Prime ideals, unique factorization and distributions -- 5.5 Ideal lattices --

6. The new encoding schemes -- 6.1 The basic graded encoding scheme -- 6.2 Setting the parameters -- 6.3 Extensions and variants --

7. Security of our constructions -- 7.1 Our hardness assumption -- 7.2 Simplistic models of attacks -- 7.2.1 Hardness of GCDH in the arithmetic straight-line program model -- 7.3 Cryptanalysis beyond the generic models -- 7.3.1 Easily computable quantities -- 7.3.2 Using averaging attacks -- 7.3.3 Cryptanalysis with extra help -- 7.4 Some countermeasures -- 7.5 Easiness of other problems --

8. Preliminaries III: computation in a number field -- 8.1 Some computational aspects of number fields and ideal lattices -- 8.2 Computational hardness assumptions over number fields --

9. Survey of lattice cryptanalysis -- 9.1 Averaging attacks -- 9.2 Gentry-Szydlo: recovering v from v.v and (v) -- 9.3 Nguyen-Regev: a gradient descent attack -- 9.4 Ducas-Nguyen: gradient descent over zonotopes and deformed parallelepipeds -- 9.5 A new algorithm for the closest principal ideal generator problem -- 9.6 Coppersmith attacks -- 9.7 Dimension halving in principal ideal lattices --

10. One-round key exchange -- 10.1 Definitions -- 10.2 Our construction --

Appendix A. Generalizing graded encoding systems -- Efficient procedures, the dream version -- Efficient procedures, the real-life version -- Hardness assumptions -- Bibliography -- Author's biography.