This library implements the techniques described in the paper "Screaming Fast Galois Field Arithmetic Using Intel SIMD Instructions," [PGM13b]. The Paper describes all of those techniques as well.

Contents

1 Introduction

Addition in a Galois Field is equal to the bitwise exclusive-or operation. That's nice and convenient. Multiplication is a little more complex, and there are many, many ways to implement it. The Paper describes them all, and the following references providemore supporting material: [Anv09, GMS08, LHy08, LD00, LBOX12, Pla97]. The intent of this library is to implement all of the techniques. That way, their performancemay be compared, and their tradeoffs may be analyzed.

When used for erasure codes, there are typically five important operations:

1. Adding two numbers in GF(2^w). That's bitwise exclusive-or.
2. Multiplying two numbers in GF(2^w). Erasure codes are usually based on matrices in GF(2^w), and constructing these matrices requires both addition and multiplication.
3. Dividing two numbers in GF(2^w). Sometimes you need to divide to construct matrices (for example, Cauchy Reed-Solomon codes [BKK⁺95, Rab89]). More often, though, you use division to invert matrices for decoding. Sometimes it is easier to find a number's inverse than it is to divide. In that case, you can divide by multiplying by an inverse.
4. adding two regions of numbers in GF(2^w), which will be explained along with...
5. Mutiplying a region of numbers in GF(2^w) by a constant in GF(2^w). Erasure coding typically boils down to performing dot products in GF(2^w). For example, you may define a coding disk using the equation:

c₀= d₀ + 2d₁ + 4d₂ + 8d₃.
That looks like three multiplications and three additions However, the way ' implemented in a disk system looks as in Figure 1. Large regions of disks are partitioned into w-bit words in GF(2^w). In the example, let us suppose that w = 8, and therefore that words are bytes. Then the regions pictured are 1 KB from each disk. The bytes on disk Di are labeled d_i,0, d_{i,1, . . . ,} d_i,1023, and the equation above is replicated 1024 times. For 0 ≤ j < 1024:

c_0,j = d_0,j + 2d_1,j + 4d_2,j + 8d_3,j .
While it's possible to implement each of these 1024 equations independently, using the single multiplication and addition operations above, it is often much more efficient to aggregate. For example, most computer architectures support bitwise exclusive-or of 64 and 128 bit words. Thus, it makes much more sense to add regions of numbers in 64 or 128 bit chunks rather than as words in GF(2^w). Multiplying a region by a constant can leverage similar optimizations.

GF-Complete supports multiplication and division of single values for all values of w ≤ 32, plus w = 64 and w = 128. It also supports adding two regions of memory (for any value of w, since addition equals XOR), and multiplying a region by a constant in GF(2⁴), GF(2⁸), GF(2¹⁶), GF(2³²), GF(2⁶⁴) and GF(2¹²⁸). These values are chosen because words in GF(2^w) fit into machine words with these values of w. Other values of w don't lend themselves to efficient multiplication of regions by constants (although see the "CAUCHY" option in section 6.1.5 for a way to multiply regions for other values of w).

2     Files in the Library

2.1     Header files in the directory "include"

• gf_complete.h: This is the header file that applications should include. It defines the gf_t type, which holds all of the data that you need to perform the various operations in GF(2^w). It also defines all of the arithmetic operations. For an application to use this library, you should include gf_complete.h and then compile with the library src/libgf_complete.la.


• gf_method.h: If you are wanting to modify the implementation techniques from the defaults, this file provides a "helper" function so that you can do it from the Unix command line.


• gf_general.h: This file has helper routines for doing basic Galois Field operations with any legal value of w. The problem is that w ≤ 32, w = 64 and w = 128 all have different data types, which is a pain. The procedures in this file try to alleviate that pain. They are used in gf_mult, gf_unit and gf_time. I'm guessing that most applications won't use them, as most applications use w ≤ 32.


• gf_rand.h: I've learned that srand48() and its kin are not supported in all C installations. Therefore, this file defines some randomnumber generators to help test the programs. The randomnumber generator is the "Mother

of All" random number generator [Mar94] which we've selected because it has no patent issues. gf_unit and gf time use these random number generators.


• gf_int.h: This is an internal header file that the various source files use. This is not intended for applications to include.


• config.xx and stamp-h1 are created by autoconf, and should be ignored by applications.

2.2     Source files in the "src" directory"

The following C files compose gf_complete.a, and they are in the direcoty src. You shouldn't have to mess with these files, but we include them in case you have to:


• gf_.c: This implements all of the procedures in both gf_complete.h and gf_int.h.


• gf_w4.c: Procedures specific to w = 4.


• gf_w8.c: Procedures specific to w = 8


• gf_w16.c: Procedures specific to w = 16


• gf_w32.c: Procedures specific to w = 32


• gf_w64.c: Procedures specific to w = 64


• gf_w128.c: Procedures specific to w = 128


• gf_wgen.c: Procedures specific to other values of w between 1 and 31


• gf_general.c: Procedures that let you manipulate general values, regardless of whether w ≤ 32, w = 64 or w = 128. (I.e. the procedures defined in gf_ general.h)


• gf_method.c: Procedures to help you switch between the various implementation techniques. (I.e. the procedures defined in gf_method.h)


• gf_ rand.c:"The Mother of all" random number generator. (I.e. the procedures defined in gf_rand.h)

2.3     Library tools files in the "tools" directory

The following are tools to help you with Galois Field arithmetic, and with the library. They are explained in greater detail elsewhere in this manual.


• gf_mult.c, gf_ div.c and gf_ add: Command line tools to do multiplication, division and addition by single numbers


• gf_time.c: A program that times the procedures for given values of w and implementation options


• time tool.sh: A shell script that helps perform rough timings of the various multiplication, division and region operations in GF-Complete


• gf_methods.c: A program that enumerates most of the implementation methods supported by GF-Complete


• gf_poly.c: A program to identify irreducible polynomials in regular and composite Galois Fields

2.4     The unit tester in the "test" directory

2.5    Example programs in the "examples" directory

3     Compilation

If you perform the install, then the header, source, tool, and library files will be moved to system locations. In particular, you may then compile the library by linking with the flag -lgf_complete, and you may use the tools from a global executable directory (like /usr/local/bin).

If you don't perform the install, then the header and tool files will be in their respective directories, and the library will be in src/libgf_complete.la.

If your system supports the various Intel SIMD instructions, the compiler will find them, and GF-Complete will use them by default.

4     Some Tools and Examples to Get You Started

4.1 Three Simple Command Line Tools: gf_mult, gf_div and gf_add

• gf_mult a b w - Multiplies a and b in GF(2^w).


• gf_div a b w - Divides a by b in GF(2^w ).


• gf_add a b w - Adds a and b in GF(2^w ).

You may use any value of w from 1 to 32, plus 64 and 128. By default, the values are read and printed in decimal; however, if you append an 'h' to w , then a, b and the result will be printed in hexadecimal. For w = 128, the 'h' is mandatory, and all values will be printed in hexadecimal.
4     SOME TOOLS AND EXAMPLES TO GET YOU STARTED 9 9




Try them out on some examples like the ones below. You of course don't need to know that, for example, 5 * 4 = 7 in GF(2⁴ ) ; however, once you know that, you know that 7/ 5 = 4 and 7/4 = 5. You should be able to verify the gf_add statements below in your head. As for the other gf_mult's, you can simply verify that division and multiplication work with each other as you hope they would.




UNIX> gf_mult 5 4 4
7
UNIX> gf_div 7 5 4
4
UNIX> gf_div 7 4 4
5
UNIX> gf_mult 8000 2 16h
100b
UNIX> gf_add f0f0f0f0f0f0f0f0 1313131313131313 64h
e3e3e3e3e3e3e3e3
UNIX> gf_mult f0f0f0f0f0f0f0f0 1313131313131313 64h
8da08da08da08da0
UNIX> gf_div 8da08da08da08da0 1313131313131313 64h
f0f0f0f0f0f0f0f0
UNIX> gf_add f0f0f0f0f0f0f0f01313131313131313 1313131313131313f0f0f0f0f0f0f0f0 128h
e3e3e3e3e3e3e3e3e3e3e3e3e3e3e3e3
UNIX> gf_mult f0f0f0f0f0f0f0f01313131313131313 1313131313131313f0f0f0f0f0f0f0f0 128h
786278627862784982d782d782d7816e
UNIX> gf_div 786278627862784982d782d782d7816e f0f0f0f0f0f0f0f01313131313131313 128h
1313131313131313f0f0f0f0f0f0f0f0
UNIX>


Don't bother trying to read the source code of these programs yet. Start with some simpler examples like the ones below.



4.2 Quick Starting Example #1: Simple multiplication and division

The source files for these examples are in the examples directory.

These two examples are intended for those who just want to use the library without getting too complex. The first example is gf_example 1, and it takes one command line argument - w, which must be between 1 and 32. It generates two random non-zero numbers in GF(2^w ) and multiplies them. After doing that, it divides the product by each number.

