Rijndael Round - Partial MixColumns - Numbas at mathcentre.ac.uk

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Created this as a copy of Rijndael Round.

Name	Status	Author	Last Modified
Rijndael Round	Ready to use	Julien Ugon	19/09/2023 07:47
Rijndael Round - Partial MixColumns	Ready to use	Julien Ugon	26/09/2023 01:28

There is one other version that you do not have access to.

Question statement

We want to encrypt a message using AES.

The output of the Rijndael round $\var{round-1}$ is $\var{AH}$ .

Run the next Rijndael round. All your answers must be given in hexadecimal.

Give any introductory information the student needs.

			Name	Type	Generated Value

KH	string	\begin{pmatrix}b0 & 85 & 84 &
Kiprev	string	\begin{pmatrix}3f & 97 & a8 &
KiH	list	Nested 4×4 list
PH	string	\begin{pmatrix}1a & 01 & 6e &
AH	string	\begin{pmatrix}4b & 6d & 13 &
BH	list	Nested 4×4 list
CH	list	Nested 4×4 list
DH	list	Nested 4×4 list
FH	list	Nested 4×4 list

			Name	Type	Generated Value

K	matrix	Matrix of size 4×4
P	matrix	Matrix of size 4×4
round	integer	5
A0	matrix	Matrix of size 4×4
A	matrix	Matrix of size 4×4
B	matrix	Matrix of size 4×4
C	matrix	Matrix of size 4×4
D	matrix	Matrix of size 4×4
F	matrix	Matrix of size 4×4
Ai	list	Nested 10×2 list

			Name	Type	Generated Value

list

Nested 11×2 list

			Name	Type	Generated Value

Elmt1	string	4b
Lhs	string	4
Rhs	string	b
key	integer	205
el	integer	79
elk	integer	130
elbin	string	<tr><td>⊕</td><td>0</td><td>1<
keybin	string	<tr><td></td><td>1</td><td>1</
elkbin	string	<tr><td>=</td><td>1</td><td>0<
roundkeyop	html	HTML node
MMc	matrix	Matrix of size 4×4
mmcop	string	\begin\{align} & x⋅(1 + x + x^
mmcopres	expression	1 + x + x^2 + x^3 + x^6
mmcop2	string	\begin\{align} & 1⋅(x^{ 2 } +
mmcopres2	expression	1

			Name	Type	Generated Value

Kiprevt	matrix	Matrix of size 4×4
wim4	string	(3f, fd, 61, dd)
wim3	string	(97, 2c, 23, b7)
wim2	string	(a8, 5b, cd, de)
wim1	string	(ae, 3b, ea, 88)
wis1	vector	vector(59,234,136,174)
wis2	vector	vector(226,135,196,228)
ri	number	16
wis3	vector	vector(242,135,196,228)
wi	list	[ 205, 122, 165, 57 ]
wip1	list	[ 90, 86, 134, 142 ]
wip2	list	[ 242, 13, 75, 80 ]
wip3	list	[ 92, 54, 161, 216 ]
rib	string	00010000

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
latex(hexmatrixdisplay(tohex(K)))

JME function reference

Description

The starting key for AES, in hexadecimal form.

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `string`

\begin{pmatrix}b0 & 85 & 84 & 76\\98 & ad & 4a & f7\\da & 79 & 09 & 5b\\7c & 3c & a3 & e7\end{pmatrix}

← Depends on:

This variable doesn't seem to be used anywhere.

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Ask the student a question, and give any hints about how they should answer this part.

Finding the next key

In Rijndael round $\var{round-1}$ we used the subkey $\var{Kiprev}$.

Find the next key in the key schedule; that is, the key for round $\var{round}$. Recall that for Round $\var{round}$, the key consists of the columns $W(\var{4round}), W(\var{4round+1}), W(\var{4round+2}), W(\var{4round+3})$.

A11	A12	A13	A14
A21	A22	A23	A24
A31	A32	A33	A34
A41	A42	A43	A44

Advice

Before we can apply a Rijndael Round, we need to find the key applicable to that round according to the AES key schedule. Then we will apply the steps in order:

The SubBytes step
The ShiftRows step
The MixColumns step
The AddRoundKey

Applying AES key schedule to find the key

In order to find the key for this round, we start from the key from the previous round: $\var{Kiprev}$.

Computing $W(\var{4*(round)})$

The formula is $W(\var{4*(round)}) = W(\var{4*(round-1)})\oplus T(W(\var{4*(round)-1}))$. From the previous key we know that $W(\var{4*(round-1)}) = \var{wim4}$ and $W(\var{4*(round)-1}) = \var{wim1}$.

Then we apply the following steps:

Shift to the left: $\var{tohexvector(wis1)}$
S-Box Substitution: $\var{tohexvector(wis2)}$
Find the round constant: $00000010^{\var{round-1}} = \var{rib}$.
x-or this round constant with the first coordinate:: $\var{tohexvector(wis3)}$

Thus $T(W(\var{4*(round)-1}))=\var{tohexvector(wis3)}$

Finally $W(\var{4*(round)})$ is obtained by $\oplus$ this vector $\var{tohexvector(wis3)}$ with $W(\var{4*(round-1)})$: $ W(\var{4*(round)}) = \var{wim4}\oplus \var{tohexvector(wis3)} = \var{tohexvector(vector(wi))}$.

Computing $W(\var{4round+1})$ to $W(\var{4(round)+3})$.

