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| {{unreferenced|date=December 2011}}
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| '''Arithmetic in a [[finite field]]''' is different from standard integer [[arithmetic]]. There are a limited number of elements in the finite field; all operations performed in the finite field result in an element within that field. | |
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| While each finite field is itself not infinite, there are infinitely many different finite fields; their number of elements (which is also called [[cardinal number|cardinality]]) is necessarily of the form ''p''<sup>''n''</sup> where ''p'' is a [[prime number]] and ''n'' is a [[positive integer]], and two finite fields of the same size are [[isomorphism|isomorphic]]. The prime ''p'' is called the [[characteristic (algebra)|characteristic]] of the field, and the positive integer ''n'' is called the [[dimension (vector space)|dimension]] of the field over its [[characteristic (algebra)#Case of fields|prime field]].
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| Finite fields are used in a variety of applications, including in classical [[coding theory]] in [[linear block code]]s such as [[BCH code]]s and [[Reed–Solomon error correction]] and in [[cryptography]] algorithms such as the [[Rijndael]] encryption algorithm.
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| ==Effective polynomial representation==
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| The finite field with ''p''<sup>''n''</sup> elements is denoted GF(''p''<sup>''n''</sup>) and is also called the '''Galois Field''', in honor of the founder of finite field theory, [[Évariste Galois]]. GF(''p''), where ''p'' is a prime number, is simply the [[ring (algebra)|ring]] of integers [[Modular arithmetic|modulo]] ''p''. That is, one can perform operations (addition, subtraction, multiplication) using the usual operation on integers, followed by reduction modulo ''p''. For instance, in GF(5), 4+3=7 is reduced to 2 modulo 5. Division is multiplication by the inverse modulo ''p'', which may be computed using the [[extended Euclidean algorithm]]. | |
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| A particular case is GF(2), where addition is [[XOR gate|exclusive OR]] (XOR) and multiplication is [[AND gate|AND]]. Since the only invertible element is 1, division is the [[identity function]].
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| Elements of GF(''p''<sup>''n''</sup>) may be represented as [[polynomial]]s of degree strictly less than ''n'' over GF(''p''). Operations are then performed modulo ''R'' where ''R'' is an [[irreducible polynomial]] of degree ''n'' over GF(''p''), for instance using [[polynomial long division]]. The addition of two polynomials ''P'' and ''Q'' is done as usual; multiplication may be done as follows: compute ''W'' =''P''.''Q'' as usual, then compute the remainder modulo ''R'' (there exist better ways to do this).
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| When the prime is 2, it is conventional to express elements of GF(''p''<sup>''n''</sup>) as [[binary numeral system|binary numbers]], with each term in a polynomial represented by one bit in the corresponding element's binary expression. Braces ( "{" and "}" ) or similar delimiters are commonly added to binary numbers, or to their hexadecimal equivalents, to indicate that the value is an element of a field. For example, the following are equivalent representations of the same value in a characteristic 2 finite field:
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| ;Polynomial<nowiki>: </nowiki> ''x''<sup>6</sup> + ''x''<sup>4</sup> + ''x'' + 1
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| ;Binary<nowiki>: </nowiki>{01010011}
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| ;Hexadecimal<nowiki>: </nowiki>{53}
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| ==Addition and subtraction==
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| Addition and subtraction are performed by adding or subtracting two of these polynomials together, and reducing the result modulo the characteristic.
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| In a finite field with characteristic 2, addition modulo 2, subtraction modulo 2, and XOR are identical. Thus,
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| ;Polynomial<nowiki>: </nowiki> (''x''<sup>6</sup> + ''x''<sup>4</sup> + ''x'' + 1) + (''x''<sup>7</sup> + ''x''<sup>6</sup> + ''x''<sup>3</sup> + ''x'') = ''x''<sup>7</sup> + ''x''<sup>4</sup> + ''x''<sup>3</sup> + 1
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| ;Binary<nowiki>: </nowiki> {01010011} + {11001010} = {10011001}
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| ;Hexadecimal<nowiki>: </nowiki> {53} + {CA} = {99}
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| Notice that under regular addition of polynomials, the sum would contain a term 2''x''<sup>6</sup>, but that this term becomes 0''x''<sup>6</sup> and is dropped when the answer is reduced modulo 2.
