Efficient arithmetic for cryptography andcryptanalysis

Édouard Rousseau

Math Innov day

My thesis

Number theory

Cryptology

Computer algebra

Algorithmic

Finite fields

SOME RECALLS

Definition (Wall)Construction used for shelter, protection,or privacy, etc.

Definition (Door)Part of a wall that can be opened to enterinside the walls.Definition (Lock and key)Device for securing a door: a locked doorcan only be opened with the associatedkey.

Definition (Cryptology)The art/science of building walls, doors, locks and keys forprotecting informations.

SOME RECALLS

Definition (Wall)Construction used for shelter, protection,or privacy, etc.

Definition (Door)Part of a wall that can be opened to enterinside the walls.Definition (Lock and key)Device for securing a door: a locked doorcan only be opened with the associatedkey.

Definition (Cryptology)The art/science of building walls, doors, locks and keys forprotecting informations.

SOME RECALLS

Definition (Wall)Construction used for shelter, protection,or privacy, etc.

Definition (Door)Part of a wall that can be opened to enterinside the walls.Definition (Lock and key)Device for securing a door: a locked doorcan only be opened with the associatedkey.

Definition (Cryptology)The art/science of building walls, doors, locks and keys forprotecting informations.

SOME RECALLS

Definition (Wall)Construction used for shelter, protection,or privacy, etc.

Definition (Door)Part of a wall that can be opened to enterinside the walls.

Definition (Lock and key)Device for securing a door: a locked doorcan only be opened with the associatedkey.

Definition (Cryptology)The art/science of building walls, doors, locks and keys forprotecting informations.

SOME RECALLS

Definition (Wall)Construction used for shelter, protection,or privacy, etc.

Definition (Door)Part of a wall that can be opened to enterinside the walls.

Definition (Lock and key)Device for securing a door: a locked doorcan only be opened with the associatedkey.

Definition (Cryptology)The art/science of building walls, doors, locks and keys forprotecting informations.

SOME RECALLS

Definition (Wall)Construction used for shelter, protection,or privacy, etc.

Definition (Door)Part of a wall that can be opened to enterinside the walls.Definition (Lock and key)Device for securing a door: a locked doorcan only be opened with the associatedkey.

Definition (Cryptology)The art/science of building walls, doors, locks and keys forprotecting informations.

SOME RECALLS

Definition (Wall)Construction used for shelter, protection,or privacy, etc.

Definition (Door)Part of a wall that can be opened to enterinside the walls.Definition (Lock and key)Device for securing a door: a locked doorcan only be opened with the associatedkey.

Definition (Cryptology)The art/science of building walls, doors, locks and keys forprotecting informations.

SOME RECALLS

Definition (Wall)Construction used for shelter, protection,or privacy, etc.

Definition (Door)Part of a wall that can be opened to enterinside the walls.Definition (Lock and key)Device for securing a door: a locked doorcan only be opened with the associatedkey.

Definition (Cryptology)The art/science of building walls, doors, locks and keys forprotecting informations.

GOAL OF THE THESIS

= Number theory

= Algorithmically hard problems

I Goal: study the material used to craft locks and keys tobetter understand how to open the doors.

I Goal: (maths) study the algebraic structures used in thehard problems to better understand how to efficientlysolve them.

GOAL OF THE THESIS

= Number theory

= Algorithmically hard problems

I Goal: study the material used to craft locks and keys tobetter understand how to open the doors.

I Goal: (maths) study the algebraic structures used in thehard problems to better understand how to efficientlysolve them.

WHAT exactly DID I STUDY?

I If ` |m, then Fp` ↪→ Fpm

I If ` |m |n, then Fp` ↪→ Fpm ↪→ Fpn

E

F

G

φE↪→F

φE↪→G

φF↪→G

φF↪→G ◦ φE↪→F?= φE↪→G

WHAT exactly DID I STUDY?

I If ` |m, then Fp` ↪→ Fpm

I If ` |m |n, then Fp` ↪→ Fpm ↪→ Fpn

E

F

G

φE↪→F

φE↪→G

φF↪→G

φF↪→G ◦ φE↪→F?= φE↪→G

Fp

Fp2

Fp4

Fp3

Fp9

Fp5

Fp25

Fp`

Fp`2

Thanks for your attention!