We just x-or iteratively:

$W(\var{4*(round)+1}) = W(\var{4*(round)}) \oplus W(\var{4*(round-1)+1}) = \var{tohexvector(vector(wi))}\oplus\var{wim3} = \var{tohexvector(vector(wip1))}$.
$W(\var{4*(round)+2}) = W(\var{4*(round)+1}) \oplus W(\var{4*(round-1)+2}) = \var{tohexvector(vector(wip1))}\oplus\var{wim2} = \var{tohexvector(vector(wip2))}$.
$W(\var{4*(round)+3}) = W(\var{4*(round)+2}) \oplus W(\var{4*(round-1)+3}) = \var{tohexvector(vector(wip2))}\oplus\var{wim1} = \var{tohexvector(vector(wip3))}$.

The new Key

the new key is $[W(\var{4*(round)}), W(\var{4*(round)+1}), W(\var{4*(round)+2}), W(\var{4*(round)+3})]=\var{latex(hexmatrixdisplay(KiH))}$

The SubBytes step

This step is a substitution step: we take every element of the matrix $\var{AH}$ and we convert it using the substitution table. Let us do it with the first element:

The element, {Elmt1}, is split in two parts. The left part, {Lhs} gives the row and the right part {Rhs} the column of the cell to look up in the S-box. Converted to decimals, we look up the S-Box in the Row {floor(A[0][0]/16)} and the column {mod(A[0][0],16)}. This gives us {BH[0][0]}.

Similarly, we replace the second element, {tohexa2(floor(A[0][1]))}, with the element on Row {tohexadecimal(floor(A[0][1]/16))} and Column {tohexadecimal(floor(mod(A[0][1],16)))} of the S-Box, which is {BH[0][1]}.

After applying this procedure to each element in the matrix $\var{AH}$ we get the matrix $\var{latex(hexmatrixdisplay(BH))}$.

The ShiftRows step

In this step the rows of the matrix are shifted to the left.

The first row remains untouched: $\var{latex(hexmatrixdisplay([CH[0]]))}$.
The second row is shifted by 1 byte to the left: $\var{latex(hexmatrixdisplay([CH[1]]))}$.
The third row is shifted by 2 bytes to the left: $\var{latex(hexmatrixdisplay([CH[2]]))}$.
The fourth row is shifted by 3 bytes to the left: $\var{latex(hexmatrixdisplay([CH[3]]))}$.

The resulting matrix is {hexmatrixdisplay(CH)}

The MixColumns step

In the MixColumn step, we multiply the matrices $\var{MMc}$ with $\var{latex(hexmatrixdisplay(CH))}$. Let us find the entries $A_{1,1}$ and $A_{3,2}$.

$A_{1,1}$ is obtained by multiplying the first row of the first matrix, $\var{transpose(MMC[0..3][0])}$, with the first column of the second matrix, $\var{latex(hexmatrixdisplay([CH[0..3][0]]))}$.

We apply the operation in GF(16), and so we have to first write each element as a polynomial. Polynomial multiplication is done $\pmod{x^8+x^4+x^3+x+1}$. Addition is XOR, and so the first element is given by:
{mmcop}

The result of this operation is $\var{mmcopres}$, which translated back to hexadecimal gives {DH[0][0]}.

$A_{3,2}$ is obtained by multiplying the third row of the first matrix, $\var{transpose(MMC[0..3][2])}$ with the second column of the second matrix, $\var{latex(hexmatrixdisplay([CH[0..3][2]]))}$. Like before we rewrite every element as a polynomial, and then proceed with the calculations: {mmcop2}.

The result of the operation is $\var{mmcopres2}$, which translated back to hexadecimal gives {DH[2][1]}.

The AddRoundKey

This step consists in combining the message and the key with a XOR operation. Let's do that with the first elements of the matrices, {DH[0][0]} and the key {KiH[0][0]}.

First we write both numbers in binary form, and apply the XOR operation on each index;

{roundkeyop}

Applying this operation for every elements of the matrices $\var{latex(hexmatrixdisplay(DH))}$ and $\var{latex(hexmatrixdisplay(KiH))}$ give the output $\var{latex(hexmatrixdisplay(FH))}$.

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This question is used in the following exam:

aes-exam by Guillermo Pineda in SIT281.

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Rijndael Round - Partial MixColumns

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Julien Ugon 1 year, 6 months ago

Julien Ugon 1 year, 6 months ago

Julien Ugon 1 year, 6 months ago

Error in variable testing condition

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Warning

Generated value: `string`

Parts

Gap-fill

Finding the next key

Applying AES key schedule to find the key

Computing $W(\var{4*(round)})$

Computing $W(\var{4round+1})$ to $W(\var{4(round)+3})$.

The new Key

The SubBytes step

The ShiftRows step

The MixColumns step

The AddRoundKey

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Error

Rijndael Round - Partial MixColumns

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Julien Ugon 1 year, 6 months ago

Julien Ugon 1 year, 6 months ago

Julien Ugon 1 year, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: string

Parts

Gap-fill

Finding the next key

Applying AES key schedule to find the key

Computing $W(\var{4*(round)})$

Computing $W(\var{4*round+1})$ to $W(\var{4*(round)+3})$.

The new Key

The SubBytes step

The ShiftRows step

The MixColumns step

The AddRoundKey

Generated value: `string`

Computing $W(\var{4round+1})$ to $W(\var{4(round)+3})$.