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| Here is a table with both the normal algebraic sum and the characteristic 2 finite field sum of a few polynomials:
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| {| class="wikitable"
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| ! p<sub>1</sub>
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| ! p<sub>2</sub>
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| ! p<sub>1</sub> + p<sub>2</sub> (normal algebra)
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| ! p<sub>1</sub> + p<sub>2</sub> in GF(2<sup>n</sup>)
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| |-
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| | x<sup>3</sup> + x + 1
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| | x<sup>3</sup> + x<sup>2</sup>
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| | 2x<sup>3</sup> + x<sup>2</sup> + x + 1
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| | x<sup>2</sup> + x + 1
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| |-
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| | x<sup>4</sup> + x<sup>2</sup>
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| | x<sup>6</sup> + x<sup>2</sup>
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| | x<sup>6</sup> + x<sup>4</sup> + 2x<sup>2</sup>
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| | x<sup>6</sup> + x<sup>4</sup>
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| |-
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| | x + 1
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| | x<sup>2</sup> + 1
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| | x<sup>2</sup> + x + 2
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| | x<sup>2</sup> + x</tr>
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| |-
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| | x<sup>3</sup> + x
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| | x<sup>2</sup> + 1
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| | x<sup>3</sup> + x<sup>2</sup> + x + 1
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| | x<sup>3</sup> + x<sup>2</sup> + x + 1
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| |-
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| | x<sup>2</sup> + x
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| | x<sup>2</sup> + x
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| | 2x<sup>2</sup> + 2x
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| | 0
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| |}
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| Note: In computer science applications, the operations are simplified for finite fields of characteristic 2, also called GF(2<sup>n</sup>) [[Galois field]]s, making these fields especially popular choices for applications.
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| ==Multiplication==
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| <!-- Please do not change the examples to use a finite field besides x8 + x4 + x3 + x + 1; the Rijndael page links here and so people are expecting to get help figuring out Rijndael's finite field form this article -->
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| Multiplication in a finite field is multiplication [[Equivalence relation|modulo]] an [[irreducible polynomial|irreducible]] reducing polynomial used to define the finite field. (I.e., it is multiplication followed by division using the reducing polynomial as the divisor—the remainder is the product.) The symbol "•" may be used to denote multiplication in a finite field.
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| === Rijndael's finite field ===
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| [[Rijndael]] uses a characteristic 2 finite field with 256 elements, which can also be called the Galois field '''GF'''(2<sup>8</sup>). It employs the following reducing polynomial for multiplication:
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| :''x''<sup>8</sup> + ''x''<sup>4</sup> + ''x''<sup>3</sup> + ''x'' + 1.
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| For example, {53} • {CA} = {01} in Rijndael's field because
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| (''x''<sup>6</sup> + ''x''<sup>4</sup> + ''x'' + 1)(''x''<sup>7</sup> + ''x''<sup>6</sup> + ''x''<sup>3</sup> + ''x'') =
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| (''x''<sup>13</sup> + ''x''<sup>12</sup> + ''x''<sup>9</sup> + '''x<sup>7</sup>''')
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| + (''x''<sup>11</sup> + ''x''<sup>10</sup> + '''x<sup>7</sup>''' + ''x''<sup>5</sup>)
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| + (''x''<sup>8</sup> + '''x<sup>7</sup>''' + ''x''<sup>4</sup> + ''x''<sup>2</sup>)
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| + ('''x<sup>7</sup>''' + ''x''<sup>6</sup> + ''x''<sup>3</sup> + ''x'') =
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| ''x''<sup>13</sup> + ''x''<sup>12</sup> + ''x''<sup>9</sup> + ''x''<sup>11</sup> + ''x''<sup>10</sup> + ''x''<sup>5</sup> + ''x''<sup>8</sup> + ''x''<sup>4</sup> + ''x''<sup>2</sup> + ''x''<sup>6</sup> + ''x''<sup>3</sup> + ''x'' =
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| ''x''<sup>13</sup> + ''x''<sup>12</sup> + ''x''<sup>11</sup> + ''x''<sup>10</sup> + ''x''<sup>9</sup> + ''x''<sup>8</sup> + ''x''<sup>6</sup> + ''x''<sup>5</sup> + ''x''<sup>4</sup> + ''x''<sup>3</sup> + ''x''<sup>2</sup> + ''x''
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| and | |
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| ''x''<sup>13</sup> + ''x''<sup>12</sup> + ''x''<sup>11</sup> + ''x''<sup>10</sup> + ''x''<sup>9</sup> + ''x''<sup>8</sup> + ''x''<sup>6</sup> + ''x''<sup>5</sup> + ''x''<sup>4</sup> + ''x''<sup>3</sup> + ''x''<sup>2</sup> + ''x'' modulo ''x''<sup>8</sup> + ''x''<sup>4</sup> + ''x''<sup>3</sup> + ''x''<sup>1</sup> + 1 = (11111101111110 mod 100011011) = {3F7E mod 11B} = {01} = 1 (decimal), which can be demonstrated through [[long division]] (shown using binary notation, since it lends itself well to the task. Notice that [[Exclusive_or#Truth_table|exclusive OR]] is applied in the example and not arithmetic subtraction, as one might use in grade-school long division.):
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| <u> </u>
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| 11111101111110 (mod) 100011011
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| <u>^100011011 </u>
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| 1110000011110
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| <u>^100011011 </u>
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| 110110101110
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| <u>^100011011 </u>
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| 10101110110
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| <u>^100011011 </u>
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| 0100011010
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| <u>^100011011 </u>
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| 00000001
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| (The elements {53} and {CA} are [[multiplicative inverse]]s of one another since their product is [[1 (number)|1]].)
