The Mastery of Computer Programming: Primary Algorithms - Sykalo Eugene 2023

Multiple precision arithmetic
Arithmetic Algorithms

Introduction to multiple precision arithmetic

Multiple precision arithmetic, also known as bignum arithmetic, is a branch of arithmetic that deals with the manipulation of integers with more digits than a computer's native word size. It is used in a variety of computing applications, such as cryptography, numerical analysis, and scientific computing, where precise calculations with large numbers are required.

In multiple precision arithmetic, integers are represented as arrays of digits, with each digit typically represented by a byte or a word. The number of digits used to represent an integer depends on the desired precision of the calculation. For example, if we want to represent a 1000-bit integer, we would need to use an array of 125 bytes, each of which can represent 8 bits.

The importance of multiple precision arithmetic lies in its ability to perform precise calculations with large numbers that are beyond the range of a computer's native word size. For example, in cryptography, large prime numbers are often used for encryption and decryption. These prime numbers can have hundreds or even thousands of digits, which makes them impossible to represent using a computer's native word size.

In the following sections, we will discuss the algorithms used in multiple precision arithmetic, including addition, subtraction, multiplication, and division. We will also cover modular arithmetic, which is used in many cryptographic algorithms. Finally, we will discuss performance analysis and optimization techniques that can be applied to improve the efficiency of multiple precision arithmetic algorithms.

Algorithms for addition and subtraction

In multiple precision arithmetic, addition and subtraction are performed in a similar way to the corresponding operations in elementary school arithmetic. The main difference is that we need to carry out the operations digit by digit, starting from the least significant digit and working our way up to the most significant digit.

To add two integers, we start by adding the least significant digits and then adding any carry from the previous digit. We continue this process for each digit until we reach the most significant digit. If the result has more digits than the original integers, we need to allocate additional digits and propagate the carry to the next digit.

Subtraction is similar to addition, but we need to borrow from the next digit if the current digit is smaller than the corresponding digit in the other integer. We continue this process for each digit until we reach the most significant digit. If the result is negative, we need to adjust the sign and perform a complement operation.

There are several algorithms for addition and subtraction that are commonly used in multiple precision arithmetic. One of the most basic algorithms is the ripple carry adder, which simply adds each pair of digits and propagates any carry to the next digit. This algorithm has a time complexity of O(n), where n is the number of digits in the integers.

A more efficient algorithm for addition is the carry look-ahead adder, which computes the carry bits in parallel and then adds the digits using the carry bits. This algorithm has a time complexity of O(log n).

For subtraction, the most common algorithm is the two's complement method, which involves taking the complement of the subtrahend and then adding it to the minuend. This algorithm has a time complexity of O(n).

Algorithms for multiplication and division

Multiplication and division in multiple precision arithmetic are more complex than addition and subtraction, and there are many different algorithms that can be used depending on the desired precision and the specific application. In this section, we will discuss some of the most common algorithms for multiplication and division.

Karatsuba algorithm

The Karatsuba algorithm is a recursive algorithm for multiplying two integers that has a time complexity of O(n^log2(3)). The algorithm works by splitting the input integers into two smaller integers and recursively computing the product of these smaller integers. The final product is then computed using these intermediate results and some additional arithmetic operations.

The Karatsuba algorithm is faster than the traditional long multiplication algorithm for sufficiently large integers, but it has a higher overhead due to the recursive calls. The algorithm is also more difficult to implement and may require more memory than other multiplication algorithms.

Toom-Cook algorithm

The Toom-Cook algorithm is another recursive algorithm for multiplying two integers that has a time complexity of O(n^log2(5)). Like the Karatsuba algorithm, the Toom-Cook algorithm splits the input integers into smaller integers and recursively computes the product of these smaller integers. The final product is then computed using these intermediate results and some additional arithmetic operations.

The Toom-Cook algorithm is faster than the Karatsuba algorithm for even larger integers, but it has an even higher overhead due to the recursive calls. The algorithm is also more difficult to implement and may require more memory than other multiplication algorithms.

Binary long division

Binary long division is a common algorithm for dividing two integers in multiple precision arithmetic. The algorithm works by dividing the most significant digits of the dividend and divisor and then subtracting the resulting product from the dividend. The process is repeated for each digit, with the quotient and remainder updated at each step.

Binary long division has a time complexity of O(n^2), where n is the number of digits in the integers. The algorithm is slower than multiplication algorithms, but it is essential for many cryptographic algorithms that rely on modular arithmetic.

