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67-822: The Schönhage–Strassen algorithm was the asymptotically fastest multiplication method known from 1971 until 2007. It is asymptotically faster than older methods such as Karatsuba and Toom–Cook multiplication , and starts to outperform them in practice for numbers beyond about 10,000 to 100,000 decimal digits. In 2007, Martin Fürer published an algorithm with faster asymptotic complexity. In 2019, David Harvey and Joris van der Hoeven demonstrated that multi-digit multiplication has theoretical O ( n log ⁡ n ) {\displaystyle O(n\log n)} complexity; however, their algorithm has constant factors which make it impossibly slow for any conceivable practical problem (see galactic algorithm ). Applications of

134-441: A ( 2 n − 1 ) 2 n − t {\displaystyle g_{t}=a^{(2^{n}-1)2^{n-t}}} . The following formula is helpful, finding a proper K (number of groups to divide N bits into) given bit size N by calculating efficiency : E = 2 N K + k n {\displaystyle E={\frac {{\frac {2N}{K}}+k}{n}}} N

201-455: A i b j = ∑ i = 0 k a i b k − i {\displaystyle c_{k}=\sum _{(i,j):i+j=k}{a_{i}b_{j}}=\sum _{i=0}^{k}{a_{i}b_{k-i}}} , we have a convolution. By using FFT ( fast Fourier transform ), used in the original version rather than NTT ( Number-theoretic transform ), with convolution rule; we get That is; C k =

268-520: A k ∙ b k {\displaystyle C_{k}=a_{k}\bullet b_{k}} , where C k {\displaystyle C_{k}} is the corresponding coefficient in Fourier space. This can also be written as: fft ( a ∗ b ) = fft ( a ) ∙ fft ( b ) {\displaystyle {\text{fft}}(a*b)={\text{fft}}(a)\bullet {\text{fft}}(b)} . We have

335-504: A Fermat number . When doing mod N = 2 M + 1 = 2 2 L + 1 {\displaystyle N=2^{M}+1=2^{2^{L}}+1} , we have a Fermat ring. Because some Fermat numbers are Fermat primes, one can in some cases avoid calculations. There are other N that could have been used, of course, with same prime number advantages. By letting N = 2 k − 1 {\displaystyle N=2^{k}-1} , one have

402-473: A fast Fourier transform . Let C ^ i = A ^ i B ^ i {\displaystyle {\widehat {C}}_{i}={\widehat {A}}_{i}{\widehat {B}}_{i}} (pointwise product), and compute the inverse transform C {\displaystyle C} of the array C ^ {\displaystyle {\widehat {C}}} , again using

469-457: A is an element that generates elements in { 1 , 2 , 4 , . . .2 n − 1 , 2 n } {\displaystyle \{1,2,4,...2^{n-1},2^{n}\}} in a cyclic manner. If N = 2 t {\displaystyle N=2^{t}} , where 1 ≤ t ≤ n {\displaystyle 1\leq t\leq n} , then g t =

536-530: A bit-wise shift instruction is usually (but not always) faster than a multiply instruction and can be used to multiply (shift left) and divide (shift right) by powers of two. Multiplication by a constant and division by a constant can be implemented using a sequence of shifts and adds or subtracts. For example, there are several ways to multiply by 10 using only bit-shift and addition. In some cases such sequences of shifts and adds or subtracts will outperform hardware multipliers and especially dividers. A division by

603-649: A fast manner. Let x {\displaystyle x} and y {\displaystyle y} be represented as n {\displaystyle n} -digit strings in some base B {\displaystyle B} . For any positive integer m {\displaystyle m} less than n {\displaystyle n} , one can write the two given numbers as where x 0 {\displaystyle x_{0}} and y 0 {\displaystyle y_{0}} are less than B m {\displaystyle B^{m}} . The product