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| Multiplication in this particular finite field can also be done using a modified version of the "[[Multiplication_algorithm#Peasant_or_binary_multiplication|peasant's algorithm]]". Each polynomial is represented using the same binary notation as above. Eight bits is sufficient because only degrees 0 to 7 are possible in the terms of each (reduced) polynomial.
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| This algorithm uses three [[Variable (programming)|variable]]s (in the computer programming sense), each holding an eight-bit representation. '''a''' and '''b''' are initialized with the multiplicands; '''p''' accumulates the product and must be initialized to 0.
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| At the start and end of the algorithm, and the start and end of each iteration, this [[invariant (computer science)|invariant]] is true: '''a''' '''b''' + '''p''' is the product. This is obviously true when the algorithm starts. When the algorithm terminates, '''a''' or '''b''' will be zero so '''p''' will contain the product.
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| * Run the following loop eight times (once per bit). It is OK to stop when '''a''' or '''b''' are zero before an iteration:
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| *# If the rightmost bit of '''b''' is set, exclusive OR the product '''p''' by the value of '''a'''. This is polynomial addition.
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| *# Shift '''b''' one bit to the right, discarding the rightmost bit, and making the leftmost bit have a value of zero. This divides the polynomial by '''x''', discarding the ''x''<sup>0</sup> term.
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| *# Keep track of whether the leftmost bit of '''a''' is set to one and call this value '''carry'''.
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| *# Shift '''a''' one bit to the left, discarding the leftmost bit, and making the new rightmost bit zero. This multiplies the polynomial by '''x''', but we still need to take account of '''carry''' which represented the coefficient of ''x''<sup>7</sup>.
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| *# If '''carry''' had a value of one, exclusive or '''a''' with the hexadecimal number <tt>0x1b</tt> (00011011 in binary). <tt>0x1b</tt> corresponds to the irreducible polynomial with the high term eliminated. Conceptually, the high term of the irreducible polynomial and '''carry''' add modulo 2 to 0.
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| * '''p''' now has the product
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| This algorithm generalizes easily to multiplication over other fields of characteristic 2, changing the lengths of '''a''', '''b''', and '''p''' and the value <tt>0x1b</tt> appropriately.
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| ==Multiplicative inverse==
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| The [[multiplicative inverse]] for an element '''a''' of a finite field can be calculated a number of different ways: | |
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| * By multiplying '''a''' by every number in the field until the product is one. This is a [[Brute-force search]].
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| * For <math>n = 1</math> one can take advantage of [[Finite_field#Analog_of_Fermat.27s_little_theorem|the analog of Fermat's little theorem]], <math>a</math><sup><math>p-1</math></sup><math> \equiv 1</math> (for <math>a \ne 0</math>), thus the inverse of <math>a</math> is <math>a</math><sup><math>p-2</math></sup>
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| * By using the [[Extended Euclidean algorithm]]
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| * By making a [[logarithm]] table of the finite field, and performing subtraction in the table. Subtraction of logarithms is the same as division.