To perform multiplication and division in GF(2^w ) , you must declare an instance of the gf_t type, and then initialize it for GF(2^w ) by calling gf_init_easy(). This is done in gf_example 1.c with the following lines:



gf_t gf;

r ...

if (!gf_init_easy(&gf, w)) {
fprintf(stderr, "Couldn't initialize GF structure.\n");
exit(0);
}

4     SOME TOOLS AND EXAMPLES TO GET YOU STARTED 10




Once gf is initialized, you may use it for multiplication and division with the function pointers multiply.w32 and divide.w32. These work for any element of GF(2^w) so long as w ≤ 32.




c = gf.multiply.w32(&gf, a, b);
printf("%u * %u = %u\n", a, b, c);

printf("%u / %u = %u\n", c, a, gf.divide.w32(&gf, c, a));
printf("%u / %u = %u\n", c, b, gf.divide.w32(&gf, c, b));



Go ahead and test this program out. You can use gf_mult and gf_div to verify the results:


UNIX> gf_example_1 4
12 * 4 = 5
5 / 12 = 4
5 / 4 = 12
UNIX> gf_mult 12 4 4
5
UNIX> gf_example_1 16
14411 * 60911 = 44568
44568 / 14411 = 60911
44568 / 60911 = 14411
UNIX> gf_mult 14411 60911 16
44568
UNIX>


gf_init_easy() (and later_gf_init_hard()) do call malloc() to implement internal structures. To release memory, call gf_free(). Please see section 6.4 to see how to call gf_init_hard() in such a way that it doesn't call malloc().



4.3      Quick Starting Example #2: Multiplying a region by a constant

The program gf_example 2 expands on gf_example 1. If w is equal to 4, 8, 16 or 32, it performs a region multiply operation. It allocates two sixteen byte regions, r1 and r2, and then multiples r1 by a and puts the result in r2 using the multiply_region.w32 function pointer:


gf.multiply_region.w32 (&gf, r1, r2, a, 16, 0);


That last argument specifies whether to simply place the product into r2 or to XOR it with the contents that are already in r2. Zero means to place the product there. When we run it, it prints the results of the multiply_region.w32 in hexadecimal. Again, you can verify it using gf mult:


UNIX> gf_example_2 4
12 * 2 = 11
11 / 12 = 2
11 / 2 = 12

multiply_region by 0xc (12)

R1 (the source): 0 2 d 9 d 6 8 a 8 d b 3 5 c 1 8 8 e b 0 6 1 5 a 2 c 4 b 3 9 3 6
R2 (the product): 0 b 3 6 3 e a 1 a 3 d 7 9 f c a a 4 d 0 e c 9 1 b f 5 d 7 6 7 e


4     SOME TOOLS AND EXAMPLES TO GET YOU STARTED 11




UNIX>	gf_example_2 16
49598	*	35999	=	19867
19867	/	49598	=	35999
19867	/	35999	=	49598


  multiply_region by 0xc1be (49598)



R1 (the source):	8c9f	b30e	5bf3	7cbb	16a9	105d	9368	4bbe
R2 (the product):	4d9b	992d	02f2	c95c	228e	ec82	324e	35e4

UNIX> gf_mult c1be 8c9f 16h
4d9b
UNIX> gf_mult c1be b30e 16h
992d
UNIX>



4.4       Quick Starting Example #3: Using w = 64

The program in gf_example 3.c is identical to the previous program, except it uses GF(2⁶⁴ ). Now a, b and c are uint64 t's, and you have to use the function pointers that have w64 extensions so that the larger types may be employed.


UNIX> gf_example_31
a9af3adef0d23242	*	61fd8433b25fe7cd	=	bf5acdde4c41ee0c
bf5acdde4c41ee0c	/	a9af3adef0d23242	=	61fd8433b25fe7cd
bf5acdde4c41ee0c	/	61fd8433b25fe7cd	=	a9af3adef0d23242


  multiply_region by a9af3adef0d23242


R1 (the source):	61fd8433b25fe7cd	272d5d4b19ca44b7	3870bf7e63c3451a	08992149b3e2f8b7
R2 (the product):	bf5acdde4c41ee0c	ad2d786c6e4d66b7	43a7d857503fd261	d3d29c7be46b1f7c
UNIX> gf_mult a9af3adef0d23242 61fd8433b25fe7cd 64h
bf5acdde4c41ee0c
UNIX>



4.5       Quick Starting Example #4: Using w = 128

Finally, the program in gf_example_4.c uses GF(2¹²⁸). Since there is not universal support for uint128 t, the library represents 128-bit numbers as arrays of two uint64 t's. The function pointers for multiplication, division and region multiplication now accept the return values as arguments:

gf.multiply.w128(&gf, a, b, c);

Again, we can use gf_mult and gf_div to verify the results:


UNIX> gf_example_4

e252d9c145c0bf29b85b21a1ae2921fa	*	b23044e7f45daf4d70695fb7bf249432	=
7883669ef3001d7fabf83784d52eb414


4     IMPORTANT INFORMATION ON ALIGNMENT WHEN MULTIPLYING REGIONS 12



multiply_region by e252d9c145c0bf29b85b21a1ae2921fa
R1 (the source): f4f56f08fa92494c5faa57ddcd874149 b4c06a61adbbec2f4b0ffc68e43008cb
R2 (the product): b1e34d34b031660676965b868b892043 382f12719ffe3978385f5d97540a13a1
UNIX> gf_mult e252d9c145c0bf29b85b21a1ae2921fa f4f56f08fa92494c5faa57ddcd874149 128h
b1e34d34b031660676965b868b892043
UNIX> gf_div 382f12719ffe3978385f5d97540a13a1 b4c06a61adbbec2f4b0ffc68e43008cb 128h
e252d9c145c0bf29b85b21a1ae2921fa
UNIX>



5      Important Information on Alignment when Multiplying Regions

In order to make multiplication of regions fast, we often employ 64 and 128 bit instructions. This has ramifications for pointer alignment, because we want to avoid bus errors, and because on many machines, loading and manipulating aligned quantities is much faster than unalinged quantities.

When you perform multiply_region.wxx(gf, source, dest, value, size, add ), there are three requirements:
1. The pointers source and dest must be aligned for w-bit words. For w = 4 and w = 8, there is no restriction; however for w = 16, the pointers must be multiples of 2, for w = 32, they must be multiples of 4, and for w ϵ {64, 128}, they must be multiples of 8.


2. The size must be a multiple of [ w / 8 .] With w = 4 and w = 8, w/ 8 = 1 and there is no restriction. The other sizes must be multiples of w / 8 because you have to be multiplying whole elements of GF(2^w ) .


3. The source and dest pointers must be aligned identically with respect to each other for the implementation chosen. This is subtle, and we explain it in detail in the next few paragraphs. However, if you'd rather not figure it out, the following recommendation will always work in GF-Complete:
If you want to be safe, make sure that source and dest are both multiples of 16. That is not a strict requirement, but it will always work!


If you want to relax the above recommendation, please read further.

When performing multiply_region.wxx() , the implementation is typically optimized for a region of bytes whose size must be a multiple of a variable s ,, and which must be aligned to a multiple of another variable t . For example, when doing multiply_region.w32() in GF(2¹⁶ ) with SSE enabled, the implementation is optimized for regions of 32 bytes, which must be aligned on a 16-byte quantity. Thus, s = 32 and t = 16. However, we don't want multiply_ region.w32() to be too restrictive, so instead of requiring source and dest to be aligned to 16-byte regions, we require that (source mod 16) equal (dest mod 16). Or, in general, that (source mod t) equal (dest mod t).

Then, multiply_region.wxx() proceeds in three phases. In the first phase, multiply.wxx() is called on successive words until (source mod t) equals zero. The second phase then performs the optimized region multiplication on chunks of s bytes, until the remaining part of the region is less than s bytes. At that point, the third phase calls multiply.wxx() on the last part of the region.

A detailed example helps to illustrate. Suppose we make the following call in GF(2¹⁶) with SSE enabled:

multiply region.w32(gf, 0x10006, 0x20006, a, 274, 0)
2     FILES IN THE LIBRARY 13






Figure 2: Example of multiplying a region of 274 bytes in GF(216) when (source mod 16) = (dest mod 16) = 6. The alignment parameters are s = 32 and t = 16. The multiplication is in three phases, which correspond to the initial unaligned region (10 bytes), the aligned region of s-byte chunks (256 bytes), and the final leftover region (8 bytes).

First, note that source and dest are aligned on two-byte quantities, which they must be in GF(2¹⁶). Second, note that size is a multiple of [ 16/ 8 ] = 2. And last, note that (source mod 16) equals (dest mod 16). We illustrate the three phases of region multiplication in Figure 2. Because (source mod 16) = 6, there are 10 bytes of unaligned words that are multiplied with five calls to multiply.w32() in the first phase. The second phase multiplies 256 bytes (eight chunks of s = 32 bytes) using the SSE instructions. That leaves 8 bytes remaining for the third phase.

When we describe the defaults and the various implementation options, we specify s and t as "alignment parameters."

One of the advanced region options is using an alternate mapping of words to memory ("ALTMAP"). These interact in a more subtle manner with alignment. Please see Section 7.9 for details.