Modular arithmetic

Modular arithmetic is a type of arithmetic that deals with integers modulo a given modulus. In multiple precision arithmetic, modular arithmetic is used in many cryptographic algorithms, such as RSA and Diffie-Hellman.

Modular addition, subtraction, multiplication, and division are performed in a similar way to their non-modular counterparts, but with the result modulo the given modulus. The Chinese remainder theorem is a useful algorithm for performing modular arithmetic with multiple moduli.

Performance analysis and optimization techniques

As with addition and subtraction, there are several techniques that can be used to optimize the performance of multiplication and division algorithms. These include using faster algorithms for smaller inputs, reducing the number of memory accesses, and using parallel processing techniques.

In practice, the choice of algorithm depends on the desired precision and the specific application. It is often necessary to implement multiple algorithms and choose the best one based on the input size and other factors.

Modular arithmetic

Modular arithmetic is a type of arithmetic that deals with integers modulo a given modulus. In multiple precision arithmetic, modular arithmetic is used in many cryptographic algorithms, such as RSA and Diffie-Hellman.

The basic idea of modular arithmetic is to perform arithmetic operations on the remainders of the integers modulo the given modulus, rather than on the integers themselves. For example, in modulo 5 arithmetic, the remainders of the integers 2 and 7 are the same, since 2 mod 5 = 7 mod 5 = 2. Therefore, we can perform arithmetic operations on the remainders rather than on the original integers.

Modular addition, subtraction, multiplication, and division are performed in a similar way to their non-modular counterparts, but with the result modulo the given modulus. For example, in modulo 5 arithmetic, 3 + 4 = 2, since 3 + 4 = 7, and 7 mod 5 = 2. Similarly, 4 - 3 = 1, since 4 - 3 = 1, and 1 mod 5 = 1.

Modular multiplication is performed by multiplying the remainders of the integers modulo the given modulus and then taking the result modulo the given modulus. For example, in modulo 5 arithmetic, 3 * 4 = 2, since 3 * 4 = 12, and 12 mod 5 = 2.

Modular division is more complex than modular multiplication, and there are several different algorithms that can be used depending on the specific application. One common algorithm is the extended Euclidean algorithm, which computes the greatest common divisor of two integers and expresses it as a linear combination of the integers. This algorithm can be used to compute the modular inverse of an integer, which is necessary for modular division.

The Chinese remainder theorem is a useful algorithm for performing modular arithmetic with multiple moduli. The theorem states that if we know the remainders of an integer modulo several coprime integers, then we can determine the remainder of the integer modulo the product of the integers. This can be useful in cryptography, where modular arithmetic with multiple moduli is often used.

Modular arithmetic is an essential part of many cryptographic algorithms, and efficient implementation is crucial for performance. The choice of modulus depends on the specific application, and it is often necessary to choose a large prime modulus to ensure security.

Performance analysis and optimization techniques

Multiplication and division algorithms in multiple precision arithmetic can be optimized in several ways to improve their performance. One approach is to use faster algorithms for smaller inputs. For example, for smaller numbers, it may be more efficient to use the traditional long multiplication algorithm instead of the Karatsuba algorithm. Another approach is to reduce the number of memory accesses by using optimized data structures and algorithms.

Parallel processing techniques can also be used to improve the performance of multiple precision arithmetic algorithms. For example, the carry look-ahead adder can be parallelized by computing the carry bits for multiple digits in parallel. Similarly, the Toom-Cook algorithm can be parallelized by splitting the input integers into more than two smaller integers and computing the product of these smaller integers in parallel.

Time and space complexity analysis can also be used to optimize multiple precision arithmetic algorithms. For example, the space complexity of the Karatsuba algorithm is O(n^log2(3)), which means that the algorithm requires more memory than other multiplication algorithms. By analyzing the space complexity of different algorithms, it is possible to choose the most efficient algorithm for a given input size.

Branch prediction and caching techniques can also be used to optimize multiple precision arithmetic algorithms. For example, by using caching techniques, it is possible to reduce the number of memory accesses and improve the performance of the algorithm. Similarly, by using branch prediction techniques, it is possible to reduce the number of conditional statements and improve the performance of the algorithm.

In practice, the choice of optimization techniques depends on the specific application and the desired precision. It is often necessary to implement multiple algorithms and choose the best one based on the input size and other factors. By using a combination of optimization techniques, it is possible to achieve significant improvements in the performance of multiple precision arithmetic algorithms.