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| ==Implementation tricks==
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| When developing algorithms for Galois field computation on small Galois fields, a common performance optimization approach is to find a [[Generating set of a group#Finitely_generated_group|generator]] g and use the identity:
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| :<math>ab = g^{\log_g(ab)} = g^{\log_g (a)+\log_g (b)}</math>
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| to implement multiplication as a sequence of table look ups for the log<sub>g</sub>(x) and g<sup>(x)</sup> functions and an integer addition operation. This exploits the property that all finite fields contain generators. In the Rijndael field example, the polynomial x + 1 (or {03}) is one such generator, since it is [[Irreducible polynomial|irreducible]].
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| This same strategy can be used to determine the multiplicative inverse with the identity:
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| :<math>a^{-1} = g^{\log_g(a^{-1})} = g^{-\log_g(a)} = g^{|g| - \log_g (a)}</math>
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| Here, the [[Order (group theory)|order]] of the generator, |g|, is the number of non-zero elements of the field. In the case of GF(2<sup>8</sup>) this is 2<sup>8</sup>-1 = 255. That is to say, for the Rijndael example: (x + 1)<sup>255</sup> = 1. So this can be performed with two look up tables and an integer subtract. Using this idea for exponentiation also derives benefit:
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| :<math>a^n = g^{\log_g(a^n)} = g^{n\log_g(a)} = g^{n\log_g (a) (mod |g|)}</math>
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| This requires two table look ups, an integer multiplication and an integer modulo operation.
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| However, in cryptographic implementations, one has to be careful with such implementations since the [[CPU cache|cache architecture]] of many microprocessors leads to variable timing for memory access. This can lead to implementations that are vulnerable to a [[timing attack]].
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| == Program examples ==
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| === C programming example ===
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| Here is some [[C (programming language)|C]] code which will add, subtract, and multiply numbers in Rijndael's finite field:
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| <syntaxhighlight lang="c">
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| /* Add two numbers in a GF(2^8) finite field */
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| uint8_t gadd(uint8_t a, uint8_t b) {
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| return a ^ b;
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| }
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| /* Subtract two numbers in a GF(2^8) finite field */
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| uint8_t gsub(uint8_t a, uint8_t b) {
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| return a ^ b;
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| }
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| /* Multiply two numbers in the GF(2^8) finite field defined
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| * by the polynomial x^8 + x^4 + x^3 + x + 1 = 0 */
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| uint8_t gmul(uint8_t a, uint8_t b) {
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| uint8_t p = 0;
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| uint8_t counter;
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| uint8_t carry;
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| for (counter = 0; counter < 8; counter++) {
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| if (b & 1)
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| p ^= a;
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| carry = a & 0x80; /* detect if x^8 term is about to be generated */
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| a <<= 1;
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| if (carry)
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| a ^= 0x001B; /* replace x^8 with x^4 + x^3 + x + 1 */
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| b >>= 1;
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| }
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| return p;
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| }
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| </syntaxhighlight>
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| | |
| === D programming example ===
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| This [[D (programming language)|D]] program will multiply numbers in Rijndael's finite field and generate a [[Netpbm_format#PGM_example|PGM]] image:
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| <syntaxhighlight lang="D">
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| /**
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| Multiply two numbers in the GF(2^8) finite field defined
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| by the polynomial x^8 + x^4 + x^3 + x + 1.
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| */
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| ubyte gMul(ubyte a, ubyte b) pure nothrow {
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| ubyte p = 0;
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| foreach (immutable ubyte counter; 0 .. 8) {
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| if (b & 1)
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| p ^= a;
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| immutable ubyte carry = a & 0x80;
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| a <<= 1;
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| if (carry != 0)
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| a ^= 0b1_0001_1011; // x^8 + x^4 + x^3 + x + 1.
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| b >>= 1;
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| }
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| return p;
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| }
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| void main() {
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| import std.stdio, std.conv;
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| enum width = ubyte.max + 1, height = width;
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| auto f = File("rijndael_finite_field_multiplication.pgm", "wb");
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| f.writefln("P5\n%d %d\n255", width, height);
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| foreach (immutable y; 0 .. height)
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| foreach (immutable x; 0 .. width) {
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| immutable char c = gMul(x.to!ubyte, y.to!ubyte);
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| f.write(c);
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| }
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| }</syntaxhighlight>
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| == External links ==
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| * [http://www.samiam.org/galois.html A description of Rijndael's finite field]
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| [[Category:Arithmetic]]
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| [[Category:Finite fields|Arithmetic]]
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| [[Category:Articles with example D code]]
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The footwear ranges are all produced in-house using the best top quality Australian 'Double-Confront footwear skins'. They have reinforced and strengthened heels, EVB ribbed soles and boots with total size rear seam. These are out there in numerous styles and shades. The Australian built sheepskin Ugg boots are of a hundred% sheepskin obtaining proper sole with tread, toe and heel reinforcement. Some of the Australian made heart-line lace boots are priced at USD $a hundred and forty and higher than.
