# 数学代写|密码学代写cryptography theory代考|CIS556

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## 数学代写|密码学代写cryptography theory代考|The Irreducible Polynomial

The Rijndael s-box is based on a specific irreducible polynomial in a specific Galois Field (Daemen and Rijmen 1999):
$$\mathrm{GF}\left(2^8\right)=\mathrm{GF}(2)[x] /\left(x^8+x^4+x^3+x+1\right)$$
In hexadecimal this is 11B, in binary it is 100011011 .
What is an irreducible polynomial? An irreducible polynomial that cannot be factored into the product of two other polynomials. In other words, it cannot be reduced. This is in reference to a specific field; in the case of the irreducible polynomial we are considering it is in reference to the Galois Field $\operatorname{GF}\left(2^8\right)$. Put more formally: A polynomial is irreducible in $\mathrm{GF}(\mathrm{p})$ if it does not factor over $\mathrm{GF}(\mathrm{p})$. Otherwise it is reducible.

Why was this specific irreducible polynomial chosen? Does it have some special property that makes it more suitable for cryptography? Well to answer that question let us consider the actual words of the inventors of Rijndael “The polynomial $\mathrm{m}$ (x) (’11B’) for the multiplication in $\operatorname{GF}\left(2^8\right)$ is the first one of the list of irreducible polynomials of degree 8” (Daemen and Rijmen 1999). In other words, they looked at a list of irreducible polynomials in a specific text and chose the first one. This is important to keep in mind. Any irreducible polynomial of the appropriate size can be used.

The text that Daemen and Rijmen consulted for their list of irreducible polynomials was “Introduction to finite fields and their applications,” Cambridge University Press, 1986. You can check the same source that was cited by the inventors of Rijndael. Here are a few irreducible polynomials from that list (in binary form, you may place them in polynomial or hex form if you wish).
100101011
100111001
10011111
101001101
101011111
101110111
110001011
You may have noticed that all of these, and the one chosen for Rijndael have 9 digits. Why use degree 8 ( 9 digits) isn’t that one too many? “Clearly, the result will be a binary polynomial of degree below 8 . Unlike for addition, there is no simple operation at byte level.” – page $3 / 4$ of the specification.

The reason an irreducible polynomial must be used, instead of just any polynomial (also called a primitive polynomial), is that we are trying to make a non-linear permutation function that has diffusion, spreading input bits to output bits in a non-linear way.

## 数学代写|密码学代写cryptography theory代考|Multiplicative Inverse

In mathematics, the reciprocal, or multiplicative inverse, of a number $x$ is the number which, when multiplied by $x$, yields 1 . The multiplicative inverse for the real numbers, for example, is $1 / x$. To avoid confusion by writing the inverse using set-specific notation, it is generally written as $x^{-1}$.

Multiplication in Galois Field, however, requires more tedious work. Suppose $f$ $(p)$ and $g(p)$ are polynomials in $g f(p n)$ and let $m(p)$ be an irreducible polynomial (or a polynomial that cannot be factored) of degree at least $\mathrm{n}$ in $g(p n)$. We want $m(p)$ to be a polynomial of degree at least $n$ so that the product of two $f(p)$ and $g(p)$ does not exceed $11111111=255$ as the product needs to be stored as a byte. If $h(p)$ denotes the resulting product then.
$$h(p)=(f(p) * g(p))(\bmod m(p))$$
On the other hand, the multiplicative inverse of $f(p)$ is given by $a(p)$ such that
$$(f(p) * a(p))(\bmod m(p))=1$$

Note that calculating the product of two polynomials and the multiplicative inverse of a polynomial requires both reducing coefficients modulo $p$ and reducing polynomials modulo $m(p)$. The reduced polynomial can be calculated easily with long division while the best way to compute the multiplicative inverse is by using Extended Euclidean Algorithm. The details on the calculations in $g f\left(2^8\right)$ are best explained in the following example.

Finite field multiplication is more difficult than addition and is achieved by multiplying the polynomials for the two elements concemed and collecting like powers of $x$ in the result. Since each polynomial can have powers of $x$ up to 7 , the result can have powers of $x$ up to 14 and will no longer fit within a single byte. This situation is handled by replacing the result with the remainder polynomial after division by a special eighth order irreducible polynomial, which, as you may recall for Rijndael, is:
$$m(x)=x 8+x 4+x 3+x+1$$
The finite field element $(00000010)$ is the polynomial $x$, which means that multiplying another element by this value increases all its powers of $x$ by 1 . This is equivalent to shifting its byte representation up by 1 bit so that the bit at position $i$ moves to position $i+1$. If the top bit is set prior to this move, it will overflow to create an $x 8$ term, in which case the modular polynomial is added to cancel this additional bit, leaving a result that fits within a single byte.

For example, multiplying (11001000) by $x$, that is $(00000010)$, the initial result is $1(10010000)$. The “overflow” bit is then removed by adding $1(00011011)$, the modular polynomial, using an exclusive-or operation to give a final result of (10001011). However, you need not calculate the multiplicative inverse manually, the table in 8-14 provides multiplicative inverses (Fig. 8.14).

# 密码学代写

## 数学代写|密码学代写密码学理论代考|不可约多项式

Rijndael s-box是基于特定伽罗瓦场中的特定不可约多项式(Daemen and Rijmen 1999):
$$\mathrm{GF}\left(2^8\right)=\mathrm{GF}(2)[x] /\left(x^8+x^4+x^3+x+1\right)$$

Daemen和Rijmen为他们的不可约多项式列表所参考的文本是“有限域及其应用导论”，剑桥大学出版社，1986年。你可以查一下Rijndael的发明者所引用的同一出处。下面是这个列表中的一些不可约多项式(以二进制形式，如果你愿意，你可以把它们放在多项式或十六进制形式)。
100101011
100111001
10011111
101001101
101011111
101110111
110001011

## 数学代写|密码学代写密码学理论代考|乘法逆

$$h(p)=(f(p) * g(p))(\bmod m(p))$$

$$(f(p) * a(p))(\bmod m(p))=1$$

$$m(x)=x 8+x 4+x 3+x+1$$

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