670-438: A guess by Schönhage and Strassen that this would be the optimal bound, although this remains a conjecture today. Integer multiplication algorithms can also be used to multiply polynomials by means of the method of Kronecker substitution . If a positional numeral system is used, a natural way of multiplying numbers is taught in schools as long multiplication , sometimes called grade-school multiplication , sometimes called

737-546: A hardware multiplier. Charles Putney implemented this for the 6502 . A line of research in theoretical computer science is about the number of single-bit arithmetic operations necessary to multiply two n {\displaystyle n} -bit integers. This is known as the computational complexity of multiplication. Usual algorithms done by hand have asymptotic complexity of O ( n 2 ) {\displaystyle O(n^{2})} , but in 1960 Anatoly Karatsuba discovered that better complexity

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804-482: A normal FFT which operates over complex numbers, one would use: However, FFT can also be used as a NTT ( number theoretic transformation ) in Schönhage–Strassen. This means that we have to use θ to generate numbers in a finite field (for example G F ( 2 n + 1 ) {\displaystyle \mathrm {GF} (2^{n}+1)} ). A root of unity under a finite field GF( r ) ,

871-537: A number of the form 2 n {\displaystyle 2^{n}} or 2 n ± 1 {\displaystyle 2^{n}\pm 1} often can be converted to such a short sequence. In addition to the standard long multiplication, there are several other methods used to perform multiplication by hand. Such algorithms may be devised for speed, ease of calculation, or educational value, particularly when computers or multiplication tables are unavailable. The grid method (or box method)

938-434: A relatively simple multiplication-only stage, before these contributions are then totalled to give the final answer in a separate addition stage. The calculation 34 × 13, for example, could be computed using the grid: followed by addition to obtain 442, either in a single sum (see right), or through forming the row-by-row totals (300 + 40) + (90 + 12) = 340 + 102 = 442. This calculation approach (though not necessarily with

1005-405: A table from 1 to 200000 by Joseph Blater in 1888. Quarter square multipliers were used in analog computers to form an analog signal that was the product of two analog input signals. In this application, the sum and difference of two input voltages are formed using operational amplifiers . The square of each of these is approximated using piecewise linear circuits. Finally the difference of

1072-441: A usefully explicit method to introduce the idea of multiple-digit multiplications; and, in an age when most multiplication calculations are done using a calculator or a spreadsheet, it may in practice be the only multiplication algorithm that some students will ever need. Lattice, or sieve, multiplication is algorithmically equivalent to long multiplication. It requires the preparation of a lattice (a grid drawn on paper) which guides

1139-639: A very similar algorithm to long multiplication in base 2, but modern processors have optimized circuitry for fast multiplications using more efficient algorithms, at the price of a more complex hardware realization. In base two, long multiplication is sometimes called "shift and add" , because the algorithm simplifies and just consists of shifting left (multiplying by powers of two) and adding. Most currently available microprocessors implement this or other similar algorithms (such as Booth encoding ) for various integer and floating-point sizes in hardware multipliers or in microcode . On currently available processors,

1206-488: Is a Mersenne number. This formula can be used to generate sets of equations, that can be used in CRT (Chinese remainder theorem): Furthermore; g 2 ( p − 1 ) n − 1 ≡ a 2 n − 1 ( mod M p , n ) {\displaystyle g^{2^{(p-1)n}-1}\equiv a^{2^{n}-1}{\pmod {M_{p,n}}}} , where

1273-699: Is a classical problem in algorithms. Having this in mind, N = 2 M + 1 {\displaystyle N=2^{M}+1} help us to group ( i , j ) {\displaystyle (i,j)} into M 2 k {\displaystyle {\frac {M}{2^{k}}}} groups for each group of subtasks in depth k in a tree with N = 2 M 2 k + 1 {\displaystyle N=2^{\frac {M}{2^{k}}}+1} Notice that N = 2 M + 1 = 2 2 L + 1 {\displaystyle N=2^{M}+1=2^{2^{L}}+1} , for some L. This makes N