6    The Defaults

GF-Complete implements a wide variety of techniques for multiplication, division and region multiplication. We have set the defaults with three considerations in mind:
1. Speed: Obviously, we want the implementations to be fast. Therefore, we choose the fastest implementations that don’t violate the other considerations. The compilation environment is considered. For example, if SSE is enabled, region multiplication in GF(2⁴ ) employs a single multiplication table. If SSE is not enabled, then a "double" table is employed that performs table lookup two bytes at a time.


2. Memory Consumption: We try to keep the memory footprint of GF-Complete low. For example, the fastest way to perform multiply.w32() in GF(2³²) is to employ 1.75 MB of multiplication tables (see Section 7.4 below). We do not include this as a default, however, because we want to keep the default memory consumption of GF-Complete low.

3.   Compatibility with "standard" implementations: While there is no de facto standard of Galois Field arithmetic, most libraries implement the same fields. For that reason, we have not selected composite fields, alternate polynomials or memory layouts for the defaults, even though these would be faster. Again, see section 7.7 for more information.

Table 1 shows the default methods used for each power-of-two word size, their alignment parameters s and t, their memory consumption and their rough performance. The performance tests are on an Intel Core i7-3770 running at 3.40 GHz, and are included solely to give a flavor of performance on a standard microprocessor. Some processors will be faster with some techniques and others will be slower, so we only put numbers in so that you can ballpark it. For other values of w between 1 and 31, we use table lookup when w ≤ 8, discrete logarithms when w ≤ 16 and "Bytwop" for w ≤ 32.

A few comments on Table 1 are in order. First, with SSE, the performance of multiply() is faster when w = 64 than when w = 32. That is because the primitive polynomial for w = 32, that has historically been used in Galois Field implementations, is sub-ideal for using carry-free multiplication (PCLMUL). You can change this polynomial (see section 7.7) so that the performance matches w = 64.

The region operations for w = 4 and w = 8 without SSE have been selected to have a low memory footprint. There are better options that consume more memory, or that only work on large memory regions (see section 6.1.5).

• You may want better performance.

• You may want a lower memory footprint.
• You may want to use a different Galois Field or even a ring.
• You only care about multiplying a region by the value two.

Our command line tools allow you to deviate from the defaults, and we have two C functions -gf_init_hard() and create_gf_from_argv() that can be called from application code to override the default methods. There are six command-line tools that can be used to explore the many techniques implemented in GF-Complete:

• gf_methods is a tool that enumerates most of the possible command-line arguments that can be sent to the other tools


• gf_mult and gf_div are explained above. You may change the multiplication and division technique in these tools if you desire


• gf_unit performs unit tests on a set of techniques to verify correctness


• gf_time measures the performance of a particular set of techniques


• time_tool.sh makes some quick calls to gf_time so that you may gauge rough performance.


• gf_poly tests the irreducibility of polynomials in a Galois Field

To change the default behavior in application code, you need to call gf_init_hard() rather than gf_init_easy(). Alternatively, you can use create_g_from_argv(), included from gf_method.h, which uses an argv-style array of strings to specify the options that you want. The procedure in gf_method.c parses the array and makes the proper gf_init_hard() procedure call. This is the technique used to parse the command line in gf_mult, gf_div, gf_unit et al.

6.1.1 Changing the Components of a Galois Field with create gf_from_argv()

• w
• Multiplication technique
• Division technique
• Region technique(s)
• Polynomial

The procedures gf_init_hard() and create_gf_from_argv() allow you to specify these parameters when you create your Galois Field instance. We focus first on create_gf_from_argv(), because that is how the tools allow you to specify the components. The prototype of create_gf_from_argv() is as follows:

For example, gf_mult.c calls create gf_from_argv() by simply passing argc and argv from its main() declaration, and setting starting to 4.

To choose defaults, argv[starting] should equal "-". Otherwise, you specify the component that you are changing with "-m" for multiplication technique, "-d" for division technique, "-r" for region technique, and "-p" for the polynomial. You may change multiple components. You end your specification with a single dash. For example, the following call multiplies 6 and 5 in GF(2⁴) with polynomial 0x19 using the "SHIFT" technique for multiplication (we'll explain these parameters later):

If create_gf_from_argv() fails, then you can call the procedure gf_error(), which prints out the reason why create_ gf_from_argv() failed.

6.1.2 Changing the Polynomial

Multiplication is based on a special polynomial that we will refer to here as the "defining polynomial." This polynomial has binary coefficients and is of degree w. You may change the polynomial with "-p" and then a number in hexadecimal (the leading "0x" is optional). It is assumed that the w-th bit of the polynomial is set - you may include it or omit it. For example, if you wish to set the polynomial for GF(2¹⁶) to x¹⁶ + x⁵ + x³ + x² + 1, rather than its default of x¹⁶ + x¹² + x³ + x + 1, you may say "-p 0x1002d," "-p 1002d," "-p 0x2d" or "-p 2d." We discuss changing the polynomial for three reasons in other sections:

• Leveraging carry-free multiplication (section 7.7).
• Defining composite fields (section 7.6).
• Implementing rings (section 7.8.1).

Some words about nomenclature with respect to the polynomial. A Galois Field requires the polynomial to be irreducible .. That means that it cannot be factored. For example, when the coefficients are binary, the polynomial x⁵+ x⁴+x+1 may be factored as (x⁴+1)(x+1). Therefore it is not irreducible and cannot be used to define a Galois Field. It may, however, be used to define a ring. Please see section 7.8.1 for a discussion of ring support in GF-Complete.

There is a subset of irreducible polynomials called primitive. These have an important property that one may enumerate all of the elements of the field by raising 2 to successive posers. All of the default polynomials in GF-Complete are primitive. However, so long as a polynomial is irreducible, it defines a Galois Field. Please see section 7.7 for a further discussion of the polynomial.

One thing that we want to stress here is that changing the polynomial changes the field, so fields with different polynomialsmay not be used interchangeably. So long as the polynomial is irreducible, it generates a Galois Field that is isomorphic to all other Galois Fields; however the multiplication and division of elements will differ. For example, the polynomials 0x13 (the default) and 0x19 in GF(2⁴) are both irreducible, so both generate valid Galois Fields. However, their multiplication differs:

6.1.3     Changing the Multiplication Technique

table lookups. For example, the following multiplies 100 and 200 in GF(2⁸) using two 4K tables, as opposed to one 64K table when you use "TABLE:"


UNIX> ./gf_mult 100 200 8 -m SPLIT 8 4 -
79
UNIX>


See section 7.4 for additional information on the arguments to "SPLIT." The SSE version typically requires SSSE3.


• "GROUP:" This implements the "left-to-right comb" technique [LBOX12]. I'm afraid we don't like that name, so we call it "GROUP," because it performs table lookup on groups of bits for shifting (left) and reducing (right). It takes two additional arguments - g_s, which is the number of bits you use while shifting (left) and g_r, which is the number of bits you use while reducing (right). Increasing these arguments can you higher computational speed, but requires more memory. SSE version exists only for w = 128 and it requires SSE4. For more description on the arguments g_s and g_r, see section 7.5. For a full description of "GROUP" algorithm, please see The Paper.


• "COMPOSITE:" This allows you to perform operations on a composite Galois Field, GF((2^l)^k) as described in [GMS08], [LBOX12] and The Paper. The field size w is equal to lk. It takes one argument, which is k, and then a specification of the base field. Currently, the only value of k that is supported is two. However, that may change in a future revision of the library.

In order to specify the base field, put appropriate flags after specifying k. The single dash ends the base field, and after that, you may continue making specifications for the composite field. This process can be continued for multiple layers of "COMPOSITE." As an example, the following multiplies 1000000 and 2000000 in GF((2¹⁶)²), where the base field uses BYTWO_p for multiplication:

./gf mult 1000000 2000000 32 -m COMPOSITE 2 -m BYTWO p - -
In the above example, the red text applies to the base field, and the black text applies to the composite field. Composite fields have two defining polynomials - one for the composite field, and one for the base field. Thus, if you want to change polynomials, you should change both. The polynomial for the composite field must be of the form x²+sx+1, where s is an element of GF(2^k). To change it, you specify s (in hexadecimal)with "-p." In the example below, we multiply 20000 and 30000 in GF((2⁸)²) , setting s to three, and using x⁸+x⁴+x³+x²+1 as the polynomial for the base field:

./gf mult 20000 30000 16 -m COMPOSITE 2 -p 0x11d - -p 0x3 -

If you use composite fields, you should consider using "ALTMAP" as well. The reason is that the region operations will go much faster. Please see section 7.6.

As with changing the polynomial, when you use a composite field, GF((2^l)^k), you are using a different field than the "standard" field for GF((2^l)^k). All Galois Fields are isomorphic to each other, so they all have the desired properties; however, the fields themselves change when you use composite fields.

With the exception of "COMPOSITE", only one multiplication technique can be provided for a given Galois Field instance. Composite fields may use composite fields as their base fields, in which case the specification will be recursive.

