构建加密大数运算：实现安全算法背后的多精度算法库

BigNum

Cryptographic

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"《大整数数学：加密多精度算术实现》是一本专门探讨在现代计算机安全算法中扮演关键角色的书籍。作者们，Tom St Denis和Greg Rose，通过LibTomProjects这一开源项目，旨在向读者传授如何有效地设计和实现大整数（BigNum）算法，这些算法在处理超出标准编程语言支持的极大数值时至关重要，以抵抗密码分析攻击。书中深入浅出地介绍了大整数数学的概念，从基础理论到实际的编程实践，涵盖了构建完整功能库的全过程。通过理论指导、伪代码示例以及实战的C源代码，读者能够掌握一系列现代大整数库中所使用的算法。这本书不仅适合正在学习数学和密码学的学生，作为学习工具，也适合需要查阅其中算法的专业人员，无论他们是初学者还是实践经验丰富的从业者。内容涵盖的领域包括但不限于：大整数运算的高效处理技术，如乘法、除法、模运算等，以及在加密算法（如RSA、椭圆曲线加密）中所需的大数运算。作者强调，由于现代编程语言对大整数的支持有限，理解并实现这些算法对于确保网络安全至关重要。书中提供的不仅仅是理论知识，更是一种实践技能，鼓励读者在遇到大规模数据处理时，运用优雅且偶尔显得奇特的算法策略。此外，作者还提醒读者，在使用电脑、网络和数据时，始终要保持谨慎，并采取备份和其他适当预防措施，以防意外损失。值得注意的是，尽管Syngress Publishing和作者们提供了这本书，但不提供任何形式的保证或保修，用户购买时应自行承担风险。并且，根据州法律的不同，关于责任限制的条款可能适用，可能不适用于某些类型的损害赔偿。《大整数数学：加密多精度算术实现》是一本实用的资源，无论你是学术研究者还是行业从业人员，都将从中受益匪浅，提升在大整数计算和加密领域的专业知识和技能。"

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Preface

The origins of this book ar e part of an interesting period of my life. A period that

saw me move from a shy and disor ganized young adult, into a software developer

who has toured various parts o f the world, and met countless new friends and

colleagues. It all began in December of 20 01, nearly ﬁve years ago. I started a

project that would later become known as LibTomCrypt, and be used by devel-

opers throughout industry worldwide.

The LibTomCrypt project was originally started as a way to focus my energies

on to something constructive, while also learning new skills. The ﬁrst year of the

project taught me quite a bit about how to organize a product, document and

support it and maintain it over time. Around the winter of 2002 I was seeking

another project to spread my time with. Realizing that the math performance of

LibTomCrypt was lacking, I set out to develop a new math library.

Hence, the LibTomMath project was bor n. It was originally merely a set of

patches against an existing project tha t quickly grew into a project of its own.

Writing the ma th library from scratch was fundamental to producing a stable and

independent product. It als o taught me what sort of algorithms are available to

do operations such as modular exponentiation. The library became fairly stable

and reliable after only a couple of months of development and was immediately

put to use.

In the summer of 2003, I was yet again looking for ano ther pr oject to g row

into. Realizing that merely implementing the math routines is not enough to

truly understand them, I set out to try and explain them myself. In doing so, I

eventually mastered the concepts behind the algo rithms. This knowledge is what

I hope will be passed on to the reader. This text is actually derived from the

public domain ar chives I maintain on my www.libtomcrypt.com Web site.

When I tell people about my LibTom projects (of which there are six) and

that I release them as public domain, they a re often puzzled. They ask w hy I

xvi www.syngress.com

did it, and especially why I continue to work on them fo r free. The best I can

explain it is, “Because I can”–which seems odd and pe rhaps too terse for a dult

conversation. I often qualify it with “I am able, I am willing,” which perhaps

explains it better. I am the ﬁrst to admit there is nothing that special with what

I have done. Perhaps others can see tha t, too, and then we would have a society

to be proud of. My LibTom projects are what I am doing to give back to society

in the form of tools and k nowledge that can help others in their endeavors.

I sta rted writing this book because it was the most logical task to further my

goal of open academia. The LibTomMa th source code itself was wr itten to be easy

to follow and learn from. There are times, however, where pure C source code

does not explain the algorithms properly– hence this book. The book literally

starts with the foundation of the library and works itself outward to the more

complicated algorithms. The use of both pseudo–code and verbatim source code

provides a duality of “theory” and “practice” the computer science students of

the world shall appreciate. I never deviate too far from relatively stra ightforward

algebra, and I hope this book ca n be a valuable learning asset.

This book, and indeed much of the LibTom projects, would not exist in its

current form if it were not for a plethora of kind people donating their time,

resources , and kind words to help supp ort my work. Writing a text of signiﬁcant

length (along with the source code) is a tiresome and lengthy proc e ss. Cur rently,

the LibTom project is ﬁve years old, composed of literally thousands of users and

over 100,000 lines of source code, T

X, and other material. People like Mads

Rassmussen and Greg Ro se were there at the beginning to encoura ge me to work

well. It is ama z ing how timely validation from others can boost morale to c ontinue

the project. Deﬁnitely, my parents were there for me by providing room and board

during the many months of work in 2003.

Both Gr eg and Mads were invaluable s ources o f support in the early stages

of this project. The initial draft of this text, released in August 2003, was the

project of several months of dedicated work. Long hours and still going to school

were a constant drain of energy that would not have lasted without support.

Of course this book would not be here if it were not for the success of the var-

ious LibTom projects. That success is not only the product of my hard work, but

also the contribution of hundreds of other people. People like Colin Percival, Sky

Schultz, Wayne Scott, J Harper, Dan Kaminsky, Lance James, Simon Johnson,

Greg Rose, Clay Culver, Jochen Ka tz, Zhi Chen, Zed Shaw, Andrew Mann, Matt

Johnston, Steven Dake, Richard Amacker, Stefan Arentz, Richard Outerbridge,

Martin Carpenter, Cra ig Schlenter, John Kuhns, Bruce Guenter, Adam Miller,

Wesley Shields, John Dirk, Jean– Luc Cooke, Michael Heyman, Nelson Bolyard,

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