Boolean functions with optimal stability of their cryptographic indicators under restriction of the inputs

-简短摘要-加密数据对确保电子通讯的安全至为重要。评估对称密码安全性的主要工具是检查其对当前已知攻击技术的抵抗力。正在使用的密码是由较小的组件构建的，其中一个组件是加密布尔函数。它对已知攻击的抵抗力是通过加密指标来量化的。我们研究了布尔函数的一种性质，尽管它与实际攻击有关，但以前很少被研究。也就是说，对于几个密码指示符，我们通过将它们限制为特定类型的输入，来确定当输入被攻击者恶意操纵时它们如何变化。理想情况下，加密指示器应该是稳定的，即不会有太大变化。我们的目标是回答这样的问题：指标的稳定性可以达到的最佳值是什么？有多少个最优函数？如何构造这样的函数？如何从这一点上测试建议或现有的功能(来自当前使用的密码)是最优的还是接近最优的？-扩展摘要-电子通信是个人和社会(例如网上购物、银行、电子政务)日常生活的重要组成部分。加密数据是实现这些通信安全的核心技术。使用两种类型的密码术：对称密码术(用于绝大多数传输的数据)和公钥密码术(主要用于密钥交换和数字签名)。在对称密码术中，发送者加密和接收者使用相同的密钥解密；密钥保密至关重要。对称密码，如AES(当前的主要标准)，以及移动电话中使用的密码，都是由几个较小的组件组成的。每个单独的组件以及整个系统都必须满足某些密码要求，这使它们能够抵抗当前已知的攻击技术。这个项目着眼于一个这样的组件，即密码布尔函数，以及随着时间的推移而开发的几个指示器，以量化它们对已知密码攻击的抵抗力。对对称密码的攻击超出了拦截加密数据和尝试确定原始数据的范围。在选定的明文攻击中，攻击者在加密之前对数据进行操作，希望相应的加密数据将揭示有关密钥的有用信息。操作输入的一个简单而有效的方法是只考虑符合给定模式的输入，例如。将数据的第一个字节设置为零，或者将第一个字节设置为与第二个字节相同的值。这些例子属于更一般的仿射子空间，这是我们在本项目中关注的。布尔函数不仅应该具有良好的密码指示符的值，而且当受到上述输入的恶意操纵时，也应该保持这些良好的值。我们将考虑几种密码学，看看对仿射空间的限制是如何影响它们的。对于每个指标，我们将从保持指标良好价值的角度来研究哪些函数是最优的，了解它们的数学性质，确定存在多少这样的函数，并制定构造它们的方法。我们还将研究现有的密码，并从这个角度确定它们的行为。理论结果将发表在研究期刊和会议上。我们将要构造的新函数以及为现有基准函数计算的新引入的参数的值将公之于众。项目中研究的新函数和性质将有助于确保未来设计的新密码以及基于它们的协议更不容易受到攻击。

-Short summary-Encrypting data is essential for ensuring the security of our electronic communications. The main tool for evaluating the security of a symmetric cipher is to examine its resistance to the currently known attack techniques. The ciphers in use are built from smaller components, one of them being a cryptographic Boolean function. Its resistance to known attacks is quantified by cryptographic indicators. We investigate a type of property of Boolean functions which was very little studied before, despite its relevance to actual attacks. Namely, for several cryptographic indicators, we determine how they change when the inputs are maliciously manipulated by an attacker by restricting them to certain types of inputs. Ideally, the cryptographic indicators should be stable, i.e. not change much. We aim to answer questions like: What are the optimal values that can be achieved for the stability of the indicators? How many optimal functions are there? How to construct such functions? How to test if proposed or existing functions (from currently used ciphers) are optimal or close to optimal from this point is view?-Extended summary-Electronic communications are an essential part of everyday life for individuals and for society (e.g. online shopping, banking, e-government). Encrypting the data is a core technique for achieving security of these communications. Two types of cryptography are used: symmetric cryptography (used for the vast majority of the transmitted data) and public-key cryptography (used mainly for key exchange and digital signatures).In symmetric cryptography, the sender encrypts and the receiver decrypts using the same key; it is crucial that the key is kept secret. The symmetric ciphers such as AES (the main current standard), as well as the ciphers used in mobile phones, are built out of several smaller components. Each individual component, as well as the overall system, must satisfy certain cryptographic requirements which makes them resistant to the currently known attack techniques. This project looks at one such component, namely cryptographic Boolean functions, and several indicators that have been developed over time to quantify their resistance to the known cryptographic attacks.Attacks on symmetric ciphers go beyond intercepting encrypted data and attempting to determine the original data. In chosen plaintext attacks, the attacker manipulates the data before encryption in the hope that the corresponding encrypted data will reveal useful information about the key.One simple but effective way to manipulate the input is to only consider inputs that conform to a given pattern, eg. setting the first byte of the data to zero, or setting the first byte to the same value as the second. These examples belong to the more general class of affine subspaces, which we are focusing on in this project.Boolean functions should not only have good values of their cryptographic indicators, but also preserve these good values when subjected to the malicious manipulation of the inputs mentioned above. We will consider several cryptographic and see how they are affected by restriction to affine spaces. For each of these indicators, we will study functions which are optimal from the point of view of maintaining good values of the indicators, understand their mathematical properties, determine how many such functions exist and devise methods of constructing them. We will also examine existing ciphers and determine their behaviour from this point of view.The theoretical results will be published in research journals and conferences. The new functions that we will construct, and the values of the newly introduced parameters computed for existing benchmark functions, will be made publicly available.The new functions and properties studied in the project will contribute to ensuring that new ciphers designed in the future, as well as the protocols based on them, will be less vulnerable to attacks.