doc: add GF(2^8) context

doc/whirlpool.tex

@@ -16,10 +16,67 @@ substitution, a column shift, a row mixing, and a round key addition.
 
 \vspace{1em}
 
-The padding scheme follows the same structure as MD5 and SHA-256: a single
+The padding scheme appends a single \texttt{1} bit, followed by \texttt{0}
-\texttt{1} bit is appended, followed by \texttt{0} bits until the message
+bits until the message length is congruent to 256 bits modulo 512. The original
-length is congruent to 448 bits modulo 512. The original message length in bits
+message length in bits is then appended as a 256-bit big-endian integer,
-is then appended as a 64-bit big-endian integer.
+bringing the total padded length to an exact multiple of 512 bits. Whirlpool
+uses a 256-bit length field (rather than the 64-bit field of MD5 and SHA-256)
+to support messages up to $2^{256} - 1$ bits in length.
+
+\vspace{1em}
+
+\textbf{Arithmetic in $\mathrm{GF}(2^8)$.}
+A \textbf{finite field} (or Galois field) is a finite set in which addition,
+subtraction, multiplication and division (by any non-zero element) are all
+well-defined and satisfy the usual algebraic laws. The simplest example is
+$\mathrm{GF}(2) = \{0, 1\}$, where addition is XOR and multiplication is AND.
+
+\vspace{1em}
+
+$\mathrm{GF}(2^8)$ extends this to 256 elements by representing each element
+as a polynomial of degree less than 8 with coefficients in $\{0, 1\}$. A byte
+$b_7 b_6 \cdots b_0$ encodes the polynomial
+$b_7 x^7 + b_6 x^6 + \cdots + b_0$; for example,
+\texttt{0b10100011} $= x^7 + x^5 + x + 1$.
+
+\vspace{1em}
+
+This construction is analogous to modular arithmetic: just as
+$\mathbb{Z}/p\mathbb{Z}$ is a field because $p$ is prime, the set of
+polynomials with binary coefficients forms a field when reduced modulo an
+\emph{irreducible} polynomial --- one that cannot be factored. Without this
+reduction, multiplying two polynomials could produce a degree greater than 7,
+stepping outside the 256-element set. Reducing modulo an irreducible polynomial
+of degree 8 keeps every result within one byte, and guarantees that every
+non-zero element has a multiplicative inverse.
+
+\vspace{1em}
+
+\textbf{Addition} is coefficient-wise addition modulo 2, equivalent to a
+bitwise XOR (there is no carry: $1 + 1 = 0$):
+\begin{align*}
+a + b = a \oplus b
+\end{align*}
+
+\textbf{Multiplication by $x$} (called \emph{xtime}) is a left shift by one
+bit, followed by a conditional reduction: if the original high bit was 1, the
+degree of the result would reach 8 and must be reduced modulo the irreducible
+polynomial $p(x) = x^8 + x^4 + x^3 + x^2 + 1$ (\texttt{0x11d}), which means
+XORing with its low byte \texttt{0x1d}:
+\begin{align*}
+\mathrm{xtime}(a) =
+\begin{cases}
+a \ll 1 & \text{if } b_7 = 0 \\
+(a \ll 1) \oplus \texttt{0x1d} & \text{if } b_7 = 1
+\end{cases}
+\end{align*}
+
+\textbf{Multiplication by an arbitrary element} decomposes the multiplier into
+powers of 2, applies xtime repeatedly for each power, then combines the results
+with XOR. For example, multiplying by \texttt{0x05} $= x^2 + 1$:
+\begin{align*}
+\texttt{0x05} \cdot a = \mathrm{xtime}(\mathrm{xtime}(a)) \oplus a
+\end{align*}
 
 \newpage
 
@@ -44,27 +101,28 @@ b_{i,j} = a_{i',\ j} \quad \text{where } i' = (i - j) \bmod 8
 
 \medskip
 
-\textbf{MixRows} multiplies each row of the state matrix by a fixed MDS matrix
+\textbf{MixRows} multiplies each row of the state matrix by a fixed circulant
-over $\mathrm{GF}(2^8)$ with irreducible polynomial $x^8 + x^4 + x^3 + x^2 +
+MDS matrix over $\mathrm{GF}(2^8)$ with irreducible polynomial
-1$, providing diffusion across the eight bytes of each row. Formally, for each
+$x^8 + x^4 + x^3 + x^2 + 1$, providing diffusion across the eight bytes of
-row $i$, each output byte $b_j$ is computed as:
+each row. The MDS matrix is fully determined by its first row
+$(c_0,\ c_1,\ \ldots,\ c_7)$; the entry at row $j$, column $k$ equals
+$c_{(j-k) \bmod 8}$. Formally, for each row $i$, each output byte $b_{i,j}$
+is computed as:
 
 \begin{align*}
-b_j = \bigoplus_{k=0}^{7} \mathrm{MDS}[(j - k) \bmod 8] \cdot a_{i,k}
+b_{i,j} = \bigoplus_{k=0}^{7} c_{(j-k) \bmod 8} \cdot a_{i,k}
 \end{align*}
 
 \noindent where $\cdot$ denotes multiplication in $\mathrm{GF}(2^8)$ and $\oplus$ denotes XOR.
 
 \medskip
 
-\textbf{AddRoundKey} XORs the state with the current round key.
+\textbf{AddRoundKey} XORs the state with the current round key:
 
 \begin{align*}
-	%TODO:  \forall i \in \mathbb{N},\ 0 \leq i < 8, 
+S \leftarrow S \oplus K[r]
 \end{align*}
 
-\newpage
-
 The S-box and the MDS matrix coefficients are fixed tables defined in the
 Whirlpool specification; their values are too large to reproduce here. The
 round constants $\mathrm{RC}[r]$, $r \in \mathbb{N},\ 1 \leq r \leq 10$, are