Computer Aided Design Projects

• Computer Aided Design Projects
  • Introduction
  • CA1: Permutation
  • Midterm: ColParity
  • CA2: Encoder
    • ColParity
    • Rotate
    • Permute
    • Revaluate
    • AddRc

Introduction

In all of the projects, a $5\times 5\times 64$ matrix is encoded to another $5\times 5\times 64$ matrix using different methods.
The encodings are implemented completely in hardware.
The $(i,\ j)$ indices for a matrix slice are as follows:

(3, 2)	(4, 2)	(0, 2)	(1, 2)	(2, 2)
(3, 1)	(4, 1)	(0, 1)	(1, 1)	(2, 1)
(3, 0)	(4, 0)	(0, 0)	(1, 0)	(2, 0)
(3, 4)	(4, 4)	(0, 4)	(1, 4)	(2, 4)
(3, 3)	(4, 3)	(0, 3)	(1, 3)	(2, 3)

CA1: Permutation

In this part, the encoding is done using the following formula:

$$\forall i,\ j \in [0,4]\ \forall k \in [0, 63]: a[i][j][k] = a[j][(2i+3j)\ %\ 5][k]$$

This means that in each slice, the bits are mapped to a different location in the slice.

Midterm: ColParity

In this part, the encoding is done using the following formula:

$$\forall i,\ j \in [0,4]\ \forall k \in [0, 63]: a[i][j][k] = a[i][j][k] \oplus a[(i-1)\ %\ 5][0..4][k] \oplus a[(i+1)\ %\ 5][0..4][(k+1)\ %\ 64]$$

This means that each bit of the matrix becomes the XOR of itself, the bit's left column, and its right column in the previous slice.

CA2: Encoder

Here, 5 different encodings are applied serially to the input 24 times.

ColParity

This part is the same function implemented in Midterm.

Rotate

The following formula is used for encoding:

$$\forall i,\ j \in [0,4]\ \forall k \in [0, 63]: a[i][j][k] = a[i][j][(k-\frac{(t+1)(t+2)}{2})\ %\ 64]$$

The number $t$ starts from 0 and goes up to 23. On each increment, an $(x,\ y)$ pair is calculated using the following formula:

$$\begin{pmatrix} 0 & 1 \ 2 & 3\end{pmatrix}^t\begin{pmatrix} 1 \ 0\end{pmatrix} \mod 5 = \begin{pmatrix} x \ y\end{pmatrix}$$

The $(x,\ y)$ pair shows which lane of the matrix should be shifted $\frac{(t+1)(t+2)}{2}$ times.

Permute

This part is the same function implemented in CA1.

Revaluate

The following formula is used for encoding:

$$\forall i,\ j \in [0,4]\ \forall k \in [0, 63]: a[i][j][k] = a[i][j][k] \oplus (\lnot a[(i+1)\ %\ 5][j][k]\ &\ a[(i+2)\ %\ 5][j][k])$$

AddRc

The following formula is used for encoding:

$$\forall k \in [0,\ 63]: a[0][0][k] = a[0][0][k] \oplus RC[t][k]$$

Where the $RC$ values are presented in the following table:

RC[0]	RC[1]	RC[2]	RC[3]
0x0000000000000001	0x0000000000008082	0x800000000000808A	0x8000000080008000
RC[4]	RC[5]	RC[6]	RC[7]
0x000000000000808B	0x0000000080000001	0x8000000080008081	0x8000000000008009
RC[8]	RC[9]	RC[10]	RC[11]
0x000000000000008A	0x0000000000000088	0x0000000080008009	0x000000008000000A
RC[12]	RC[13]	RC[14]	RC[15]
0x000000008000808B	0x800000000000008B	0x8000000000008089	0x8000000000008003
RC[16]	RC[17]	RC[18]	RC[19]
0x8000000000008002	0x8000000000000080	0x000000000000800A	0x800000008000000A
RC[20]	RC[21]	RC[22]	RC[23]
0x8000000080008081	0x8000000000008080	0x0000000080000001	0x8000000080008008

And $t$ is the iteration number where the function is being executed. (the encoder module executes the 5 functions 24 times)