SaTC: CORE: Small: Applications of Galois Theory to the Search for Non-Linear Functions

SaTC：核心：小：伽罗瓦理论在搜索非线性函数中的应用

基本信息

批准号：
2127742
负责人：
Giacomo Micheli
金额：
$ 50万
依托单位：
University of South Florida
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-15 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2127742&HistoricalAwards=false
关键词：
SaTC CORE Small Applications Galois

项目摘要

Block ciphers and hash functions are a foundational building block of secret key Cryptography. Almost Perfect Nonlinear (APN) functions, i.e. functions defined over a finite field that behave as non-linearly as possible, are used to design block ciphers and hash functions secure against differential attacks. This project supports research on the construction of APN functions that impact the design of the next generation of ciphers which will need to operate in constraint environments (lightweight cryptography), and feature large keys to offer high levels of bit security against quantum adversaries. The broader impacts of this project include the organization of workshops bringing together Academia and Industry, the development of new cryptography curriculum at USF, the organization of Summer camps, and the design of animated videos featuring important concepts in cryptography.To achieve the construction of new APN functions, this project supports the development of Galois theoretical methods to decide whether APN functions (more specifically APN permutations) exist for given parameters such as the size of the finite field and the degree of the APN function. In particular, the problem of finding an APN function is converted into the problem of solving a polynomial system of equations arising from group theory. This generates multiple polynomial systems, each one allowing either the construction of a given APN function for a chosen degree, or the proof of non-existence of APN permutations for the target degree. The methods developed as part of this project are constructive and they can be efficiently implemented via some standard computational algebra software such as Magma or Sage.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

块密码和哈希功能是秘密密钥密码学的基础基础。几乎完美的非线性（APN）函数，即在有限的字段上定义的函数，其行为尽可能地非线性，用于设计阻止密码和哈希函数可抗差分攻击。该项目支持对影响下一代密码设计的APN功能的构建的研究，这些密码需要在约束环境（轻量级加密）中操作，并具有大键，以提供针对量子对手的高水平安全性。该项目的更广泛的影响包括组织的组织组织，将学术界和行业汇集在一起，在USF的新密码学课程的开发，夏令营的组织以及动画视频的设计以及具有密码学重要概念的动画视频的设计。要实现新的APN功能的构建，该项目支持APN ASS的特定范围（该项目）是否具有APN ASS的开发（是否有特定的APN函数），以实现APN ASS的发展。有限场和APN函数的程度。特别是，找到APN函数的问题被转换为解决群体理论引起的多项式方程系统的问题。这会生成多个多项式系统，每个系统都允许为所选程度构建给定的APN函数，或者允许目标度不存在APN排列的证据。作为该项目的一部分开发的方法是建设性的，可以通过一些标准的计算代数软件（例如Magma或Sage）有效地实施它们。该奖项反映了NSF的法定任务，并认为值得通过基金会的知识分子优点和更广泛的影响通过评估来获得支持。

项目成果

期刊论文数量（7）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

R-fat Linearized Polynomials over Finite Fields

DOI：
10.1016/j.jcta.2022.105609
发表时间：
2020-12
期刊：
J. Comb. Theory A
影响因子：
0
作者：
D. Bartoli;Giacomo Micheli;Giovanni Zini;Ferdinando Zullo
通讯作者：
D. Bartoli;Giacomo Micheli;Giovanni Zini;Ferdinando Zullo

Advanced signature functionalities from the code equivalence problem

DOI：
10.1080/23799927.2022.2048206
发表时间：
2022-03
期刊：
International Journal of Computer Mathematics: Computer Systems Theory
影响因子：
0
作者：
Alessandro Barenghi;Jean-François Biasse;Tra Ngô;Edoardo Persichetti;Paolo Santini
通讯作者：
Alessandro Barenghi;Jean-François Biasse;Tra Ngô;Edoardo Persichetti;Paolo Santini

New results on permutation binomials of finite fields

有限域置换二项式的新结果