From 4521358a6365e0247e87044c3efe6f7ca3408ee2 Mon Sep 17 00:00:00 2001 From: Fabrice Mouhartem Date: Fri, 16 Feb 2018 15:52:29 +0100 Subject: [PATCH] Add explanations --- chap-structures.tex | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/chap-structures.tex b/chap-structures.tex index 37953f8..51e468d 100644 --- a/chap-structures.tex +++ b/chap-structures.tex @@ -6,7 +6,14 @@ Beside \emph{information theory-base cryptography}, most hardness assumptions ar For instance the discrete logarithm assumption (Definition~\ref{de:DLP}) is based on a cyclic group structure. That is, in some groups it is assumed that computing the discrete logarithm is an intractable problem for any probabilistic polynomial time algorithms. -In this chapter, we focus on describing the different structures on which the cryptography we build in this thesis are based on, namely bilinear groups and lattices. +The existence of these structures proves useful when it comes to design protocols. +For that, constructions takes advantage of the mathematical properties of the structure to allow the functionality. +An example is the multiplicative homomorphism of the ElGamal cryptosystem which is possible using the structure of the underlying cyclic group $\GG$ on which the scheme is built upon. +%Namely, an El Gamal encryption of a message $M$ under the public key $h = g^\alpha_{} \in \GG$ is a couple $(c_1^{}, c_2^{}) = (g^r_{}, M \cdot h^r_{}) \in \GG^2_{}$, which can be decrypted with the knowledge of the secret key $\alpha \in \Zp$: $M = c_2^{} \cdot c_1^{-\alpha}$. +%Then, the cyclic group structure of $\GG$ leads to the ability to compute a valid ciphertext for $M \cdot M'$ given ciphertexts $(c_1^{}, c_2^{})$ and $(c'_1, c'_2)$ of $M$ and $M'_{}$ respectively. +%The resulting ciphertext is $(c_1^{} \cdot c'_1, c_2^{} \cdot c'_2) = (g^{r \cdot r'_{}}, M \cdot M' \cdot h^{r \cdot r'_{}})$ + +In this chapter, we describe the different structures on which the cryptography primitives we design in this thesis are based on, namely bilinear groups and lattices. \section{Pairing-Based Cryptography} \label{se:pairing}