Miracl图书馆用户手册：多精度计算指南

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"Miracl 用户帮助文档是 Shamus Software Ltd. 出版的一份详细指南，涵盖了 Miracl 库版本 5.0 的安装、优化、升级以及多线程等内容。Miracl 库提供了超过100个用于处理多精度算术的函数，包括大整数(big)和浮点分数(flash)运算。大整数算法基于 Knuth 的经典著作《计算机程序设计艺术》中的算法，而浮点分数运算则是 Matula 和 Kornerup 提出的用以处理近似分数的方法。所有的函数都经过了深度优化，以保证速度和效率，并且编写成标准、可移植的 C 语言。此外，还提供了针对英特尔 80x86 系列处理器的快速汇编语言替代方案。文档还包括 C++ 接口的使用说明，并附带完整源代码。" Miracl 是一个强大的数学库，专注于高精度计算，尤其适合在需要处理大整数和浮点分数运算的应用场景中。以下是对该库关键知识点的详细说明： 1. **大整数算法**：Miracl 使用 Knuth 的算法实现大整数运算，这些算法确保了在处理大整数时的准确性和效率。大整数算法可以进行加法、减法、乘法、除法等基本运算，同时也支持更复杂的数学操作，如模幂、因数分解等。 2. **浮点分数运算**：Miracl 引入了 flash 数据类型来处理浮点分数，这是一种近似分数的表示方式，允许在保持高精度的同时，执行高效的浮点运算。这种数据结构特别适用于需要高精度浮点计算的科学或工程应用。 3. **优化与可移植性**：所有函数都是用标准 C 语言编写，保证了代码在不同平台上的可移植性。同时，对于关键性能敏感的函数，还提供了汇编语言实现，以提升特定处理器架构下的运行速度，特别是针对 Intel 80x86 系列处理器。 4. **C++ 接口**：除了基础的 C 语言接口，Miracl 还提供了一个 C++ 接口，使得用户可以在 C++ 项目中更方便地集成和使用 Miracl 库，利用面向对象编程的特性提高代码的组织和复用性。 5. **安装与升级**：文档中详细介绍了如何安装 Miracl 库，包括如何进行优化设置以提升性能，以及从旧版本（例如3.x）升级到新版本（5.0）的步骤。对于多线程应用，文档也提供了相关的指导，以充分利用多核处理器的优势。 6. **源代码**：Miracl 提供完整的源代码，这意味着用户可以深入理解其内部工作原理，根据需要自定义和扩展功能，或者调试和修复可能遇到的问题。 Miracl 库的这些特性使其成为需要高精度计算的软件开发者的强大工具，尤其在密码学、金融计算、物理模拟等领域有广泛应用。通过深入学习和使用这份用户手册，开发者可以有效地利用 Miracl 实现复杂计算任务，同时确保代码的高效性和跨平台兼容性。

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appropriate time before new threads are forked off from the main thread. They must be

called before any thread calls mirsys either explicitly, or implicitly by creating a thread-

specific instance of the class Miracl.

It is strongly recommended that program development be carried out without support

for threads. Only when a program is fully tested and debugged should it be converted

into a thread.

Threaded programming may require other OS-specific measures, in terms of linking to

special libraries, or access to special heap routines. In this regard it is worth pointing

out that all MIRACL heap accesses are via the module mralloc.c.

See the example program threadwn.cpp for an example of Windows C++

multithreading. Read the comments in this program – it can be compiled and run from

a Windows Command prompt. Similarly see threadux.cpp for an example of Unix

multi-threading.

2.4 Constrained environments

In version 5 of MIRACL there is new support for implementations in very small and

constrained environments. Using the config utility it is now possible to allow various

time/space trade-offs, but the main innovation is the possibility of building and using

MIRACL in an environment which does not support a heap. Normally space for big

variables is obtained from the heap, but by specifying in the configuration header

MR_STATIC, a version of the library is built which will always attempt to allocate

space not from the heap, but from static memory or from the stack.

The main downside to this is that the maximum size of big variables must be set at

compile time, when the library is being created. As always it is best to let the config

utility guide you through the process of creating a suitable mirdef.h configuration

header.

For the C programmer, the allocation of memory from the stack for big variables

proceeds as follows.

big x,y,z;

char mem[MR_BIG_RESERVE(3)];

memset(mem,0, MR_BIG_RESERVE(3));

This allocates space for 3 big variables on the stack, and set that memory to zero. Each

individual big variable is then initialised as

x=mirvar_mem(mem,0);

y=mirvar_mem(mem,1);

z=mirvar_mem(mem,2);

Allocating all the space for multiple big variables from a single chunk of memory

makes sense, as it leads to a faster initialization, and also gives complete control over

3-12

The initial call to mirsys also initialises the error tracing system which is integrated

with the MIRACL package. Whenever an error is detected the sequence of routine

calls down to the routine which generated the error is reported, as well as the error

itself. A typical error message might be

MIRACL error from routine powltr

called from isprime

called from your program

Raising integer to a negative power

Such an error report facilitates debugging, and assisted us during the development of

these routines. An associated instance variable TRACER, initialised to OFF, if set by

the user to ON, will cause a trace of the program's progress through the MIRACL

routines to be output to the computer screen.

An instance flag ERNUM, initialised to zero, records the number of the last internal

MIRACL error to have occurred. If the flag ERCON is set to FALSE (the default), an

error message is directed to stdout and the program aborts via a call to the system

routine exit(0). If your system does not supply such a routine, the programmer must

provide one instead. If ERCON is set to TRUE no error message is emitted and

instead the onus is on the programmer to detect and handle the error. In this case

execution continues. The programmer may choose to deal with the error, and reset

ERNUM to zero. However errors are usually fatal, and if ERNUM is non-zero all

MIRACL routines called subsequently will “fall-through” and exit immediately. See

miracl.h for a list of all possible errors.

Every big or flash variable in the users program must be initialised by a call to the

routine mirvar, which also allows the variable to be given an initial small integer value.

The full set of arithmetic and number-theoretic routines declared in miracl.h may be

used on these variables. Full flexibility is (almost always) allowed in parameter usage

with these routines. For example the call multiply(x,y,z), multiplies the big variable x

by the big variable y to give the result as big variable z. Equally valid would be

multiply(x,y,x), multiply(y,y,x), or multiply(x,x,x). This last simply squares x. Note

that the first parameters are by convention always (usually) the inputs to the routines.

Routines are provided not only to allow arithmetic on big and flash numbers, but also

to allow these variables to perform arithmetic with the built-in integer and double

precision data-types.

Conversion routines are provided to convert from one type to another. For details of

each routine see the relevant documentation in Chapter 9 and the example programs of

Chapter 8.

Input and output to a file or I/O device is handled by the routines innum, otnum

cinnum and cotnum. The first two use the fixed number base specified by the user in

the initial call of mirsys. The latter pair work in conjunction with the instance variable

IOBASE which can be assigned dynamically by the user. A simple rule is that if the

program is CPU bound, or involves changes of base, then set the base initially to

MAXBASE (or 0 if a full-width base is possible - see Chapter 4) and use cinnum and

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