search
HomeBackend DevelopmentC++How Can I Optimize My Number Theoretic Transform (NTT) and Modular Arithmetic for Faster Big Number Squaring?

How Can I Optimize My Number Theoretic Transform (NTT) and Modular Arithmetic for Faster Big Number Squaring?

Modular arithmetics and NTT (finite field DFT) optimizations

Original question:

I wanted to use NTT for fast squaring (see Fast bignum square computation), but the result is slow even for really big numbers .. more than 12000 bits.

My question is:

  1. Is there a way to optimize my NTT transform?
    I did not mean to speed it by parallelism (threads); this is low-level layer only.
  2. Is there a way to speed up my modular arithmetics?

Code:

//---------------------------------------------------------------------------
class fourier_NTT                                    // Number theoretic transform
    {

public:
    DWORD r,L,p,N;
    DWORD W,iW,rN;
    fourier_NTT(){ r=0; L=0; p=0; W=0; iW=0; rN=0; }

    // main interface
    void  NTT(DWORD *dst,DWORD *src,DWORD n=0);               // DWORD dst[n] = fast  NTT(DWORD src[n])
    void iNTT(DWORD *dst,DWORD *src,DWORD n=0);               // DWORD dst[n] = fast INTT(DWORD src[n])

    // Helper functions
    bool init(DWORD n);                                       // init r,L,p,W,iW,rN
    void  NTT_fast(DWORD *dst,DWORD *src,DWORD n,DWORD w);    // DWORD dst[n] = fast  NTT(DWORD src[n])

    // Only for testing
    void  NTT_slow(DWORD *dst,DWORD *src,DWORD n,DWORD w);    // DWORD dst[n] = slow  NTT(DWORD src[n])
    void iNTT_slow(DWORD *dst,DWORD *src,DWORD n,DWORD w);    // DWORD dst[n] = slow INTT(DWORD src[n])

    // DWORD arithmetics
    DWORD shl(DWORD a);
    DWORD shr(DWORD a);

    // Modular arithmetics
    DWORD mod(DWORD a);
    DWORD modadd(DWORD a,DWORD b);
    DWORD modsub(DWORD a,DWORD b);
    DWORD modmul(DWORD a,DWORD b);
    DWORD modpow(DWORD a,DWORD b);
    };

//---------------------------------------------------------------------------
void fourier_NTT:: NTT(DWORD *dst,DWORD *src,DWORD n)
    {
    if (n>0) init(n);
    NTT_fast(dst,src,N,W);
//    NTT_slow(dst,src,N,W);
    }

//---------------------------------------------------------------------------
void fourier_NTT::INTT(DWORD *dst,DWORD *src,DWORD n)
    {
    if (n>0) init(n);
    NTT_fast(dst,src,N,iW);
    for (DWORD i=0;i<n dst intt_slow bool fourier_ntt::init n p else ntt overflow can ocur r="2;" if>0x10000000)) { r=0; L=0; p=0; W=0; iW=0; rN=0; N=0; return false; } L=0x30000000/n; // 32:30 bit best for unsigned 32 bit
//    r=2; p=0x78000001; if ((n0x04000000)) { r=0; L=0; p=0; W=0; iW=0; rN=0; N=0; return false; } L=0x3c000000/n; // 31:27 bit best for signed 32 bit
//    r=2; p=0x00010001; if ((n0x00000020)) { r=0; L=0; p=0; W=0; iW=0; rN=0; N=0; return false; } L=0x00000020/n; // 17:16 bit best for 16 bit
//    r=2; p=0x0a000001; if ((n0x01000000)) { r=0; L=0; p=0; W=0; iW=0; rN=0; N=0; return false; } L=0x01000000/n; // 28:25 bit
     N=n;                // size of vectors [DWORDs]
     W=modpow(r,    L);    // Wn for NTT
    iW=modpow(r,p-1-L);    // Wn for INTT
    rN=modpow(n,p-2  );    // scale for INTT
    return true;
    }

//---------------------------------------------------------------------------
void fourier_NTT:: NTT_fast(DWORD *dst,DWORD *src,DWORD n,DWORD w)
    {
    if (n>1,w2=modmul(w,w);
    // reorder even,odd
    for (i=0,j=0;i<n2 dst for j="1;i<n" recursion ntt_fast even odd restore results a0="src[i];" a1="modmul(src[j],w2);" void fourier_ntt:: ntt_slow n w dword i a="0;" wi="modmul(wi,wj);" wj="modmul(wj,w);" fourier_ntt::intt_slow fourier_ntt::shl return fourier_ntt::shr>>1)&0x7FFFFFFF; }

//---------------------------------------------------------------------------
DWORD fourier_NTT::mod(DWORD a)
    {
    DWORD bb;</n2></n>

The above is the detailed content of How Can I Optimize My Number Theoretic Transform (NTT) and Modular Arithmetic for Faster Big Number Squaring?. For more information, please follow other related articles on the PHP Chinese website!