6.1.4       Changing the Division Technique

If you use "-d", you must also specify the multiplication technique with "-m."

To force Euclid's algorithm instead of the defaults, you may specify it with "-d EUCLID." If instead, you would rather convert elements of a Galois Field to a binary matrix and find an element's inverse by inverting the matrix, then specify "-d MATRIX." In all of our tests, "MATRIX" is slower than "EUCLID." "MATRIX" is also not defined for w > 32.

6.1.5     Changing the Region Technique

• "SSE:" Use SSE instructions. Initialization will fail if the instructions aren't supported. Table 2 details the multiplication techniques which can leverage SSE instructions and which versions of SSE are required.



Multiplication Technique	multiply()	multiply_region()	SSE Version	Comments
"TABLE"	-	Yes	SSSE3	Only for GF(24).
"SPLIT"	-	Yes	SSSE3	Only when the second argument equals 4.
"SPLIt"	-	Yes	SSE4	When w = 64 and not using "ALTMAP".
"BYTWO p"	-	Yes	SSE2
"BYTWO p"	-	Yes	SSE2



Table 2: Multiplication techniques which can leverage SSE instructions when they are available.


• "NOSSE:" Force non-SSE version


• "DOUBLE:" Use a table that is indexed on two words rather than one. This applies only to w = 4, where the table is indexed on bytes rather than 4-bit quantities, and to w = 8, where the table is indexed on shorts rather than bytes. In each case, the table lookup performs two multiplications at a time, which makes region multiplication faster. It doubles the size of the lookup table.


• "QUAD:" Use a table that is indexed on four words rather than two or one. This only applies to w = 4, where the table is indexed on shorts. The "Quad" table may be lazily created or created ahead of time (the default). If the latter, then it consumes 2⁴ × 2¹⁶ × 2 = 2 MB of memory.


• "LAZY:" Typically it's clear whether tables are constructed upon initialization or lazily when a region operation is performed. There are two times where it is ambiguous: "QUAD" when w = 4 and "DOUBLE" when w = 8. If you don't specify anything, these tables are created upon initialization, consuming a lot of memory. If you specify "LAZY," then the necessary row of the table is created lazily when you call "multiply_region().

• "ALTMAP:" Use an alternate mapping, where words are split across different subregions of memory. There are two places where this matters. The first is when implementing "SPLIT w 4" using SSE when w > 8. In these cases, each byte of the word is stored in a different 128-bit vector, which allows the implementation to better leverage 16-byte table lookups. See section 7.4 for examples, and The Paper or [PGM13b] for detailed explanations.
  
  
The second place where it matters is when using "COMPOSITE." In this case, it is advantageous to split each memory region into two chunks, and to store half of each word in a different chunk. This allows us to call region_multiply() recursively on the base field, which is much faster than the alternative. See Section 7.6 for examples, and The Paper for an explanation.

It is important to note that with "ALTMAP," the words are not "converted" from a standard mapping to an alternate mapping and back again. They are assumed to always be in the alternate mapping. This typically doesn't matter, so long as you always use the same "ALTMAP" calls. Please see section 7.9 for further details on "ALTMAP," especially with respect to alignment.


• "CAUCHY:" Break memory into w subregions and perform only XOR's as in Cauchy Reed-Solomon coding [BKK⁺95] (also described in The Paper). This works for any value of w ≤ 32, even those that are not powers of two. If SSE2 is available, then XOR's work 128 bits at a time. For "CAUCHY" to work correctly, size must be a multiple of w .

It is possible to combine region multiplication options. This is fully supported as long as gf_methods has the combination listed. If multiple region options are required, they should be specified independently (as flags for gf_init_hard() and independent options for command-line tools and create_gf_from_argv()).

6.2    Determining Supported Techniques with gf methods

• "B:" This only prints out "basic" methods that are useful for the given value of w . It omits "SHIFT" and other methods that are never really going to be useful.


• "A:" In constrast, this specifies to print "all" methods.


• "D:" This includes the "EUCLID" and "MATRIX" methods for division. By default, they are not included.


• "C:" This includes the "CAUCHY" methods for region multiplication. By default, it is not included.

You may specify multiple of these as the second argument. If you include both "B" and "A," then it uses the last one specified.

The last argument determines the output format of gf_methods. If it is "L," then it simply lists methods. If it is "U," then the output contains gf_unit commands for each of the methods. For the others, the output contains gf_time_tool.sh commands for M ultiplication,Division,Region multiplications with multiple buffer sizes, and the Best region multiplication.

gf_methods enumerates combinations of flags, and calls create_gf_from_argv() to see if the combinations are supported. Although it enumerates a large number of combinations, it doesn't enumerate all possible parameters for "SPLIT," "GROUP" or "COMPOSITE."

Some examples of calling gf_methods are shown below in section 6.3.2.

6.3 Testing with gf_unit , gf_time , and time_tool.sh

• 'M': Single multiplications


• 'D': Single divisions


• 'I': Single inverses


• 'G': Region multiplication of a buffer by a random constant


• '0': Region multiplication of a buffer by zero (does nothing andbzero())


• '1': Region multiplication of a buffer by one (does memcpy() and XOR)


• '2': Region multiplication of a buffer by two – sometimes this is faster than general multiplication


• 'S': All three single tests


• 'R': All four region tests


• 'A': All seven tests

Here are some examples of calling gf_unit and gf_time to verify that "-m SPLIT 32 4 -r ALTMAP -" works in GF(2³²), and to get a feel for its performance. First, we go to the test directory and call gf_unit:

The first column of output displays the name of the test performed. Region tests will test with and without the XOR flag being set (see Section 4.3 for an example). The second column displays the total time the test took to complete measured in seconds (s). The third column displays the size of the test measured in millions of operations (Mops) for single tests and in Megabytes (MB) for the region tests. The final column displays the speed of the tests calculated from the second and third columns, and is where you should look to get an idea of a method's performance.

If the output of gf_unit and gf_time are to your satisfaction, you can incorporate the method into application code using create gf_from_argv() or gf_init hard().

The performance of "Region-By-Zero" and "Region-By-One" will not change from test to test, as all methods make the same calls for these. "Region-By-Zero" with "XOR: 1" does nothing except set up the tests. Therefore, you may use it as a control.

6.3.1       time tool.sh

The values for the first argument are MDRB, for Multiplication, Division,Region multiplications with multiple buffer sizes, and the Best region multiplication. For the example above, let's call time_tool.sh to get a rough idea of performance:

We say that time_tool.sh is "rough" because it tries to limit each test to 5 ms or less. Thus, the time granularity is fine, which means that the numbers may not be as precise as they could be were the time granularity to be course. When in doubt, you should make your own calls to gf_time with a lot of iterations, so that startup costs and roundoff error may be minimized.

6.3.2       An example of gf methods and time tool.sh

You'll note, this is on my old Macbook Pro, which doesn't support (PCLMUL), so "CARRY FREE" is not included as an option. Now, let's run the unit tester on these to make sure they work, and to see their memory consumption:

As anticipated, "SPLIT 8 8" consumes quite a bit of memory! Now, let's see how well they perform with both single multiplications and region multiplications:

The default is quite a bit slower than the best performing methods for both single and region multiplication. So why are the defaults the way that they are? As detailed at the beginning of this chapter, we strive for lower memory consumption, so we don't use "SPLIT 8 8," which consumes 1.75MB.We don't implement alternate fields by default, which is why we don't use "COMPOSITE." Finally, we don't implement alternate mappings of memory by default, which is why we don't use "-m SPLIT 32 4 -r ALTMAP -."

Of course, you may change these defaults if you please.

Test question: Given the numbers above, it would appear that "COMPOSITE" yields the fastest performance of single multiplication, while "SPLIT 32 4" yields the fastest performance of region multiplication. Should I use two gf t's in my application – one for single multiplication that uses "COMPOSITE," and one for region multiplication that uses "SPLIT 32 4?"

The answer to this is "no." Why? Because composite fields are different from the "standard" fields, and if you mix these two gf_t's, then you are using different fields for single multiplication and region multiplication. Please read section 7.2 for a little more information on this.

6.4      Calling gf_init_hard()

You can mix the region types with bitwise or. The arguments to GF_MULT_GROUP,GF_MULT_SPLIT_TABLE and GF_MULT_COMPOSITE are specified in arg1 and arg2. GF_MULT_COMPOSITE also takes a base field in base_gf. The base field is itself a gf_t, which should have been created previously with create_gf_fro_argv(), gf_init_easy() or gf_init_hard(). Note that this base_gf has its own base_gf member and can be a composite field itself.

You can specify an alternate polynomial in prim_poly. For w ≤ 32, the leftmost one (the one in bit position w) is optional. If you omit it, it will be added for you. For w = 64, there's no room for that one, so you have to leave it off. For w = 128, your polynomial can only use the bottom-most 64 bits. Fortunately, the standard polynomial only uses those bits. If you set prim_poly to zero, the library selects the "standard" polynomial.

