efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard...
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![Page 1: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/1.jpg)
Efficient arithmetic for cryptography andcryptanalysis
Édouard Rousseau
Math Innov day
![Page 2: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/2.jpg)
My thesis
Number theory
Cryptology
Computer algebra
Algorithmic
Finite fields
![Page 3: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/3.jpg)
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.
![Page 4: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/4.jpg)
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.
![Page 5: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/5.jpg)
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.
![Page 6: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/6.jpg)
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.
![Page 7: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/7.jpg)
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.
![Page 8: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/8.jpg)
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.
![Page 9: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/9.jpg)
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.
![Page 10: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/10.jpg)
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.
![Page 11: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/11.jpg)
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.
![Page 12: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/12.jpg)
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.
![Page 13: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/13.jpg)
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
![Page 14: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/14.jpg)
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
![Page 15: Efficient arithmetic for cryptography and cryptanalysis · 2020-05-21 · cryptanalysis Édouard Rousseau Math Innov day. My thesis Number theory Cryptology Computer algebra Algorithmic](https://reader034.vdocuments.mx/reader034/viewer/2022042803/5f49845921b065199c6d3608/html5/thumbnails/15.jpg)
Fp
Fp2
Fp4
Fp3
Fp9
Fp5
Fp25
Fp`
Fp`2
Thanks for your attention!