1340-468: Is a primitive D {\displaystyle D} th root of unity modulo 2 n ′ + 1 {\displaystyle 2^{n'}+1} . We now take the discrete Fourier transform of the arrays A , B {\displaystyle A,B} in the ring Z / ( 2 n ′ + 1 ) Z {\displaystyle \mathbb {Z} /(2^{n'}+1)\mathbb {Z} } , using

1407-1171: Is an element a such that θ r − 1 ≡ 1 {\displaystyle \theta ^{r-1}\equiv 1} or θ r ≡ θ {\displaystyle \theta ^{r}\equiv \theta } . For example GF( p ) , where p is a prime number , gives { 1 , 2 , … , p − 1 } {\displaystyle \{1,2,\ldots ,p-1\}} . Notice that 2 n ≡ − 1 {\displaystyle 2^{n}\equiv -1} in GF ⁡ ( 2 n + 1 ) {\displaystyle \operatorname {GF} (2^{n}+1)} and 2 ≡ − 1 {\displaystyle {\sqrt {2}}\equiv -1} in GF ⁡ ( 2 n + 2 + 1 ) {\displaystyle \operatorname {GF} (2^{n+2}+1)} . For these candidates, θ N ≡ − 1 {\displaystyle \theta ^{N}\equiv -1} under its finite field, and therefore act

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1474-463: Is an introductory method for multiple-digit multiplication that is often taught to pupils at primary school or elementary school . It has been a standard part of the national primary school mathematics curriculum in England and Wales since the late 1990s. Both factors are broken up ("partitioned") into their hundreds, tens and units parts, and the products of the parts are then calculated explicitly in

1541-735: Is bit size (the one used in 2 N + 1 {\displaystyle 2^{N}+1} ) at outermost level. K gives N K {\displaystyle {\frac {N}{K}}} groups of bits, where K = 2 k {\displaystyle K=2^{k}} . n is found through N, K and k by finding the smallest x , such that 2 N / K + k ≤ n = K 2 x {\displaystyle 2N/K+k\leq n=K2^{x}} If one assume efficiency above 50%, n 2 ≤ 2 N K , K ≤ n {\displaystyle {\frac {n}{2}}\leq {\frac {2N}{K}},K\leq n} and k

1608-537: Is common to use long multiplication with the base set to 2 , where w is the number of bits in a word, for multiplying relatively small numbers. To multiply two numbers with n digits using this method, one needs about n operations. More formally, multiplying two n -digit numbers using long multiplication requires Θ ( n ) single-digit operations (additions and multiplications). When implemented in software, long multiplication algorithms must deal with overflow during additions, which can be expensive. A typical solution

1675-459: Is conjectured to be the best possible algorithm, but lower bounds of Ω ( n log ⁡ n ) {\displaystyle \Omega (n\log n)} are not known. Karatsuba multiplication is an O( n ) ≈ O( n ) divide and conquer algorithm, that uses recursion to merge together sub calculations. By rewriting the formula, one makes it possible to do sub calculations / recursion. By doing recursion, one can solve this in

1742-403: Is evaluated. It is therefore advantageous to select the parameters D , M {\displaystyle D,M} so that this pointwise product can be performed efficiently, either because it is a single machine word or using some optimized algorithm for multiplying integers of a (ideally small) number of words. Selecting the parameters D , M {\displaystyle D,M}

1809-525: Is found in overview through Multiplication algorithm#Computational complexity of multiplication A multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Numerous algorithms are known and there has been much research into the topic. The oldest and simplest method, known since antiquity as long multiplication or grade-school multiplication , consists of multiplying every digit in

1876-451: Is important to strike the right balance between the parameters M , k {\displaystyle M,k} . In any case, this algorithm will provide a way to multiply two positive integers, provided n {\displaystyle n} is chosen so that a b < 2 n + 1 {\displaystyle ab<2^{n}+1} . Let n = D M {\displaystyle n=DM} be

1943-400: Is not necessary to store the digits of a , b {\displaystyle a,b} to arbitrary precision, but rather only up to n ′ + 1 {\displaystyle n'+1} bits, which gives a more efficient machine representation of the arrays A , B {\displaystyle A,B} . Secondly, it is clear that the multiplications in

