Privacy-preserving computation (PPC) refers to a series of information technologies for analyzing and computing data under the premise of ensuring that the data provider does not disclose the original data</font> and guaranteeing that the data in the process of circulation and fusion is The "available invisible" . Gartner's Strategic Trends in Frontier Technology 2021 lists privacy computing (which it calls privacy-enhanced computing) as one of the nine trends in technology development over the next few years. (The need for data flow is driving the momentum for privacy computing.) But there are still many obstacles.

The year 2021 has been called the year of privacy computing, and the technology is a very comprehensive field involving many directions, such as <font color=greem> cryptography, mathematics, big data, real-time computing, high-performance computing, distributed, traditional machine learning frameworks and algorithms, deep learning frameworks and algorithms, and </font> and so on. gt; and so on. This series of articles focuses on the relevant cryptography used in privacy computing.

Cryptography has been an important part of human history, covering all aspects of human activity. For example, in spy movies, we often see underground workers use encrypted messages to transmit important information, such as in the Internet surfing time TLS/SSL is also to protect our information security, it can be said that the impact of cryptography on human beings and the role of the far-reaching, it is difficult to imagine that there is no cryptographic protection of the day is what! So what exactly is cryptography? How to accurately define cryptography? This series of articles will be relatively detailed introduction, welcome to watch.

First of all, the precise definition of cryptography is still quoted from the authorities, the following is from Baidu Encyclopedia:

The function of cryptography at the beginning was to protect the two sides of the communication security in the environment of the existence of malicious attackers, and now it is used to protect the information security of the core technology.

The basic requirements of modern information security:

The specific operation of encryption and decryption is determined by two parts:

Divided by time, cryptographic algorithms prior to 1976 belong to classical cryptography, which is basically used in the field of military secrets and diplomacy, which is characterized by a simple process of encryption and decryption, which is generally done manually or mechanically, and classical cryptography is now rarely The following is a classical cryptographic codebook that provides one-to-one mapping transformations. The main advantage of having this codebook is that it is easy to encrypt and decrypt information by hand.

Common encryption algorithms in modern cryptography can be roughly divided into the following categories:

This series of articles will focus on describing symmetric and asymmetric encryption, from the mathematical principles, encryption algorithms derivation, encryption algorithms principle will be introduced. The content of this article involves mathematical knowledge of number theory, for encryption algorithms will be used in the knowledge, this chapter will do some appropriate introduction to the number of students interested in the theory of tea related to the "theory of number" books for the refinement of the derivation, if there are not rigorous place, welcome students to help correct.

Encryption using a single-key cryptosystem, the same key can be used as both encryption and decryption of information, this encryption method is called symmetric encryption, also known as single-key encryption.

There are numerous algorithms for symmetric encryption, and a number of implementations of these algorithms have appeared throughout history, with different algorithms assuming important roles at particular times. Although some of these algorithms are no longer relevant and have security vulnerabilities, they did play a very important role in the arithmetic situation at the time. Giving people a security barrier to protect information security. Here is a brief comparison of the more famous centralized symmetric encryption algorithms.

In fact, many people have expressed the view that symmetric encryption is insecure and has the risk of being cracked. In fact, I think this is not a very strict statement, simple symmetric encryption such as AES is still very safe, but an important feature of symmetric encryption is that encryption and decryption use the same secret key, for the preservation of the secret key and the delivery of the secret key is a very headache, once the leakage will cause great risk.

So, is there any way to further reduce the risk of leakage? The answer is asymmetric encryption. Here's an introduction to asymmetric encryption.

Before the formal introduction of the RSA algorithm, because the algorithm requires more mathematical knowledge, as the saying goes, "Sharpening the knife is not a good idea, ten thousand feet from the ground", so follow the steps below to explain.

So this section mainly introduces the basics of number theory to provide a theoretical basis for the subsequent derivation and proof of the RSA algorithm.