Finally, scratch_memory is there in case you don't want gf_init_hard() to call malloc(). Youmay call gf_scratch_size() to find out how much extra memory each technique uses, and then you may pass it a pointer for it to use in scratc_memory. If you set scratch memory to NULL, then the extra memory is allocated for you with malloc(). If you use gf_init_easy() or create_gf_from_argv(), or you use gf_init_hard() and set scratch_memory to NULL, then you should call gf_free() to free memory. If you use gf_init_hard() and use your own scratch_memory you can still call gf_free(), and it will not do anything.

Both gf_init_hard() and gf_scratch_size() return zero if the arguments don't specify a valid gf_t. When that happens, you can call gf_error() to print why the call failed.

We'll give you one example of calling gf_ init_hard(). Suppose you want to make a gf_ init_hard() call to be equivalent to "-m SPLIT 16 4 -r SSE -r ALTMAP -" and you want to allocate the scratch space yourself. Then you'd do the following:

6.5     gf_size()

7   Further Information on Options and Algorithms

7.1   Inlining Single Multiplication and Division for Speed

When w = 4, you may inline multiplication and division as follows. The following procedures return pointers to the multiplication and division tables respectively:

The macro Gf_W4_INLINE_MULTDIV (table, a, b) then multiplies or divides a by b using the given table. This of course only works if the multiplication technique is "TABLE," which is the default for w = 4. If the multiplication technique is not "TABLE," then gf_w4_get_mult_table() will return NULL.

When w = 8, the procedures gf_w8_et_mult_table() and gf_ w8_get_div_table(), and the macro

When w = 16, the following procedures return pointers to the logarithm table, and the two inverse logarithm tables respectively:

The first inverse logarithm table works for multiplication, and the second works for division. They actually point to the same table, but to different places in the table. You may then use the macro GF_W16_INLINE_MULT(log, alog, a, b ) to multiply a and b, and the macro GF_W16_INLINE_DIV (log, alog, a, b ) to divide a and b. Make sure you use the alog table returned by gf_w16_get_mult_alog_table() for multiplication and the one returned by gf_w16_get_div_alog_table() for division. Here are some timings:

7.2     Using different techniques for single and region multiplication

• "COMPOSITE" implements a different Galois Field.


• If you change a field's polynomial, then the resulting Galois Field will be different.

• If you are using "ALTMAP" to multiply regions, then the contents of the resulting regions of memory will depend on the multiplication technique, the size of the region and its alignment. Please see section 7.9 for a detailed explanation of this.
• If you are using "CAUCHY" to multiply regions, then like "ALTMAP," the contents of the result regions of memory the multiplication technique and the size of the region. You don't have to worry about alignment.

7.3     General w

The library supports Galois Field arithmetic with 2 < w ≤ 32. Values of w which are not whole number powers of 2 are handled by the functions in gf_wgen.c . For these values of w , the available multiplication types are "SHIFT," "BYTw O p," "BYTw O b," "GROUP," "TABLE" and "LOG." "LOG" is only valid for w < 28 and "TABLE" is only valid for w < 15. The defaults for these values of w are "TABLE" for w < 8, "LOG" for w < 16, and "BYTw O p" for w < 32.



7.4 Arguments to "SPLIT"

The "SPLIT" technique is based on the distributive property of multiplication and addition:

a * (b + c) = (a * b) + (a * c).
This property allow s us to, for example, split an eight bit w ord into tw o four-bit components and calculate the product by performing tw o table lookups in 16-element tables on each of the compoents, and adding the result. There is much more information on "SPLIT" in The Paper. Here w e describe the version of "SPLIT" implemented in GF-Complete.

"SPLIT" takes tw o arguments, w hich are the number of bits in each component of a, w hich w e call w _a, and the number of bits in each component of b, w hich w e call w _b. If the tw o differ, it does not matter w hich is bigger - the library recognizes this and performs the correct implementation. The legal values of w _a and w _b fall into five categories:



1. w _a is equal to w and w _b is equal to four. In this case, b is broken up into w /4 four-bit w ords w hich are used in 16-element lookup tables. The tables are created on demand in multiply_region() and the SSSE3 instruction mm_shuffle_epi8() is leveraged to perform 16 lookups in parallel. Thus, these are very fast implementations. w hen w ≥ 16, you should combine this w ith "ALTMAP" to get the best performance (see The Paper or [PGM13b] for explanation). If you do this please see section 7.9 for information about "ALTMAP" and alignment.
   
   If you don't use "ALTMAP," the implementations for w ∈ {16, 32, 64} convert the standard representation into "ALTMAP," perform the multiplication w ith "ALTMAP" and then convert back to the standard representation. The performance difference using "ALTMAP" can be significant:
   
   
   

   gf time 16 G 0 1048576 100 -m SPLIT 16 4 -	Speed = 8,389 MB/s
   gf time 16 G 0 1048576 100 -m SPLIT 16 4 -r ALTMAP -	Speed = 8,389 MB/s
   gf time 32 G 0 1048576 100 -m SPLIT 32 4 -	Speed = 5,304 MB/s
   gf time 32 G 0 1048576 100 -m SPLIT 32 4 -r ALTMAP -	Speed = 7,146 MB/s
   gf time 64 G 0 1048576 100 -m SPLIT 64 4 -	Speed = 2,595 MB/s
   gf time 64 G 0 1048576 100 -m SPLIT 64 4 -r ALTMAP -	Speed = 3,436 MB/s

   
   6     FURTHER INFORMATION ON OPTIONS AND ALGORITHMS 29
   
   
   

   1. 2.   w_a is equal to w and w_b is equal to eight. Now, b is broken into bytes, each of these is used in its own 256-element lookup table. This is typically the best w_ay to perform multiply_region() without SSE.
   Because this is a region optimization, when you specify these options, you get a default multiply() see Table 1 for a listing of the defaults. See section 7.2 for using a different multiply() than the defaults.
   
   
   2. 3.   w_a is equal to w and w _b is equal to 16. This is only valid for w = 32 and w = 64. Now , b is broken into shorts, each of these is used in its own 64K-element lookup table. This is typically slower than when w _b equals 8, and requires more amortization (larger buffer sizes) to be effective.
   
   
   3. 4.   w _a and w _b are both equal to eight. Now both a and b are broken into bytes, and the products of the various bytes are looked up in multiple 256 × 256 tables. In GF(2¹⁶), there are three of these tables. In GF(232), there are seven, and in GF(2⁶⁴) there are fifteen. Thus, this implementation can be a space hog. How ever, for w = 32, this is the fastest way to perform multiply() on some machines. when this option is employed, multiply_region() is implemented in an identical fashion to when w _a = w and w _b = 8.
   
   
   4. 5.  w_a = 32 and w_b = 2. (w = 32 only). I was playing with a different way to use mm_shuffle_epi8(). It works, but it's slower than when w_b = 4.

7.5    Arguments to "GROUP"

1. For w ≤ 32 and w = 64, any values of g_s and g_r may be used, so long as they are less than or equal to w and so long as the tables fit into memory. There are four exceptions to this, listed below .


2. For w = 4, "GROUP" is not supported.


3. For w = 8, "GROUP" is not supported.


4. For w = 16, "GROUP" is only supported for gs = gr = 4.


5. For w = 128 "GROUP" only supports g_s = 4 and g_r ∈ {4, 8, 16}.

The way that gs and gr impact performance is as follows. The "SHIFT" implementation works by performing a carry-free multiplication in w steps, and then performing reduction in w steps. In "GROUP," the carry-free multiplication is reduced to w /g_ssteps, and the reduction is reduced to w /g_r . Both require tables. The table for the carry-free multiplication must be created at the beginning of each multiply() or multiply_region(), while the table for reduction is created when the gf_t is initialized. For that reason, it makes sense for g_r to be bigger than g_s.

To give a flavor for the impact of these arguments, Figure 3 show s the performance of varying g_s and g_r for single multiplication and region multiplication respectively, in GF(2³²) and GF(2⁶⁴). As the graphs demonstrate, multiply() performs better w ith smaller values of gs, w hile multiply region() amortizes the creation of the shifting table, and can tolerate larger values of g_s. w hen g_s equals g_r, there are some optimizations that we hand-encode. These can be seen clearly in the multiply_region() graphs.

7.6  Considerations with "COMPOSITE"

There is support for performing multiply() inline for the "TABLE" implementations for w ∈ {4, 8} and for the "LOG" implementation for w = 16 (see section 7.1). These are leveraged by multiply() in "COMPOSITE," and by multiply_region() if you are not using "ALTMAP." To demonstrate this, in the table below, you can see that the performance of multiply() with "SPLIT 8 4" is 88 percent as fast than the default in w = 8 (which is "TABLE"). When you use each as a base field for "COMPOSITE" with w = 16, the one with "SPLIT 8 4" is now just 37 percent as fast. The difference is the inlining of multiplication in the base field when "TABLE" is employed:

Please see section 7.8.1 for a discussion of polynomials in composite fields.

7.7       "CARRY FREE" and the Primitive Polynomial

This is relevant for w = 16 and w = 32, where the "standard" polynomials are sub-optimal with respect to "CARRY FREE." For w = 16, the polynomial 0x1002d has the desired property. It’s less important, of course, with w = 16, because "LOG" is so much faster than CARRY FREE.