2010-408: Is that it takes more steps than long multiplication, so it can be unwieldy for large numbers. On paper, write down in one column the numbers you get when you repeatedly halve the multiplier, ignoring the remainder; in a column beside it repeatedly double the multiplicand. Cross out each row in which the last digit of the first number is even, and add the remaining numbers in the second column to obtain

2077-403: Is thus an important area for further optimization of the method. Every number in base B, can be written as a polynomial: Furthermore, multiplication of two numbers could be thought of as a product of two polynomials: Because, for B k {\displaystyle B^{k}} : c k = ∑ ( i , j ) : i + j = k

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2144-458: Is to represent the number in a small base, b , such that, for example, 8 b is a representable machine integer. Several additions can then be performed before an overflow occurs. When the number becomes too large, we add part of it to the result, or we carry and map the remaining part back to a number that is less than b . This process is called normalization . Richard Brent used this approach in his Fortran package, MP. Computers initially used

2211-401: Is very small compared to rest of formula; one get This means: When something is very effective; K is bound above by 2 N {\displaystyle 2{\sqrt {N}}} or asymptotically bound above by N {\displaystyle {\sqrt {N}}} Following alogithm, the standard Modular Schönhage-Strassen Multiplication algorithm (with some optimizations),

2278-533: The Chinese remainder theorem , after splitting M into smaller different types of N , one can find the answer of multiplication xy Fermat numbers and Mersenne numbers are just two types of numbers, in something called generalized Fermat Mersenne number (GSM); with formula: In this formula, M 2 , 2 k {\displaystyle M_{2,2^{k}}} is a Fermat number, and M p , 1 {\displaystyle M_{p,1}}

2345-483: The Standard Algorithm : multiply the multiplicand by each digit of the multiplier and then add up all the properly shifted results. It requires memorization of the multiplication table for single digits. This is the usual algorithm for multiplying larger numbers by hand in base 10. A person doing long multiplication on paper will write down all the products and then add them together; an abacus -user will sum

2412-1085: The Toom-3 algorithm. Using many parts can set the exponent arbitrarily close to 1, but the constant factor also grows, making it impractical. In 1968, the Schönhage-Strassen algorithm , which makes use of a Fourier transform over a modulus , was discovered. It has a time complexity of O ( n log ⁡ n log ⁡ log ⁡ n ) {\displaystyle O(n\log n\log \log n)} . In 2007, Martin Fürer proposed an algorithm with complexity O ( n log ⁡ n 2 Θ ( log ∗ ⁡ n ) ) {\displaystyle O(n\log n2^{\Theta (\log ^{*}n)})} . In 2014, Harvey, Joris van der Hoeven , and Lecerf proposed one with complexity O ( n log ⁡ n 2 3 log ∗ ⁡ n ) {\displaystyle O(n\log n2^{3\log ^{*}n})} , thus making

2479-442: The implicit constant explicit; this was improved to O ( n log ⁡ n 2 2 log ∗ ⁡ n ) {\displaystyle O(n\log n2^{2\log ^{*}n})} in 2018. Lastly, in 2019, Harvey and van der Hoeven came up with an galactic algorithm with complexity O ( n log ⁡ n ) {\displaystyle O(n\log n)} . This matches

2546-905: The modular multiplicative inverse . In Schönhage–Strassen algorithm, N = 2 M + 1 {\displaystyle N=2^{M}+1} . This should be thought of as a binary tree, where one have values in 0 ≤ index ≤ 2 M = 2 i + j {\displaystyle 0\leq {\text{index}}\leq 2^{M}=2^{i+j}} . By letting K ∈ [ 0 , M ] {\displaystyle K\in [0,M]} , for each K one can find all i + j = K {\displaystyle i+j=K} , and group all ( i , j ) {\displaystyle (i,j)} pairs into M different groups. Using i + j = k {\displaystyle i+j=k} to group ( i , j ) {\displaystyle (i,j)} pairs through convolution