A prime number is a magic number in mathematics, and many mathematical theorems are based on prime numbers.

<font color=Greem> A prime number, also known as a prime number</font>, is a natural number greater than 1 that is not divisible by any other natural number except 1 and the number itself (which can also be defined as a number that has only two positive factors, 1 and the number itself). Natural numbers greater than 1 that are not prime are called composite numbers (also known as synthetic numbers)</font>. For example, 5 is a prime number because its positive constants are only 1 and 5, while 6 is a composite number because, in addition to 1 and 6, 2 and 3 are also its positive constants.

<font color=greem> Two integers are called mutually prime if their covenant is only 1. </font> Two natural numbers with a convention of only 1 are called mutually prime natural numbers, the latter being a special case of the former.

For example:

Regarding the mutual prime relation, to summarize, there are the following properties

When it comes to Euler, it's important to say a few more words, Leonhard? Euler (Leonhard Euler, April 5, 1707 - September 18, 1783) was the son of a pastor near Basel, Switzerland, who, in addition to studying theology, studied mathematics with great success. He has been described by some historians of mathematics as one of the two greatest mathematicians in history (the other being Carl? Friedrich? Gauss). Many terms in mathematics are named after Euler, such as Euler's constant, Euler's equation, Euler's constant, Euler's schematic number, and so on.

Euler is recognized as one of the most accomplished mathematicians in human history. Many constants, formulas, and theorems named after Euler can be found in mathematics and in many branches, and his work has brought mathematics closer to its present form. He not only contributed to the mathematical world, but also pushed mathematics into almost the entire field of physics. Euler was also involved in the fields of architecture, ballistics and navigation. Charles Kleiber, the Swiss Secretary of State for Education and Research, once said, "Without Euler's many scientific discoveries, we would be living a completely different life today." French mathematician Laplace, on the other hand, argued, Read Euler, he is the teacher of all.

Then in number theory for <font color=greem> positive integers n</font>, the logic of Euler's function φ(n) is the number of positive integers less than n that are mutually prime with n.

Above we introduced primes, mutual prime relations, and Euler's function, and based on this knowledge, we continue with Euler's theorem.

If <font color=greem> two positive integers a and n are mutually prime</font>, then Euler's function φ(n) for n allows the following equation to hold, which can be understood as a

The Euler function φ(n) of n allows the following equation to hold, which can be interpreted as the remainder of the division of a by n.

In layman's terms, this means that the remainder of the division of a by n is 1, and it can also be interpreted as the division of a by n plus one.

Suppose positive integer a is mutually prime with <font color=greem> prime p</font> because φ(p) of prime p is equal to p-1, then Euler's theorem can be written as

It can also be expressed as

The modulo inverse element, also called the modulo inverse operation, is defined as follows: if two <font color=greem> primes p are equal to n, then Euler's theorem can be written as

The modulo inverse element, also known as modulo inverse, is defined as follows. color=greem> positive integers a and n are prime to each other</font>, then it must be possible to find an integer b such that ab-1 is divisible by n, or ab is divisible by n by 1. In this case, b is called the "modulo inverse element" of a.

For example, if you want to find an integer b such that ab-1 is divisible by n, or if ab is divisible by n by 1, then you have to find a modulo inverse element.

For example, if 3 and 14 are mutually prime, then the modulo inverse of 3 is 5, because (3 × 5)-1 is divisible by 14. Obviously, there is more than one modulo inverse element; 5 plus or minus an integer multiple of 14 is a modulo inverse element of 3 {... ,-23, -9, 5, 19, 33,...} , i.e., if b is the modulo inverse of a, then b+kn are all modulo inverse of a.

At this point, the math for deriving and proving RSA encryption algorithms is complete, and subsequent chapters will continue to introduce RSA-related knowledge and application scenarios.

<font color=Blue> All the law, such as dreams and bubbles, such as dew and electricity, should be viewed as such</font>

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