7.8   More on Primitive Polynomials

7.8.1   Primitive Polynomials that are not Primitive

If a polynomial is reducible, then it does not define a Galois Field, but instead a ring. GF-Complete attempts to work here where it can; however certain parts of the library will not work:

1. Division is a best effort service. The problemis that often quotients are not unique. If divide() returns a non-zero number, then that number will be a valid quotient, but it may be one of many. If the multiplication technique is "TABLE," then if a quotient exists, one is returned. Otherwise, zero is returned. Here are some examples - the polynomial x⁴ + 1 is reducible, and therefore produces a ring. Below, we see that with this polynomal, 1*6 = 6 and 14*6 = 6. Therefore, 6/6 has two valid quotients: 1 and 14. GF-Complete returns 14 as the quotient:


UNIX> gf_mult 1 6 4 -p 0x1 -
6
UNIX> gf_mult 14 6 4 -p 0x1 -
6
UNIX> gf_div 6 6 4 -p 0x1 -
14
UNIX>


2. When "EUCLID" is employed for division, it uses the extended Euclidean algorithm for GCD to find a number's inverse, and then it multiplies by the inverse. The problem is that not all numbers in a ring have inverses. For example, in the above ring, there is no number a such that 6a = 1. Thus, 6 has no inverse. This means that even though 6/6 has quotients in this ring, "EUCLID" will fail on it because it is unable to find the inverse of 6. It will return 0:


UNIX> gf_div 6 6 4 -p 0x1 -m TABLE -d EUCLID -
0
UNIX>



3. Inverses only work if a number has an inverse. Inverses may not be unique.


4. "LOG" will not work. In cases where the default would be "LOG," "SHIFT" is used instead.

Due to problems with division, gf_unit may fail on a reducible polynomial. If you are determined to use such a polynomial, don't let this error discourage you.

7.8.2 Default Polynomials for Composite Fields

• w = 8 and the default polynomial (0x13) is employed for GF(2⁴)


• w = 16 and the default polynomial (0x11d) is employed for GF(2⁸)


• w = 32 and the default polynomial (0x1100b) is employed for GF(2¹⁶)


• w = 32 and 0x1002d is employed for GF(2¹⁶)


• w = 32 and the base field for GF(w16) is a composite field that uses a default polynomial


• w = 64 and the default polynomial (0x100400007) is employed for GF(2³²)


• w = 64 and 0x1000000c5 is employed for GF(2³²)


• w = 64 and the base field for GF(w³²) is a composite field that uses a default polynomial


• w = 128 and the default polynomial (0x1b) is employed for GF(2⁶⁴)


• w = 128 and the base field for GF(w⁶⁴ ) is a composite field that uses a default polynomial

7.8.3 The Program gf poly for Verifying Irreducibility of Polynomials

You can use it to test for irreducible polynomials with binary coefficients by specifying w = 1. For example, from the discussion above, we know that x⁴ +x+1 and x⁴ +x³ +x² +x+1 are both irreducible, but x⁴ +1 is reducible. gf_poly confirms:

For composite fields GF((2^l)²), we are looking for a value s such that x² + sx + 1 is irreducible. That value depends on the base field. For example, for the default field GF(2³²), a value of s = 2 makes the polynomial irreducible. However, if the polynomial 0xc5 is used (so that PCLMUL is fast - see section 7.7), then s = 2 yields a reducible polynomial, but s = 3 yields an irreducible one. You can use gf_poly to help verify these things, and to help define s if you need to stray from the defaults:

gf_unit does random sampling to test for problems. In particular, it chooses a random a and a random b, multiplies them, and then tests the result by dividing it by a and b. When w is large, this sampling does not come close to providing complete coverage to check for problems. In particular, if the polynomial is reducible, there is a good chance that gf_unit won't discover any problems. For example, the following gf_unit call does not flag any problems, even though the polynomial is reducible.

How can we demonstrate that this particular field has a problem? Well, when the polynomial is 0xc5, we can factor x² + 2x + 1 as (x + 0x7f6f95f9)(x + 0x7f6f95fb). Thus, in the composite field, when we multiply 0x17f6f95f9 by 0x17f6f95fb, we get zero. That's the problem:

7.9 "ALTMAP" considerations and extract_word()

1. When using "SPLIT" and w_b = 4.
2. When using "COMPOSITE."

When you use alternate memory mappings, the exact mapping of words in GF(2^w ) to memory depends on the situation, the size of the region, and the alignment of the pointers. To help you figure things out, we have included the procedures extract_word.wxx() as part of the gf_t struct. This procedure takes four parameters:

• A pointer to the gf_t.
• The beginning of the memory region.
• The number of bytes in the memory region.
• The desired word number: n.

It then returns the n-th word in memory. When the standard mapping is employed, this simply returns the n- th contiguous word in memory. With alternate mappings, each word may be split over several memory regions, so extract_word() grabs the relevant parts of each memory region to extract the word. Below, we go over each of the above situations in detail. Please refer to Figure 2 in Section 5 for reference.

7.9.1 Alternate mappings with "SPLIT"

In other words, it is multiplying a region a of 60 bytes (30 words) by the constant 0x1234 in GF(2¹⁶), and placing the result into b. The pointers a and b have been set up so that they are not multiples of 16. The first line of output prints a and b:

1. 4 bytes starting at 0x1001008c / 0x10010015c.
2. 32 bytes starting at 0x10010090 / 0x100100160.
3. 24 bytes starting at 0x100100b0 / 0x100100180.

In the first and third parts, the bytes are laid out according to the standard mapping. However, the second part is split into two 16-byte regions- one that holds the high bytes of each word and one that holds the low bytes. To help illustrate, the remainder of the output prints the 30 words of a and b as they appear in memory, and then the 30 return values of extract_word.w32():

When we reach word 22, we are in the third region of memory, and words are once again identical to how they appear in memory.

While this is confusing, we stress that that so long as you call multiply_region() with pointers of the same alignment and regions of the same size, your results with ALTMAP will be consistent. If you call it with pointers of

When w = 32, the middle region is a multiple of 64, and each word in the middle region is broken into bytes, each of which is in a different 16-byte region. When w = 64, the middle region is a multiple of 128, and each word is stored in eight 16-byte regions. And finally, whenw = 128, the middle region is a multiple of 128, and each word is stored in 16 16-byte regions.

7.9.2   Alternate mappings with "COMPOSITE"

As with "SPLIT," using multiply_region() with "COMPOSITE" and "ALTMAP" will be consistent only if the alignment of pointers and region sizes are identical.

7.9.3 The mapping of "CAUCHY"

The program prints the three bytes of a and b in hexadecimal and in binary. To see how words are broken up, consider word 0, which is the lowest bit of each of the three bytes of a (and b). These are the bits 1, 1 and 0 in a, and 0, 0, and 1 in b. Accordingly, the word is 3 in a, and 3*5 = 4 in b. Similarly, word 7 is the high bit in each byte: 0, 1, 1 (6) in a, and 1, 1, 0 (3) in b.

With "CAUCHY," multiply_region()may be implemented exclusively with XOR operations. Please see [BKK⁺95] for more information on the motivation behind "CAUCHY."

8   Thread Safety

• All "GROUP" implementations are not thread safe for either region_multiply() or multiply(). Other than "GROUP," multiply() is always thread-safe.

• For w = 4, region_multiply.w32() is unsafe in in "-m TABLE -r QUAD -r LAZY."


• For w = 8, region_multiply.w32() is unsafe in in "-m TABLE -r DOUBLE -r LAZY."


• For w = 16, region_multiply.w32() is unsafe in in "-m TABLE."


• For w ∈ {32, 64, 128}, all "SPLIT" implementations are unsafe for region_multiply(). This means that if the default uses "SPLIT" (see Table 1 for when that occurs), then region_multiply() is not thread safe.


• The "COMPOSITE" operations are only safe if the implementations of the underlying fields are safe.

9  Listing of Procedures

• GF_W16_INLINE_DIV() in gf_complete.h: This is a macro for inline division when w = 16. See section 7.1.


• GF_W16_INLINE_MULT() in gf_complete.h: This is a macro for inline multiplication when w = 16. See section 7.1.


• GF_W4_INLINE_MULTDIV() in gf_complete.h: This is a macro for inline multiplication/division when w = 4. See section 7.1.


• GF_W8_INLINE_MULTDIV() in gf_complete.h: This is a macro for inline multiplication/division when w = 8. See section 7.1.


• MOA_Fill_Random_Region() in gf_rand.h: Fills a region with random numbers.


• MOA_Random_128() in gf_rand.h: Creates a random 128-bit number.


• MOA_Random_32() in gf_rand.h: Creates a random 32-bit number.


• MOA_Random_64() in gf_rand.h: Creates a random 64-bit number.


• MOA_Random_W() in gf_rand.h: Creates a random w-bit number, where w ≤ 32.


• MOA_Seed() in gf_rand.h: Sets the seed for the random number generator.


• gf_errno in gf_complete.h: This is to help figure out why an initialization call failed. See section 6.1.