2613-934: The FFT of the polynomial interpolation of each C k {\displaystyle C_{k}} , one can determine the desired coefficients. This algorithm uses the divide-and-conquer method to divide the problem into subproblems. By letting: where θ N = − 1 {\displaystyle \theta ^{N}=-1} is the n root, one sees that: This mean, one can use weight θ i {\displaystyle \theta ^{i}} , and then multiply with θ − k {\displaystyle \theta ^{-k}} after. Instead of using weight, as θ N = − 1 {\displaystyle \theta ^{N}=-1} , in first step of recursion (when n = N {\displaystyle n=N} ), one can calculate: In

2680-535: The Ottoman Empire. Napier's bones , or Napier's rods also used this method, as published by Napier in 1617, the year of his death. As shown in the example, the multiplicand and multiplier are written above and to the right of a lattice, or a sieve. It is found in Muhammad ibn Musa al-Khwarizmi 's "Arithmetic", one of Leonardo's sources mentioned by Sigler, author of "Fibonacci's Liber Abaci", 2002. The pictures on

2747-500: The Schönhage–Strassen algorithm include large computations done for their own sake such as the Great Internet Mersenne Prime Search and approximations of π , as well as practical applications such as Lenstra elliptic curve factorization via Kronecker substitution , which reduces polynomial multiplication to integer multiplication. This section has a simplified version of the algorithm, showing how to compute

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2814-512: The calculation and separates all the multiplications from the additions . It was introduced to Europe in 1202 in Fibonacci 's Liber Abaci . Fibonacci described the operation as mental, using his right and left hands to carry the intermediate calculations. Matrakçı Nasuh presented 6 different variants of this method in this 16th-century book, Umdet-ul Hisab. It was widely used in Enderun schools across

2881-420: The difference of which is 27, which is the product of 9 and 3. In prehistoric time, quarter square multiplication involved floor function ; that some sources attribute to Babylonian mathematics (2000–1600 BC). Antoine Voisin published a table of quarter squares from 1 to 1000 in 1817 as an aid in multiplication. A larger table of quarter squares from 1 to 100000 was published by Samuel Laundy in 1856, and

2948-455: The elements of the arrays A , B {\displaystyle A,B} as (arbitrary precision) integers modulo 2 n ′ + 1 {\displaystyle 2^{n'}+1} . Observe that since 2 n ′ + 1 ≥ 2 2 M + k + 1 = D 2 2 M + 1 {\displaystyle 2^{n'}+1\geq 2^{2M+k}+1=D2^{2M}+1} ,

3015-405: The explicit grid arrangement) is also known as the partial products algorithm . Its essence is the calculation of the simple multiplications separately, with all addition being left to the final gathering-up stage. The grid method can in principle be applied to factors of any size, although the number of sub-products becomes cumbersome as the number of digits increases. Nevertheless, it is seen as

3082-410: The first number by every digit in the second and adding the results. This has a time complexity of O ( n 2 ) {\displaystyle O(n^{2})} , where n is the number of digits. When done by hand, this may also be reframed as grid method multiplication or lattice multiplication . In software, this may be called "shift and add" due to bitshifts and addition being

3149-485: The forward transforms are simple bit shifts. With some care, it is also possible to compute the inverse transform using only shifts. Taking care, it is thus possible to eliminate any true multiplications from the algorithm except for where the pointwise product C ^ i = A ^ i B ^ i {\displaystyle {\widehat {C}}_{i}={\widehat {A}}_{i}{\widehat {B}}_{i}}

3216-410: The integral part of squares divided by 4 like in the following example. Below is a lookup table of quarter squares with the remainder discarded for the digits 0 through 18; this allows for the multiplication of numbers up to 9×9 . If, for example, you wanted to multiply 9 by 3, you observe that the sum and difference are 12 and 6 respectively. Looking both those values up on the table yields 36 and 9,

