C/C++编程中加密算法详解（第二版）

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"Cryptography in C and C++.2Ed" 是一本专著，专注于在C和C++编程语言中实现加密技术的教程。本书将复杂的密码学概念以易于理解的方式介绍给读者，特别关注算术和数论在密码学中的基础应用。作者通过清晰的章节结构，引导学习者逐步探索关键概念。第1章“Arithmetic and Number Theory in C”为整本书奠定了基础，介绍数学原理如何在C语言中体现，包括大数的表示和处理。数字格式的讨论涉及不同方法来高效地表示和操作大数值，这对于加密算法中的大数运算至关重要。在第2章中，作者详细讲解了数字格式的表示，探讨了如何用C语言精确地表示和处理超出计算机原生数据类型范围的大数值。这不仅关乎效率，也关系到数据安全，因为大数操作常用于密钥生成和计算。第3章探讨接口语义，确保程序员在使用加密函数时能正确理解和设计接口，这对于代码的稳定性和互操作性至关重要。这一章节可能会涉及类型系统、错误处理以及内存管理。第4章深入剖析基本的算术运算，如加法、减法、乘法和除法，特别是针对大数的高效算法。作者提到了多种方法，如grade school方法、平方优化（如快速幂）和Karatsuba算法，这些对于实现快速、安全的加密算法是不可或缺的。第5章介绍了模运算，即计算余数，这是模数密码学的基础，如RSA加密算法就依赖于它。通过模运算，可以处理整数除法问题，并且保持在特定模数下的安全性。第6章聚焦于模指数运算，这是许多加密算法的核心，包括RSA、ElGamal和Diffie-Hellman等。书中讨论了不同的计算方法，如逐次乘法、M-ary算法、窗口方法，以及 Montgomery 减轻法和快速模指数运算，这些都是加密性能的关键优化手段。第7章进一步探讨了位操作和逻辑函数，如移位和布尔运算，这些在实现高级加密算法时，如AES（高级加密标准）的线性反馈移位寄存器（LFSR）等，扮演着至关重要的角色。这本书旨在帮助C和C++程序员掌握密码学基础，并将其应用到实际的软件开发中。无论是初学者还是经验丰富的开发者，都能从中受益，了解如何在安全的编程环境中利用C和C++进行加密和解密操作。同时，它还涵盖了与现代密码学实践密切相关的最新技术和优化策略。

Part I

Arithmetic and

Number Theory in C

How necessary arithmetic and the entire art of mathematics are can be easily

measured, in that nothing can be created that is not connected with precise

number and measurement, and no independent art can exist without its

measures and proportions.

—Adam Ries: Book of Calculation, 1574

Typographical rules for manipulating numerals are actually arithmetical

rules for operating on numbers.

—D. R. Hofstadter: Gödel, Escher, Bach: An Eternal Golden Braid

The human brain would no longer be burdened with anything that needed to

be calculated! Gifted people would again be able to think instead of scribbling

numbers.

—Sten Nadolny: The Discovery of Slowness, trans. Ralph Freedman

APress/Authoring/2005/04/10:10:11 Page 1

CHAPTER 1

Introduction

God created the integers. All the rest is the work of man.

—Leopold Kronecker

If you look at zero you see nothing; but look through it and you will see the

world.

—Robert Kaplan, The Nothing That Is: A Natural History of Zero

TO BE INVOLVED WITH MODERN cryptography is to dive willy-nilly into number

theory, that is, the study of the natural numbers, one of the most beautiful areas

of mathematics. However, we have no intention of becoming deep-sea divers who

raise sunken treasure from the mathematical ocean ﬂoor, which in any case is

unnecessary for cryptographic applications. Our goals are much more modest.

On the other hand, there is no limit to the depth of involvement of number theory

with cryptography, and many signiﬁcant mathematicians have made important

contributions to this area.

The roots of number theory reach back to antiquity. The Pythagoreans—the

Greek mathematician and philosopher Pythagoras and his school—were already

deeply involved in the sixth century

B.C.E. with relations among the integers,

and they achieved signiﬁcant mathematical results, for example the famed

Pythagorean theorem, which is a part of every school child’s education. With

religious zeal they took the position that all numbers should be commensurate

with the natural numbers, and they found themselves on the horns of a serious

dilemma when they discovered the existence of “irrational” numbers such as

√

which cannot be expressed as the quotient of two integers. This discovery threw

the world view of the Pythagoreans into disarray, to the extent that they sought

to suppress knowledge of the irrational numbers, a futile form of behavior oft

repeated throughout human history.

Two of the oldest number-theoretic algorithms, which have been passed

down to us from the Greek mathematicians Euclid (third century

B.C.E.) and

Eratosthenes (276–195

B.C.E.), are closely related to the most contemporary

encryption algorithms that we use every day to secure communication across

the Internet. The “Euclidean algorithm” and the “sieve of Eratosthenes” are both

quite up-to-date for our work, and we shall discuss their theory and application

in Sections 10.1 and 10.5 of this book.