• gf_create_gf_from_argv() in gf method.h: Creates a gf t using C style argc/argv. See section 6.1.1.


• gf_division_type_t in gf_complete.h: the different ways to specify division when using gf_init_hard(). See section 6.4.


• gf_error() in gf_complete.h: This prints out why an initialization call failed. See section 6.1.


• gf_extract in gf_complete.h: This is the data type of extract_word() in a gf t. See section 7.9 for an example of how to use extract word().

9     LISTING OF PROCEDURES 39



• gf_free() in gf_complete.h: If gf_init easy(), gf_init hard() or create_gf_from_argv() allocated memory, this frees it. See section 6.4.
• gf_func_a_b in gf_complete.h: This is the data type of multiply() and divide() in a gf_t. See section 4.2 for examples of how to use multiply() and divide()


• gf_func_a_b in gf_complete.h: This is the data type of multiply() and divide() in a gf_t. See section 4.2 for examples of how to use multiply() and divide()


• gf_func_a in gf_complete.h: This is the data type of inverse() in a gf_t


• gf_general_add() in gf_general.h: This adds two gf_general_t's


• gf_general_divide() in gf_general.h: This divides two gf_general t's


• gf_general_do_region_check() in gf_general.h: This checks a region multiply of gf_general_t's


• gf_general_do_region_multiply() in gf_general.h: This does a region multiply of gf_general_t's


• gf_general_do_single_timing_test() in gf_general.h: Used in gf_time.c


• gf_general_inverse() in gf_general.h: This takes the inverse of a gf_general_t


• gf_general_is_one() in gf_general.h: This tests whether a gf_general_t is one


• gf_general_is_two() in gf_general.h: This tests whether a gf_general_t is two


• gf_general_is_zero() in gf_general.h: This tests whether a gf_general_t is zero


• gf_general_multiply() in gf_general.h: This multiplies two gf_general_t's. See the implementation of gf_mult.c for an example


• gf_general_s_to_val() in gf_general.h: This converts a string to a gf_general t. See the implementation of gf_mult.c for an example


• gf_general_set_one() in gf_general.h: This sets a gf_general_t to one


• gf_general_set_random() in gf_general.h: This sets a gf_general_t to a random number


• gf_general_set_two() in gf_general.h: This sets a gf_general_t to two


• gf_general_set_up_single_timing_test() in gf_general.h: Used in gf_time.c


• gf_general_set_zero() in gf_general.h: This sets a gf_general_t_to_zero


• gf_general_t_in .gf_general.h: This is a general data type for all values of w. See the implementation of gf_mult.c for examples of using these


• gf_general_val_to_s() ingf_general.h: This converts a gf_general_t to a string. See the implementation of gf_mult.c for an example


• gf_init_easy() in gf_complete.h: This is how you initialize a default gf_t. See 4.2 through 4.5 for examples of calling gf_init_easy()



9     LISTING OF PROCEDURES 40



• gf_init hard() in gf_complete.h: This allows you to initialize a gf_t without using the defaults. See 6.4. We recommend calling create gf_from argv() when you can, instead of gf_ init_hard()


• gf_ mult_type_t in gf_complete.h: the different ways to specify multiplication when using gf_init hard(). See section 6.4


• gf_region_type_t in gf_complete.h: the different ways to specify region multiplication when using gf_init_hard(). See section 6.4


• gf_region_in gf_complete.h: This is the data type of multiply_region() in a gf_t. See section 4.3 for an example of how to use multiply_region()


• gf_scratch_size() in gf_complete.h: This is how you calculate how much memory a gf_t needs. See section 6.4.


• gf_size() in gf_complete.h: Returns the memory consumption of a gf_t. See section 6.5.


• gf_ val_128_t in gf_complete.h: This is how you store a value where w ≤ 128. It is a pointer to two 64-bit unsigned integers. See section 4.4


• gf_val_32_t in gf_ complete.h: This is how you store a value where w ≤ 32. It is equivalent to a 32-bit unsigned integer. See section 4.2


• gf_ val_64_t in gf_complete.h: This is how you store a value where w ≤ 64. It is equivalent to a 64-bit unsigned integer. See section 4.5


• gf_w16_get_div_alog_table() in gf_ complete.h: This returns a pointer to an inverse logarithm table that can be used for inlining division when w = 16. See section 7.1


• gf_w16_get_log_table() in gf_complete.h: This returns a pointer to a logarithm table that can be used for inlining when w = 16. See section 7.1


• gf_w16_get_mult_alog_table() in gf_complete.h: This returns a pointer to an inverse logarithm table that can be used for inlining multiplication when w = 16. See section 7.1


• gf_ w4 get div table() in gf_complete.h: This returns a pointer to a division table that can be used for inlining when w = 4. See section 7.1


• gf_w4_get_mult_table() in gf_complete.h: This returns a pointer to a multiplication table that can be used for inlining when w = 4. See section 7.1


• gf_w8_get_div_table() in gf_complete.h: This returns a pointer to a division table that can be used for inlining when w = 8. See section 7.1


• gf_w8_get_mult_table() in gf_complete.h: This returns a pointer to a multiplication table that can be used for inlining when w = 8. See section 7.1



10     TROUBLESHOOTING 41



• SSE support. Leveraging SSE instructions requires processor support as well as compiler support. For example, the Mac OS 10.8.4 (and possibly earlier versions) default compile environment fails to properly compile PCLMUL instructions. This issue can be fixed by installing an alternative compiler; see Section 3 for details


• Initialization segfaults. You have to already have allocated your gf_t before you pass a pointer to it in bgf_init_easy(), create_gf_ from_argv(), or bgf_ini_hard()


• GF-Complete is slower than it should be. Perhaps your machine has SSE, but you haven't specified the SSE compilation flags. See section 3 for how to compile using the proper flags


• Bad alignment. If you get alignment errors, see Section 5


• Mutually exclusive region types. Some combinations of region types are invalid. All valid and implemented combinations are printed by bgf_methods.c


• Incompatible division types. Some choices of multiplication type constrain choice of divide type. For example, "COMPOSITE" methods only allow the default division type, which divides by finding inverses (i.e., neither "EUCLID" nor "MATRIX" are allowed). For each multiplication method printed by gf_methods.c, the corresponding valid division types are also printed


• Arbitrary "GROUP" arguments. The legal arguments to "GROUP" are specified in section 7.5


• Arbitrary "SPLIt" arguments. The legal arguments to "SPLIt" are specified in section 7.4


• Threading problems. For threading questions, see Section 8


• No default polynomial. If you change the polynomial in a base field using "COMPOSITE," then unless it is a special case for which GF-Complete finds a default polynomial, you'll need to specify the polynomial of the composite field too. See 7.8.2 for the fields where GF-Complete will support default polynomials


• Encoding/decoding with different fields. Certain fields are not compatible. Please see section 7.2 for an explanation


• "ALTMAP" is confusing. We agree. Please see section 7.9 for more explanation.
• I used "ALTMAP" and it doesn't appear to be functioning correctly. With 7.9, the size of the region and its alignment both matter in terms of how "ALTMAP" performs multiply_region(). Please see section 7.9 for detailed explanation


• Where are the erasure codes?. This library only implements Galois Field arithmetic, which is an underlying component for erasure coding. Jerasure will eventually be ported to this library, so that you can have fast erasure coding



11     Timings

We don't want to get too detailed with timing, because it is quite machine specific. However, here are the timings on an Intel Core i7-3770 CPU running at 3.40 GHz, with 4 × 256 KB L2 caches and an 8MB L3 cache. All timings are obtained with gf_time or gf_inline_time, in user mode with the machine dedicated solely to running these jobs.
10     TROUBLESHOOTING 41



Figure 4: Speed of doing single multiplications for w ∈ {4, 8, 16}.

11.1   Multiply()

The performance of multiply() is displayed in Figures 4 for w ∈ {4, 8, 16} and 5 for w ∈ {32, 64, 128}. These numbers were obtained by calling gf_time with the size and iterations both set to 10240. We plot the speed in megaops per second.

As would be anticipated, the inlined operations (see section 7.1) outperform the others. Additionally, in all cases with the exception of w = 32, the defaults are the fastest performing implementations. With w = 32, "CARRY FREE" is the fastest with an alternate polynomial (see section 7.7). Because we require the defaults to use a "standard" polynomial, we cannot use this implementation as the default.

11.2   Divide()

For the "TABLE" and "LOG" implementations, the performance of division is the same as multiplication. This means that for w ∈ {4, 8, 16}, it is very fast indeed. For the other implementations, division is implemented with Euclid's method, and is several factors slower than multiplication. In Figure 6, we plot the speed of a few implementations of the larger word sizes. Compared to the "TABLE" and "LOG" implemenations for the smaller word sizes, where the speeds are in the hundreds of mega-ops per second, these are very slow. Of note is the "COMPOSITE" implementation for w = 32, which is much faster than the others
10     TROUBLESHOOTING 43



Figure 5: Speed of doing single multiplications for w ∈ {32, 64, 128}.
because it uses a special application of Euclid's method, which relies on division in GF(2¹⁶), which is very fast.



