Daira Emma Hopwood
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# Implementing AES with Lookups [Plookup](https://eprint.iacr.org/2020/315) is a modification of [PLONK](https://eprint.iacr.org/2019/953) to allow efficient table lookups in a circuit. We consider the cost of implementing the [AES](https://www.nist.gov/publications/advanced-encryption-standard-aes) block cipher using Plookup. ## Sparse representations Definition: The $b$-sparse representation of an $n$-bit integer $x = \sum\limits^{n-1}_{i=0} x_i 2^i$ is $\mathsf{sparse}_b(x) = \sum\limits^{n-1}_{i=0} x_i b^i$. Definition: A $k$-ary operation on integers $\odot$ is "sparse-friendly" if there exists $f_\odot$ such that $\bigodot\limits^{k-1}_{j=0} a_j = f_\odot\left(\sum\limits^{k}_{j=0} \mathsf{sparse}_{k+1}(a_j)\right)$. In other words, the result of $\odot$ depends only on the bitwise sum of the inputs. Fact: bitwise $k$-ary XOR is sparse-friendly. (As are [N]AND, [N]OR, and NXOR, but we won't need those.) This technique was invented by Vitalik Buterin. We'd considered $4$-sparse representation at the Boulder Zcash summit in November 2014, but discarded it then because there was no efficient way to do lookups in R1CS-based proof systems. ## AES round function The AES state is a $4$ by $4$ matrix with elements in $GF(2^8)$. A $GF(2^8)$ element is represented in polynomial basis $GF(2)[X]/(X^8 + X^4 + X^3 + X + 1)$, where the polynomial $p(X) = \sum\limits^{7}_{i=0} c_i X^i$ is typically identified with its integer evaluation $p(2) = \sum\limits^{7}_{i=0} c_i 2^i$. Following the AES specification, we use the notation $S_{i,j}$ for the state element at (zero-based) row $i$ and column $j$. The AES round function is defined as $\mathsf{AddRoundKey}_{RK} \circ \mathsf{MixColumns} \circ \mathsf{ShiftRows} \circ \mathsf{SubBytes}$: $\mathsf{SubBytes}$ — a non-linear substitution step where each byte is replaced with another according to a lookup function $\mathsf{SBox}$. $\mathsf{ShiftRows}$ — a transposition step where the last three rows of the state are shifted cyclically a certain number of steps. $\mathsf{MixColumns}$ — a linear mixing operation which operates on the columns of the state, combining the four bytes in each column. $\mathsf{AddRoundKey}_{RK}$ — each byte of the state is combined with a byte of the round key $RK$ using bitwise XOR (equivalently, polynomial addition). The $\mathsf{MixColumns}$ operation transforms each column $0 \leq j \leq 3$ as follows: $\begin{bmatrix} S'_{0,j} \\ S'_{1,j} \\ S'_{2,j} \\ S'_{3,j} \end{bmatrix} = \begin{bmatrix} 2 & 3 & 1 & 1 \\[0.7ex] 1 & 2 & 3 & 1 \\[0.7ex] 1 & 1 & 2 & 3 \\[0.7ex] 3 & 1 & 1 & 2 \end{bmatrix} \begin{bmatrix} S_{0,j} \\[0.5ex] S_{1,j} \\[0.5ex] S_{2,j} \\[0.5ex] S_{3,j} \end{bmatrix}$ (as polynomials, $2$ means $X$ and $3$ means $X + 1$). ## A possible implementation Since $\mathsf{ShiftRows}$ commutes with $\mathsf{SubBytes}$, we can write the round function as $\mathsf{AddRoundKey}_{RK} \circ \mathsf{MixColumns} \circ \mathsf{SubBytes} \circ \mathsf{ShiftRows}$. When each byte is represented separately (no matter what representation is used), $\mathsf{ShiftRows}$ is free, and so we are left with implementing $\mathsf{AddRoundKey}_{RK} \circ \mathsf{MixColumns} \circ \mathsf{SubBytes}$. Let $S^{in}_{i,j}$ be the input element at $(i, j)$, after applying $\mathsf{ShiftRows}$ to the state from the previous round. Suppose we use a $4$-sparse representation for each byte of the state. We will use $\underline{underlining}$ for values in $4$-sparse representation and functions that return such values. Let $\underline{f_{\oplus}}$ be the function that applies $(0, 1, 2, 3) \mapsto (0, 1, 0, 1)$ to each base-$4$ digit, and let $f_{\oplus}$ be the function that applies $\underline{f_{\oplus}}$ and then converts back to binary form (i.e. squashing out the zero bits). We can implement $3$-ary bitwise XOR as $x \oplus y \oplus z = f_{\oplus}(\underline{x} + \underline{y} + \underline{z})$, or $\underline{x \oplus y \oplus z} = \underline{f_{\oplus}}(\underline{x} + \underline{y} + \underline{z})$ to leave the result in $4$-sparse form. Let $\underline{F_z}(\underline{x}) = \mathsf{sparse}_4\big(z\;\mathsf{SBox}(f_{\oplus}(\underline{x}))\big)$ for $1 \leq z \leq 3$. In each round we can write $\mathsf{AddRoundKey}_{RK} \circ \mathsf{MixColumns} \circ \mathsf{SubBytes}$ as $S^{out}_{0,j}$ $= 2\;\mathsf{SBox}(S^{in}_{0,j}) \oplus 3\;\mathsf{SBox}(S^{in}_{1,j}) \oplus 1\;\mathsf{SBox}(S^{in}_{2,j}) \oplus 1\;\mathsf{SBox}(S^{in}_{3,j}) \oplus RK_{0,j}$ $S^{out}_{1,j}$ $= 1\;\mathsf{SBox}(S^{in}_{0,j}) \oplus 2\;\mathsf{SBox}(S^{in}_{1,j}) \oplus 3\;\mathsf{SBox}(S^{in}_{2,j}) \oplus 1\;\mathsf{SBox}(S^{in}_{3,j}) \oplus RK_{1,j}$ $S^{out}_{2,j}$ $= 1\;\mathsf{SBox}(S^{in}_{0,j}) \oplus 1\;\mathsf{SBox}(S^{in}_{1,j}) \oplus 2\;\mathsf{SBox}(S^{in}_{2,j}) \oplus 3\;\mathsf{SBox}(S^{in}_{3,j}) \oplus RK_{2,j}$ $S^{out}_{3,j}$ $= 3\;\mathsf{SBox}(S^{in}_{0,j}) \oplus 1\;\mathsf{SBox}(S^{in}_{1,j}) \oplus 1\;\mathsf{SBox}(S^{in}_{2,j}) \oplus 2\;\mathsf{SBox}(S^{in}_{3,j}) \oplus RK_{3,j}$ which in $4$-sparse form can be implemented as $\underline{S^{out}_{0,j}}$ $= \underline{f_{\oplus}}\Big(\underline{F_2}\big(\underline{S^{in}_{0,j}}\big) + \underline{F_3}\big(\underline{S^{in}_{1,j}}\big) + \underline{F_1}\big(\underline{S^{in}_{2,j}}\big)\Big) + \underline{F_1}\big(\underline{S^{in}_{3,j}}\big) + \underline{RK_{0,j}}$ $\underline{S^{out}_{1,j}}$ $= \underline{f_{\oplus}}\Big(\underline{F_1}\big(\underline{S^{in}_{0,j}}\big) + \underline{F_2}\big(\underline{S^{in}_{1,j}}\big) + \underline{F_3}\big(\underline{S^{in}_{2,j}}\big)\Big) + \underline{F_1}\big(\underline{S^{in}_{3,j}}\big) + \underline{RK_{1,j}}$ $\underline{S^{out}_{2,j}}$ $= \underline{f_{\oplus}}\Big(\underline{F_1}\big(\underline{S^{in}_{0,j}}\big) + \underline{F_1}\big(\underline{S^{in}_{1,j}}\big) + \underline{F_2}\big(\underline{S^{in}_{2,j}}\big)\Big) + \underline{F_3}\big(\underline{S^{in}_{3,j}}\big) + \underline{RK_{2,j}}$ $\underline{S^{out}_{3,j}}$ $= \underline{f_{\oplus}}\Big(\underline{F_3}\big(\underline{S^{in}_{0,j}}\big) + \underline{F_1}\big(\underline{S^{in}_{1,j}}\big) + \underline{F_1}\big(\underline{S^{in}_{2,j}}\big)\Big) + \underline{F_2}\big(\underline{S^{in}_{3,j}}\big) + \underline{RK_{3,j}}$ Note that there are two uses of each $\underline{F_1}(\underline{S^{in}_{i,j}})$ term, so there are $12$ unique $F_z$ lookups here, not $16$. So, in total we have $64$ lookups per round ($4$ $\underline{f_{\oplus}}$ and $12$ $\underline{F_z}$ per column). The $S^{out}_{i,j}$ are then permuted by $\mathsf{ShiftRows}$ to become the $S^{in}_{i,j}$ of the following round. Before the first round, we need $16$ lookups to convert each byte to $4$-sparse form. At that point we add the initial round key before computing the first round function as above. After the last round, we need $16$ more $f_{\oplus}$ lookups to convert back to binary form. So, * AES-128 (with $10$ rounds) requires $672$ lookups. * AES-192 (with $12$ rounds) requires $800$ lookups. * AES-256 (with $14$ rounds) requires $928$ lookups. We need five tables $(f_{\oplus}, \underline{f_{\oplus}}, \underline{F_1}, \underline{F_2}, \underline{F_3})$, each with $4^8 = 2^{16}$ entries, and one table $\mathsf{sparse}_4$ with $2^8$ entries. This does not include computing round keys or the cost of additions. ## With fewer large tables Define $G(\underline{x}) = \mathsf{SBox}(f_{\oplus}(\underline{x}))$ and $\underline{F'_z}(x) = \mathsf{sparse}_4(z x)$ for $1 \leq z \leq 3$. Then we have $\underline{F_z}(\underline{x}) = \underline{F'_z}(G(\underline{x}))$ for $1 \leq z \leq 3$, which we substitute in the round function above. This doubles the number of lookups needed for $\underline{F_z}$, so we have $112$ lookups per round or $1152$ in total for AES-128. But we only need three $2^{16}$-sized tables ($f_{\oplus}$, $\underline{f_{\oplus}}$, and $G$) rather than five. ## With smaller ($3^8$-sized) tables We could use $3$-sparse representations throughout. This means that we can only use binary XOR rather than $3$-ary XOR before normalizing. For each byte we have, for example $\underline{S^{out}_{0,j}}$ $= \underline{f_{\oplus}}\Bigg(\underline{f_{\oplus}}\bigg(\underline{f_{\oplus}}\Big(\underline{F_2}\big(\underline{S^{in}_{0,j}}\big) + \underline{F_3}\big(\underline{S^{in}_{1,j}}\big)\Big) + \underline{F_1}\big(\underline{S^{in}_{2,j}}\big)\bigg) + \underline{F_1}\big(\underline{S^{in}_{3,j}}\big)\Bigg) + \underline{RK_{0,j}}$ So, we need $12$ $\underline{f_{\oplus}}$ lookups and $12$ $\underline{F_z}$ lookups per column, i.e. $96$ lookups per round. We still need $16$ lookups to convert to $3$-sparse form before the first round, and $16$ lookups to convert back to binary form after the last round. So in this variant, * AES-128 (with $10$ rounds) requires $992$ lookups. * AES-192 (with $12$ rounds) requires $1184$ lookups. * AES-256 (with $14$ rounds) requires $1376$ lookups. This has smaller overall table size (five tables of size $3^8 = 6561$ and one of size $2^8$), and better performance than the "fewer large tables" variant above, so the "fewer large tables" variant can be discarded. ## Other Rijndael block sizes Rijndael also supports $192$-bit and $256$-bit blocks. This is achieved by increasing the number of columns to $6$ or $8$ in a straightforward way. The number of lookups scales linearly with the block size. The resulting lookup counts for $4$-sparse representation are: \begin{array}{|c|ccc|} \hline & & \text{Block size} & \\ \text{Rounds} & 128 & 192 & 256 \\ \hline 10 & 672 & 1008 & 1344 \\ 12 & 800 & 1200 & 1600 \\ 14 & 928 & 1392 & 1856 \\ \hline \end{array} and for $3$-sparse representation: \begin{array}{|c|ccc|} \hline & & \text{Block size} & \\ \text{Rounds} & 128 & 192 & 256 \\ \hline 10 & 992 & 1488 & 1984 \\ 12 & 1184 & 1776 & 2368 \\ 14 & 1376 & 2064 & 2752 \\ \hline \end{array}

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