3283-413: The inverse: depending whether data needs to be normalized. One multiplies by 2 − m {\displaystyle 2^{-m}} to normalize FFT data into a specific range, where 1 n ≡ 2 − m mod N ( n ) {\displaystyle {\frac {1}{n}}\equiv 2^{-m}{\bmod {N}}(n)} , where m is found using

3350-529: The left and bottom sides. Then those sums are totaled as shown. The binary method is also known as peasant multiplication, because it has been widely used by people who are classified as peasants and thus have not memorized the multiplication tables required for long multiplication. The algorithm was in use in ancient Egypt. Its main advantages are that it can be taught quickly, requires no memorization, and can be performed using tokens, such as poker chips , if paper and pencil aren't available. The disadvantage

3417-536: The maximal number in a binary number with k + 1 {\displaystyle k+1} bits. N = 2 k − 1 {\displaystyle N=2^{k}-1} is a Mersenne number, that in some cases is a Mersenne prime. It is a natural candidate against Fermat number N = 2 2 L + 1 {\displaystyle N=2^{2^{L}}+1} Doing several mod calculations against different N , can be helpful when it comes to solving integer product. By using

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3484-820: The modulus is large enough to accommodate any carries that can result from multiplying a {\displaystyle a} and b {\displaystyle b} . Thus, the product a b {\displaystyle ab} (modulo 2 n + 1 {\displaystyle 2^{n}+1} ) can be calculated by evaluating the convolution of A , B {\displaystyle A,B} . Also, with g = 2 2 M ′ {\displaystyle g=2^{2M'}} , we have g D / 2 ≡ − 1 ( mod 2 n ′ + 1 ) {\displaystyle g^{D/2}\equiv -1{\pmod {2^{n'}+1}}} , and so g {\displaystyle g}

3551-472: The number of bits in the signals a {\displaystyle a} and b {\displaystyle b} , where D = 2 k {\displaystyle D=2^{k}} is a power of two. Divide the signals a {\displaystyle a} and b {\displaystyle b} into D {\displaystyle D} blocks of M {\displaystyle M} bits each, storing

3618-685: The only two operations needed. In 1960, Anatoly Karatsuba discovered Karatsuba multiplication , unleashing a flood of research into fast multiplication algorithms. This method uses three multiplications rather than four to multiply two two-digit numbers. (A variant of this can also be used to multiply complex numbers quickly.) Done recursively , this has a time complexity of O ( n log 2 ⁡ 3 ) {\displaystyle O(n^{\log _{2}3})} . Splitting numbers into more than two parts results in Toom-Cook multiplication ; for example, using three parts results in

3685-427: The process of above multiplication. It keeps only one row to maintain the sum which finally becomes the result. Note that the '+=' operator is used to denote sum to existing value and store operation (akin to languages such as Java and C) for compactness. Some chips implement long multiplication, in hardware or in microcode , for various integer and floating-point word sizes. In arbitrary-precision arithmetic , it

3752-626: The product a b {\displaystyle ab} of two natural numbers a , b {\displaystyle a,b} , modulo a number of the form 2 n + 1 {\displaystyle 2^{n}+1} , where n = 2 k M {\displaystyle n=2^{k}M} is some fixed number. The integers a , b {\displaystyle a,b} are to be divided into D = 2 k {\displaystyle D=2^{k}} blocks of M {\displaystyle M} bits, so in practical implementations, it

3819-763: The product. This example uses peasant multiplication to multiply 11 by 3 to arrive at a result of 33. Describing the steps explicitly: The method works because multiplication is distributive , so: A more complicated example, using the figures from the earlier examples (23,958,233 and 5,830): This formula can in some cases be used, to make multiplication tasks easier to complete: In the case where x {\displaystyle x} and y {\displaystyle y} are integers, we have that because x + y {\displaystyle x+y} and x − y {\displaystyle x-y} are either both even or both odd. This means that and it's sufficient to (pre-)compute