11.3   Multiply_Region()

Tables 3 through 8 show the performance of the various region operations. It should be noted that for GF(2¹⁶ ) through GF(2¹²⁸), the default is not the fastest implementation of multiply_region(). The reasons for this are outlined in section 6

For these tables, we performed 1GB worth of multiply_region() calls for all regions of size 2i bytes for 10 ≤ i ≤ 30. In the table, we plot the fastest speed obtained.

We note that the performance of "CAUCHY" can be improved with techniques from [LSXP13] and [PSR12].


REFERENCES 44



Figure 6: Speed of doing single divisions for w ∈ {32, 64, 128}.


Method	Speed (MB/s)
-m TABLE (Default) -	11879.909
-m TABLE -r CAUCHY -	9079.712
-m BYTWO b -	5242.400
-m BYTWO p -	4078.431
-m BYTWO b -r NOSSE -	3799.699
-m TABLE -r QUAD -	3014.315
-m TABLE -r DOUBLE -	2253.627
-m TABLE -r NOSSE -	2021.237
-m TABLE -r NOSSE -	1061.497
-m LOG -	503.310
m SHIFT -	157.749
-m CARRY FREE -	86.202




Table 3: Speed of various calls to multiply_region() for w = 4.

References

[Anv09] H. P. Anvin. The mathematics of RAID-6. http://kernel.org/pub/linux/kernel/people/hpa/ raid6.pdf, 2009.

[BKK⁺95] J. Blomer, M. Kalfane, M. Karpinski, R. Karp, M. Luby, and D. Zuckerman. An XOR-based erasureresilient coding scheme. Technical Report TR-95-048, International Computer Science Institute, August 1995.

[GMS08] K. Greenan, E. Miller, and T. J. Schwartz. Optimizing Galois Field arithmetic for diverse processor architectures and applications. In MASCOTS 2008: 16th IEEE Symposium on Modeling, Analysis and Simulation of Computer and Telecommunication Systems, Baltimore, MD, September 2008.

[GP97] S. Gao and D. Panario. Tests and constructions of irreducible polynomials over finite fields. In Foundations of Computational Mathematics, pages 346–361. Springer Verlag, 1997.
REFERENCES 45




Method	Speed (MB/s)
-m SPLIT 8 4 (Default)	13279.146
-m COMPOSITE 2 - -r ALTMAP -	5516.588
-m TABLE -r CAUCHY -	4968.721
-m BYTWO b -	2656.463
-m TABLE -r DOUBLE -	2561.225
-m TABLE -	1408.577
-m BYTWO b -r NOSSE -	1382.409
-m BYTWO p -	1376.661
-m LOG ZERO EXT -	1175.739
-m LOG ZERO -	1174.694
-m LOG -	997.838
-m SPLIT 8 4 -r NOSSE -	885.897
-m BYTWO p -r NOSSE -	589.520
-m COMPOSITE 2 - -	327.039
-m SHIFT -	106.115
-m CARRY FREE -	104.299




Table 4: Speed of various calls to multiply region() for w = 4.

[LBOX12] J. Luo, K. D. Bowers, A. Oprea, and L. Xu. Efficient software implementations of large finite fields GF(2ⁿ) for secure storage applications. ACM Transactions on Storage, 8(2), February 2012.

[LD00] J. Lopez and R. Dahab. High-speed software multiplication in f_2^m. In Annual International Conference on Cryptology in India, 2000.

[LHy08] H. Li and Q. Huan-yan. Parallelized network coding with SIMD instruction sets. In International Symposium on Computer Science and Computational Technology, pages 364-369. IEEE, December 2008.

[LSXP13] J. Luo, M. Shrestha, L. Xu, and J. S. Plank. Efficient encoding schedules for XOR-based erasure codes. IEEE Transactions on Computing,May 2013.

[Mar94] G. Marsaglia. The mother of all random generators. ftp://ftp.taygeta.com/pub/c/mother. c, October 1994.
[PGM13a] J. S. Plank, K. M. Greenan, and E. L. Miller. A complete treatment of software implementations of finite field arithmetic for erasure coding applications. Technical Report UT-CS-13-717, University of Tennessee, September 2013.

[PGM13b] J. S. Plank, K. M. Greenan, and E. L. Miller. Screaming fast Galois Field arithmetic using Intel SIMD instructions. In FAST-2013: 11th Usenix Conference on File and Storage Technologies, San Jose, February 2013.

[Pla97] J. S. Plank. A tutorial on Reed-Solomon coding for fault-tolerance in RAID-like systems. Software - Practice & Experience, 27(9):995-1012, September 1997.
REFERENCES 46




Method	Speed (MB/s)
-m SPLIT 16 4 -r ALTMAP -	10460.834
-m SPLIT 16 4 -r SSE (Default) -	8473.793
-m COMPOSITE 2 - -r ALTMAP -	5215.073
-m LOG -r CAUCHY -	2428.824
-m TABLE -	2319.129
-m SPLIT 16 8 -	2164.111
-m SPLIT 8 8 -	2163.993
-m SPLIT 16 4 -r NOSSE -	1148.810
-m LOG -	1019.896
-m LOG ZERO -	1016.814
-m BYTWO b -	738.879
-m COMPOSITE 2 - -	596.819
-m BYTWO p -	560.972
-m GROUP 4 4 -	450.815
-m BYTWO b -r NOSSE -	332.967
-m BYTWO p -r NOSSE -	249.849
-m CARRY FREE -	111.582
-m SHIFT -	95.813




Table 5: Speed of various calls to multiply region() for w = 4.

[PMG⁺13] J. S. Plank, E. L. Miller, K. M. Greenan, B. A. Arnold, J. A. Burnum, A. W. Disney, and A. C. McBride. GF-Complete: A comprehensive open source library for Galois Field arithmetic. version 1.0. Technical Report UT-CS-13-716, University of Tennessee, September 2013.

[PSR12] J. S. Plank, C. D. Schuman, and B. D. Robison. Heuristics for optimizing matrix-based erasure codes for fault-tolerant storage systems. In DSN-2012: The International Conference on Dependable Systems and Networks, Boston, MA, June 2012. IEEE.

[Rab89] M. O. Rabin. Efficient dispersal of information for security, load balancing, and fault tolerance. Journal of the Association for Computing Machinery, 36(2):335-348, April 1989.
REFERENCES 47




Method	Speed (MB/s)
-m SPLIT 32 4 -r SSE -r ALTMAP - -m SPLIT 32 4 (Default) -m COMPOSITE 2 -m SPLIT 16 4 -r ALTMAP - -r ALTMAP - -m COMPOSITE 2 - -r ALTMAP - -m SPLIT 8 8 -m SPLIT 32 8 -m SPLIT 32 16 -m SPLIT 8 8 -r CAUCHY -m SPLIT 32 4 -r NOSSE -m CARRY FREE -p 0xc5 -m COMPOSITE 2 - -m BYTWO b -m BYTWO p -m GROUP 4 8 -m GROUP 4 4 -m CARRY FREE -m BYTWO b -r NOSSE -m BYTWO p -r NOSSE -m SHIFT	7185.440 5063.966 4176.440 3360.860 1345.678 1340.656 1262.676 1143.263 480.859 393.185 332.964 309.971 258.623 242.076 227.399 226.785 143.403 111.956 52.295




Table 6: Speed of various calls to multiply region() w = 4.



Method	Speed (MB/s)
-m SPLIT 64 4 -r ALTMAP - -m SPLIT 64 4 -r SSE (Default) - -m COMPOSITE 2 -m SPLIT 32 4 -r ALTMAP - -r ALTMAP - -m COMPOSITE 2 - -r ALTMAP - -m SPLIT 64 16 - -m SPLIT 64 8 - -m CARRY FREE - -m SPLIT 64 4 -r NOSSE - -m GROUP 4 4 - -m GROUP 4 8 - -m BYTWO b - -m BYTWO p - -m SPLIT 8 8 - -m BYTWO p -r NOSSE - -m COMPOSITE 2 - - -m BYTWO b -r NOSSE - -m SHIFT -	3522.798 2647.862 2461.572 1860.921 1066.490 998.461 975.290 545.479 230.137 153.947 144.052 124.538 98.892 77.912 77.522 36.391 25.282




Table 7: Speed of various calls to multiply region() for w = 4.


REFERENCES 48




Method	Speed (MB/s)
-m SPLIT 128 4 -r ALTMAP- -m COMPOSITE 2 -m SPLIT 64 4 -r ALTMAP - -r ALTMAP- -m COMPOSITE 2 - -r ALTMAP- -m SPLIT 128 8 (Default)- -m CARRY FREE - -m SPLIT 128 4 - -m COMPOSITE 2 - -m GROUP 4 8 - -m GROUP 4 4 - -m BYTWO p - -m BYTWO b - -m SHIFT -	1727.683 1385.693 1041.456 872.619 814.030 500.133 289.207 133.583 116.187 25.162 25.157 14.183




Table 8: Speed of various calls to multiply region() for w = 4.