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16

While the algorithm you mention appears in Knuth's TAOCP, it is certainly not due to Knuth, and is more widely known as the Schönhage–Strassen algorithm; Knuth even attributes this algorithm to them in the text. This algorithm indeed runs in linear time in the so-called RAM machine, in which variables are allowed to hold integers of size $O(\log n)$, where $... 12 Modular exponentiation is a well-known algorithm. It is routinely available in libraries and languages that can manipulate large integers, including Wolfram Alpha. When making computations modulo a large number, one does not first make the whole computation in$\mathbb{N}$and then take the remainder of the result, because for something like an ... 6 The Galois field$GF(2^{128})$has many different "concrete" representations. One popular representation is using polynomials in$GF(2)[x]$(i.e. with coefficients in$GF(2)$) modulo some irreducible polynomial of degree$128$, say$x^{128}+x^7+x^2+x^1+x^0$. The magic constants$7,2,1,0$come from this particular irreducible polynomial. You don't see$0$... 6 In multiplier design an (n,k) counter takes an n bit input and produces a k bit output which is the binary representation of the number of input bits that are 1s. That is: it counts the number of input bits set to 1. A (3,2) counter is just a full adder. A (7,3) counter is a circuit with 7 input bits and 3 output bits. The 3 output bits tell you (in ... 6 Strassen, in his paper describing Strassen's algorithm (Gaussian elimination is not optimal) mentions the result of Klyuyev and Kokovkin-Shcherbak [1] that Gaussian elimination for solving a system of linear equations is optimal if one restricts oneself to operations upon rows and columns as a whole. The reference is to Klyuyev, V. V., and N. I. ... 6 I don't see the relevance. New multiplication algorithms only offer a practical improvement for numbers with huge numbers of bits and probably don't beat existing algorithms until you're dealing with numbers with thousands or even millions of bits. For an extreme example, Fürer's algorithm, which runs in time$O(n\log n\, 2^{O(\log^*n)})$will only ... 5 Here,$B = 10$and$n = 3$. The Karatsuba method works for any$m < n$. With$m = 2$:$a = 2 * 10^2 + 34$and$b = 8 * 10^2 + 57$; thus,$a_1 = 2$,$a_0 = 34$,$b_1 = 8$and$b_0 = 57$. It follows that$ab = z_2 B^{2m} + z_1 B^m + z_0$where$z_2 = a_1 b_1 = 16$,$z_1 = a_1 b_0 + a_0 b_1 = 386$and$z_0 = a_0 b_0 = 1938$. Thus,$ab = 16 \cdot 10^{2*2}...

5

According to the Wikipedia article, at each step the length of the integers is reduced from $N$ to (roughly) $\sqrt{N}$, and there are (roughly) $\sqrt{N}$ of them, and so the additions only cost $O(N)$. There is a detailed analysis of the running time in the final paragraph of the linked section, copied here in case it changes: In the recursive step, the ...

5

The sticking point is the size of the integers in the recursive step, which is not quite $\sqrt{n}$, but rather twice as large, since the product of two $t$-bit integers has size $2t$. Assuming that we break the original $n$-bit integer into $\sqrt{n}$ parts of length $\sqrt{n}$, the running time recursion is $$T(n) = n\log n + \sqrt{n}T(2\sqrt{n}),$$ and ...

5

Your operation is multiplication of polynomials over $GF(2)$, i.e., multiplication in the polynomial ring $GF(2)[x]$. For instance, if $p=101$ and $q=1101$, you can represent them as $p(x)=x^2+1$, $q(x)=x^3+x^2+1$, and their product as polynomials is $p(x) \times q(x) = x^5+x^4+x^3+1$, so $p \otimes q = 111001$. If $p,q$ are $r$ bits long, this polynomial ...

5

If you want to use lookup tables, and you have 4GB of memory, you'll only be able to use a lookup table with about $2^{32}$ entries or fewer, so you'll only be able to handle multiplication of numbers that are at most 16 bits long. If you want to multiply larger numbers, you won't be able to use lookup tables for multiplication. Typically we want to ...

4

The asymptotic complexity will not change with relation to how the matrices are laid out in memory, but the actual running time of the matrix multiplication will be very dependent upon the memory layout. There are two papers that I know of that go into detail about this, one by McKellar in 1969 and another by Prokop in 1999. Prokop's paper defines the ...

4

No, the complexity remains the same. The main difference between row-major and column-major order is memory access patterns - for example if you iterate by column then row-major order would be jumping around memory, which is bad for CPUs because they cannot read-ahead/cache memory. This is because transferring data from RAM to the CPU on modern processors ...

4

It's not faster (asymptotically). You can reduce general matrix multiplication down to three "adjoint-squarings". Suppose we're given an adjoint-squaring function $\mathfrak{F}$ where $\mathfrak{F}(M) = M \cdot M^\dagger$. Consider that: $\mathfrak{F}(A + B) = (A+B) \cdot (A+B)^\dagger = AA^\dagger + BB^\dagger + A B^\dagger + B A^\dagger$ $\mathfrak{F}(A ... 4 The paper All pairs shortest paths using bridging sets and rectangular matrix multiplication by Uri Zwick shows that the APSP problem can be solved in subcubic time, given a bound on the edge-weights. The APSP problem uses a$(\min,+)$-matrix product. Since$\max_{A} \sum_{a\in A} a = -\min_{A} -\sum_{a\in A}a = -\min_A \sum_{a\in A}-a$, we can express a$(\...