Statement
The content of this article is voluntarily contributed by netizens, and the copyright belongs to the original author. This site does not assume corresponding legal responsibility. If you find any content suspected of plagiarism or infringement, please contact admin@php.cn
C# vs. C  : History, Evolution, and Future ProspectsC# vs. C : History, Evolution, and Future ProspectsApr 19, 2025 am 12:07 AM

The history and evolution of C# and C are unique, and the future prospects are also different. 1.C was invented by BjarneStroustrup in 1983 to introduce object-oriented programming into the C language. Its evolution process includes multiple standardizations, such as C 11 introducing auto keywords and lambda expressions, C 20 introducing concepts and coroutines, and will focus on performance and system-level programming in the future. 2.C# was released by Microsoft in 2000. Combining the advantages of C and Java, its evolution focuses on simplicity and productivity. For example, C#2.0 introduced generics and C#5.0 introduced asynchronous programming, which will focus on developers' productivity and cloud computing in the future.

C# vs. C  : Learning Curves and Developer ExperienceC# vs. C : Learning Curves and Developer ExperienceApr 18, 2025 am 12:13 AM

There are significant differences in the learning curves of C# and C and developer experience. 1) The learning curve of C# is relatively flat and is suitable for rapid development and enterprise-level applications. 2) The learning curve of C is steep and is suitable for high-performance and low-level control scenarios.

C# vs. C  : Object-Oriented Programming and FeaturesC# vs. C : Object-Oriented Programming and FeaturesApr 17, 2025 am 12:02 AM

There are significant differences in how C# and C implement and features in object-oriented programming (OOP). 1) The class definition and syntax of C# are more concise and support advanced features such as LINQ. 2) C provides finer granular control, suitable for system programming and high performance needs. Both have their own advantages, and the choice should be based on the specific application scenario.

From XML to C  : Data Transformation and ManipulationFrom XML to C : Data Transformation and ManipulationApr 16, 2025 am 12:08 AM

Converting from XML to C and performing data operations can be achieved through the following steps: 1) parsing XML files using tinyxml2 library, 2) mapping data into C's data structure, 3) using C standard library such as std::vector for data operations. Through these steps, data converted from XML can be processed and manipulated efficiently.

C# vs. C  : Memory Management and Garbage CollectionC# vs. C : Memory Management and Garbage CollectionApr 15, 2025 am 12:16 AM

C# uses automatic garbage collection mechanism, while C uses manual memory management. 1. C#'s garbage collector automatically manages memory to reduce the risk of memory leakage, but may lead to performance degradation. 2.C provides flexible memory control, suitable for applications that require fine management, but should be handled with caution to avoid memory leakage.

Beyond the Hype: Assessing the Relevance of C   TodayBeyond the Hype: Assessing the Relevance of C TodayApr 14, 2025 am 12:01 AM

C still has important relevance in modern programming. 1) High performance and direct hardware operation capabilities make it the first choice in the fields of game development, embedded systems and high-performance computing. 2) Rich programming paradigms and modern features such as smart pointers and template programming enhance its flexibility and efficiency. Although the learning curve is steep, its powerful capabilities make it still important in today's programming ecosystem.

The C   Community: Resources, Support, and DevelopmentThe C Community: Resources, Support, and DevelopmentApr 13, 2025 am 12:01 AM

C Learners and developers can get resources and support from StackOverflow, Reddit's r/cpp community, Coursera and edX courses, open source projects on GitHub, professional consulting services, and CppCon. 1. StackOverflow provides answers to technical questions; 2. Reddit's r/cpp community shares the latest news; 3. Coursera and edX provide formal C courses; 4. Open source projects on GitHub such as LLVM and Boost improve skills; 5. Professional consulting services such as JetBrains and Perforce provide technical support; 6. CppCon and other conferences help careers

C# vs. C  : Where Each Language ExcelsC# vs. C : Where Each Language ExcelsApr 12, 2025 am 12:08 AM

C# is suitable for projects that require high development efficiency and cross-platform support, while C is suitable for applications that require high performance and underlying control. 1) C# simplifies development, provides garbage collection and rich class libraries, suitable for enterprise-level applications. 2)C allows direct memory operation, suitable for game development and high-performance computing.

See all articles

Hot AI Tools

Undresser.AI Undress

Undresser.AI Undress

AI-powered app for creating realistic nude photos

AI Clothes Remover

AI Clothes Remover

Online AI tool for removing clothes from photos.

Undress AI Tool

Undress AI Tool

Undress images for free

Clothoff.io

Clothoff.io

AI clothes remover

Video Face Swap

Video Face Swap

Swap faces in any video effortlessly with our completely free AI face swap tool!

Hot Tools

SublimeText3 Chinese version

SublimeText3 Chinese version

Chinese version, very easy to use

DVWA

DVWA

Damn Vulnerable Web App (DVWA) is a PHP/MySQL web application that is very vulnerable. Its main goals are to be an aid for security professionals to test their skills and tools in a legal environment, to help web developers better understand the process of securing web applications, and to help teachers/students teach/learn in a classroom environment Web application security. The goal of DVWA is to practice some of the most common web vulnerabilities through a simple and straightforward interface, with varying degrees of difficulty. Please note that this software

Dreamweaver Mac version

Dreamweaver Mac version

Visual web development tools

EditPlus Chinese cracked version

EditPlus Chinese cracked version

Small size, syntax highlighting, does not support code prompt function

Atom editor mac version download

Atom editor mac version download

The most popular open source editor