The children's footwear is intended for toddler care. They aid to maintain the toes heat and are ordinarily not meant for going for walks. This variety of footwear is manufactured out of toddler sheepskin in sixteen mm wool pile. They have gentle sole and are obtainable mainly in pink and blue shades. This sort of footwear also tends to make exceptional gift things. Automobile seat covers are made of a hundred% A1 excellent 25 mm deep pile and are out there in 5 shades and three specific kinds. The sheepskins are comfy in all climates and give the cars with a spanking new physical appearance. The auto seat addresses retains one awesome on incredibly hot days and heat during chilly weather. These covers ordinarily in shape most typical automobile and appear with or with out facet air bags. They are known to be extremely tough.
The 'throw over' sheepskin seat handles constitute of sheepskin on the again rest and head rest of the seat. All the handles are lined to safeguard the original seats. The professional medical sheepskin are specially handled for hygienic needs and used in hospitals, nursing homes, clinics and on wheelchair addresses. These covers present comfort and aid to the aged, sick and bedridden individuals especially people struggling from bed and pressure sores. These products are conveniently washable and can be managed even beneath superior temperature. They are perfect for people shelling out extended durations in mattress or on wheelchairs as these merchandise lessen stress and absorbs dampness. The comfortable snug and long lasting professional medical sheepskin comes in organic yellow shades.
Australian wool is extensively made use of in wool mats and wool blankets. The shorter woolen fibers stand up from the floor and give the fabric a furry touch. This purely natural fiber is bio degradable, flame resistant and absorbs h2o. The worsted woolen fabric area is clean to touch and is suitable for earning child rugs. The short wool sheepskin is also suited for toddler play mats, beneath lays and flooring mats.
The infant care brief wool sheepskins present a cushion of higher density delicate and springy fibers that help pounds to be distributed around a substantial area while even now enabling air to flow into through out the fibers. The Australian sheepskin items for infants have the capability to absorb up to 36% of its dry excess weight in moisture with no sensation damp. These products are out there in a variety of shades and are wonderful for the toddlers to relaxation and perform. Tender and extremely relaxed, they roll up and make touring pleasing.
The newborn rugs and sheepskin rugs are device washable. They continue to keep toddler pores and skin dry, absorbs perspiration, controls temperature and humidity so that the newborn skin remains warm in winter and amazing in summertime. The child rugs are made of Australian lamb's wool and chrome tanned. It lasts for about 20 decades beneath appropriate care and upkeep. The rugs are of the greatest excellent getting involving 2 inches to 3 feet in thickness. They are washable and accessible in white shades only. Additional massive rugs are also offered in chosen colors in pink, black, ivory and brown. These delicate snugly Australian sheepskin rugs make infants rest deeply and soundly.
The sheepskin mats, much too, are created to match all applications. They can be washed and are out there in eye-catching colors. These merchandise are created of quick wool Australian sheepskins and are particularly well known and are greatest value for funds. The cot underneath lays for toddlers and the pram liner can be utilised on the flooring as engage in mats for toddlers. The pram liner, as well, is made of short wool. These items now out offer the previous styles owing to its toughness. The wool mats and wool blankets are washable and are exported environment large.
The Australian sheepskin merchandise are common as they are all-natural products and solutions, sanitized for hygienic reasons and promote snooze and human body leisure. All the goods are processed below the newest technological innovation. This signifies that all sheepskin products are washable, lengthy long lasting and really hard donning. They may possibly be obtainable in a dyed color finish or in a purely natural un-dyed end. Shades might fluctuate relying on the dye batch. The vast array of solutions suits all age teams ranging from new born toddlers to the disabled and aged folks.
Now, the most well-known sheepskin merchandise are the professional medical products and solutions, brief boots, floor mats, blankets and the significant temperature resistant underneath lays. These particular person merchandise have received very favorable ratings from their glad prospects.
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