4

It's hard to tell what anyone means by "comparable" in a specific context. I would take it to mean "of the same order", so something like $n^2$, maybe $100\cdot n^2$ or $0.0001\cdot n^2$, but not $n$ or $n^3$. The final addition is doing (roughly) $n$ additions of numbers of (roughly) non-$\_$ $n$ digits, and each of those additions takes (roughly) $n$ ...

4

Remainder modulo 10 instead of last digit What is the last digit of -206? It is 6 by convention. It is not 4, the least positive remainder of -206 divided by 10. For simplicity, we will compute the value of polynomial modulo 10. We will just say "the remainder of some number" to mean the least positive remainder of that number divided by 10. (The idea and ...

4

It should be "10 superscript{n/2}" meaning $10^{n/2}$. For example, if $n=4$, then "6#2" would represent $6\cdot10^2=600$. All he's doing is shifting a decimal number to the left.

4

From context it's just exponentiation. I don't know why it's written with a subscript instead of a superscript, but with older papers you sometimes had a professional typesetter who might not have understood the material. (Of course nowadays typically do their own typesetting.)

4

No, it's not helping. You can do n bit multiplication in O (log n) time, not O (n log n). The only problem is that you need an enormous amount of hardware to do this, growing as O (n^2), while this paper describes multiplication with a fixed amount of hardware. But since a CPU typically just has only a 64 bit multiply unit, that O (n^2) space is no big ...

3

Let's say we have inputs $x, y$ and $c$, where $c$ is either 0 or 1, 0 = no swap, 1 = swap. We can make a conditional swap function like this: notc = 1 + (-1)*c x' = notc*x + c*y y' = c*x + notc*y Even if you have no negative integers, as long as your integers are modulo $2^n$, you can use $-1 \equiv 2^n - 1 \mod 2^n$ and everything works out.

3

There is nothing special about 2×2 matrices. In fact you can do much better using larger matrices. The reason that you are only being explained the 2×2 algorithm is that it is simple to describe. The special thing about 2×2 matrices are that they are the smallest square matrices which are larger than 1×1. You can read more in this recent survey by Dumas and ...

3

For completeness, I'll flesh out Yuval's answer a bit more through an example of multiplication of two polynomials $A$ and $B$ in $\text{GF}(2^{16})$. Let $$A = [0001|1100|1110] = x^8 + x^7 + x^6 + x^3 + x^2 + x,$$ and $$B = [0100|0101|0111] = x^{10} + x^6 + x^4 + x^2 + x + 1.$$ The multiplication of $A$ and $B$ over $\text{GF}(2)$ is then $$C = [0000|0000|... 3 The issue is that the expansion factor inside the recursive term is not less than 2, and that causes the bound to fail. Define: T_c(n) = n \log n + \sqrt n T(c \sqrt n) I claim that T_{2-\epsilon} is O(n \log n), but T_{2} isn't. This is probably most obvious when trying to prove the n \log n bound by induction. Assume that T_c(n_0) \leq 100 n_0 ... 3 There's an excellent explanation of exactly what's going on, including how the size of the field goes down as you progress, in GMP's documentation: 15.1.6 FFT Multiplication. A more theoretically in-depth overview starts on page 55 of Modern Computer Arithmetic (pdf) (well... the browser says page 71 but the page itself says 55). It explains the algorithm's ... 3 Your question hits at the main part of the faster algorithm presented in the paper. Many people know the usual (base-2) "repeated squaring" algorithm to compute g^n is to write n in binary:$$ n=\sum_{i=0}^m e_i2^i\qquad e_i\in\{0,1\} $$so$$ g^n=g^{e_02^0+e_12^1+e_22^2+\dotsm}=(g^{2^0})^{e_0}(g^{2^1})^{e_1}(g^{2^2})^{e_2}\dotsm  and observe that $g^{... 3 In the reference linked, in order to get product of these two polynomials,$c(x) = p(x)q(x)$, via brute force, we have to compute new coefficients via convolution$\left(c_k = \sum_{i=0}^k a_i b_{k-i} \right)$which takes$\mathcal{\Theta}(n^2)$time. That's not what it says. It says firstly that convolution is equivalent to polynomial multiplication (i.e. ... 3 Given two polynomials$P_n(x)$,$Q_n(x)$, you can obviously calculate the value$P_n(x) · Q_n(x)$in$O(n)$for every x. The document asks about calculating the coefficients, which is a very different question. If all you want is to evaluate$P_n(x) · Q_n(x)\$ for various values of x, then calculating the coefficients of the product polynomial isn't actually ...

3

You want to multiply x by y, where both are 32 bit numbers. The result is the sum of 32 numbers. The first number is x if bit #31 of y is 1, and 0 if bit #31 of y is 0. This number is shifted by 31 bits. The second number is x if bit #30 of y is 1, and 0 if bit #30 of y is 0. This number is shifted by 30 bits. And so on. The last number is x if bit #0 of y ...

3

The most common approach is to take the longest number, and divide it in half (rounding an odd number of digits arbitrarily). So for x = 12345 y = 2478 you would get a=12, b=345, c=2, d=478. Since the number of digits in x is not even, we are free to choose whether to split into a=12 and b=345 or a=123 and b=45; it makes no difference to the running time. ...

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