APress/Authoring/2005/04/10:10:11 Page 3

Chapter 1

Among the most important founders of modern number theory are to be

counted Pierre de Fermat (1601–1665), Leonhard Euler (1707–1783), Adrien

Marie Legendre (1752–1833), Carl Friedrich Gauss (1777–1855), and Ernst Eduard

Kummer (1810–1893). Their work forms the basis for the modern development of

this area of mathematics and in particular the interesting application areas such as

cryptography, with its asymmetric procedures for encryption and the generation

of digital signatures (cf. Chapter 17). We could mention many more names of

important contributors to this ﬁeld, who continue to this day to be involved in

often dramatic developments in number theory, and to those interested in a

thrilling account of the history of number theory and its protagonists, I heartily

recommend the book Fermat’s Last Theorem, by Simon Singh.

Considering that already as children we learned counting as something to be

taken for granted and that we were readily convinced of such facts as that two

plus two equals four, we must turn to surprisingly abstract thought constructs

to derive the theoretical justiﬁcation for such assertions. For example, set theory

allows us to derive the existence and arithmetic of the natural numbers from

(almost) nothing. This “almost nothing” is the empty (or null) set

∅ := {},

that is, the set that has no elements. If we consider the empty set to correspond

to the number

0, then we are able to construct additional sets as follows. The

successor

of 0 is associated with the set 0

:= {0 } = {∅ }, which contains

a single element, namely the null set. We give the successor of

0 the name 1,and

for this set as well we can determine a successor, namely

:= {∅, {∅ }}.

The successor of

1, which contains 0 and 1 as its elements, is given the name 2.

The sets thus constructed, which we have rashly given the names

0, 1, and 2,we

identify—not surprisingly—with the well-known natural numbers

0, 1,and2.

This principle of construction, which to every number

x associates a

successor

:= x ∪{x } by adjoining x to the previous set, can be continued to

produce additional numbers. Each number thus constructed, with the exception

0, is itself a set whose elements constitute its predecessors. Only 0 has no

predecessor. To ensure that this process continues ad inﬁnitum, set theory

formulates a special rule, called the axiom of inﬁnity: There exists a set that

contains

0 as well as the successor of every element that it contains.

From this postulated existence of (at least) one so-called successor set, which,

beginning with

0, contains all successors, set theory derives the existence of a

minimal successor set

N, which is itself a subset of every successor set. This

minimal and thus uniquely determined successor set

N is called the set of natural

numbers, in which we expressly include zero as an element.

It was not decisive for this choice that according to standard DIN 5473 zero belongs to the

natural numbers. From the point of view of computer science, however, it is practical to begin

counting at zero instead of 1, which is indicative of the important role played by zero as the

neutral element for addition (additive identity).

APress/Authoring/2005/04/10:10:11 Page 4

Introduction

The natural numbers can be characterized by means of the axioms of

Giuseppe Peano (1858–1932), which coincide with our intuitive understanding of

the natural numbers:

(I) The successors of two unequal natural numbers are unequal: From

n = m

it follows that n

= m

,foralln, m ∈ N.

(II) Every natural number, with the exception of

0, has a predecessor:

= N \{0 }.

(III) The principle of complete induction: If

S ⊂ N, 0 ∈ S,andn ∈ S always

imply

∈ S,thenS = N.

The principle of complete induction makes it possible to derive the arithmetic

operations with natural numbers in which we are interested. The fundamental

operations of addition and multiplication can be deﬁned recursively as follows.

We begin with addition:

For every natural number

n ∈ N there exists a function s

from N to N such

that

(i)

(0) = n,

(ii)





=(s

(x))

for all natural numbers x ∈ N.

The value of the function

(x) is called the sum n + x of n and x.

The existence of such functions

for all natural numbers n ∈ N must,

however, be proved, since the inﬁnitude of natural numbers does not a priori

justify such an assumption. The existence proof goes back to the principle of

complete induction, corresponding to Peano’s third axiom above (see [Halm],

Chapters 11–13). For multiplication one proceeds analogously:

For every natural number

n ∈ N there exists a function p

from N to N such

that

(i)

(0)=0,

(ii)





= p

(x)+n for all natural numbers x ∈ N.

The value of the function

(x) is called the product n · x of n and x.

As expected, multiplication is deﬁned in terms of addition. For the arithmetic

operations thus deﬁned one can prove, through repeated application of complete

induction on

x in accordance with Axiom III, such well-known arithmetic laws as

associativity, commutativity, and distributivity (cf. [Halm], Chapter 13). Although

we usually use these laws without further ado, we shall help ourselves to them as

much as we please in testing our FLINT functions (see Chapters 13 and 18).

In a similar way we obtain a deﬁnition of exponentiation, which we give here

in view of the importance of this operation in what follows.

APress/Authoring/2005/04/10:10:11 Page 5

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