3886-432: The products as soon as each one is computed. This example uses long multiplication to multiply 23,958,233 (multiplicand) by 5,830 (multiplier) and arrives at 139,676,498,390 for the result (product). In some countries such as Germany , the above multiplication is depicted similarly but with the original product kept horizontal and computation starting with the first digit of the multiplier: Below pseudocode describes

3953-548: The resulting blocks as arrays A , B {\displaystyle A,B} (whose entries we shall consider for simplicity as arbitrary precision integers). We now select a modulus for the Fourier transform, as follows. Let M ′ {\displaystyle M'} be such that D M ′ ≥ 2 M + k {\displaystyle DM'\geq 2M+k} . Also put n ′ = D M ′ {\displaystyle n'=DM'} , and regard

4020-409: The right show how to calculate 345 × 12 using lattice multiplication. As a more complicated example, consider the picture below displaying the computation of 23,958,233 multiplied by 5,830 (multiplier); the result is 139,676,498,390. Notice 23,958,233 is along the top of the lattice and 5,830 is along the right side. The products fill the lattice and the sum of those products (on the diagonal) are along

4087-555: The right. For 8-bit integers the table of quarter squares will have 2 −1=511 entries (one entry for the full range 0..510 of possible sums, the differences using only the first 256 entries in range 0..255) or 2 −1=511 entries (using for negative differences the technique of 2-complements and 9-bit masking, which avoids testing the sign of differences), each entry being 16-bit wide (the entry values are from (0²/4)=0 to (510²/4)=65025). The quarter square multiplier technique has benefited 8-bit systems that do not have any support for

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4154-401: The root of unity g {\displaystyle g} for the Fourier basis, giving the transformed arrays A ^ , B ^ {\displaystyle {\widehat {A}},{\widehat {B}}} . Because D = 2 k {\displaystyle D=2^{k}} is a power of two, this can be achieved in logarithmic time using

4221-668: The root of unity g {\displaystyle g} . The array C {\displaystyle C} is now the convolution of the arrays A , B {\displaystyle A,B} . Finally, the product a b ( mod 2 n + 1 ) {\displaystyle ab{\pmod {2^{n}+1}}} is given by evaluating a b ≡ ∑ j C j 2 M j mod 2 n + 1. {\displaystyle ab\equiv \sum _{j}C_{j}2^{Mj}\mod {2^{n}+1}.} This basic algorithm can be improved in several ways. Firstly, it

4288-528: The same coefficients due to linearity under the Fourier transform, and because these polynomials only consist of one unique term per coefficient: Convolution rule: f ^ ( X ∗ Y ) =   f ^ ( X ) ∙ f ^ ( Y ) {\displaystyle {\hat {f}}(X*Y)=\ {\hat {f}}(X)\bullet {\hat {f}}(Y)} We have reduced our convolution problem to product problem, through FFT. By finding

4355-436: The two squares is formed and scaled by a factor of one fourth using yet another operational amplifier. In 1980, Everett L. Johnson proposed using the quarter square method in a digital multiplier. To form the product of two 8-bit integers, for example, the digital device forms the sum and difference, looks both quantities up in a table of squares, takes the difference of the results, and divides by four by shifting two bits to

4422-539: The way we want . Same FFT algorithms can still be used, though, as long as θ is a root of unity of a finite field. To find FFT/NTT transform, we do the following: First product gives contribution to c k {\displaystyle c_{k}} , for each k . Second gives contribution to c k {\displaystyle c_{k}} , due to ( i + j ) {\displaystyle (i+j)} mod N ( n ) {\displaystyle N(n)} . To do

4489-550: Was possible (with the Karatsuba algorithm ). Currently, the algorithm with the best computational complexity is a 2019 algorithm of David Harvey and Joris van der Hoeven , which uses the strategies of using number-theoretic transforms introduced with the Schönhage–Strassen algorithm to multiply integers using only O ( n log ⁡ n ) {\displaystyle O(n\log n)} operations. This

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