From a681a053f7da13eea143a1b7b679f674b6be8119 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 6 Jan 2026 14:50:10 +0100 Subject: [PATCH] [SPCA] Integers --- .../{05_integers.c => 05_pointers.c} | 0 .../00_c/00_basics/06_pointers.c | 0 .../spca/parts/00_c/01_basics/06_integers.tex | 32 +++++++++++++++++- .../spca/parts/00_c/01_basics/07_pointers.tex | 1 + semester3/spca/spca-summary.pdf | Bin 407312 -> 5499 bytes 5 files changed, 32 insertions(+), 1 deletion(-) rename semester3/spca/code-examples/00_c/00_basics/{05_integers.c => 05_pointers.c} (100%) delete mode 100644 semester3/spca/code-examples/00_c/00_basics/06_pointers.c diff --git a/semester3/spca/code-examples/00_c/00_basics/05_integers.c b/semester3/spca/code-examples/00_c/00_basics/05_pointers.c similarity index 100% rename from semester3/spca/code-examples/00_c/00_basics/05_integers.c rename to semester3/spca/code-examples/00_c/00_basics/05_pointers.c diff --git a/semester3/spca/code-examples/00_c/00_basics/06_pointers.c b/semester3/spca/code-examples/00_c/00_basics/06_pointers.c deleted file mode 100644 index e69de29..0000000 diff --git a/semester3/spca/parts/00_c/01_basics/06_integers.tex b/semester3/spca/parts/00_c/01_basics/06_integers.tex index be56d02..9524cd8 100644 --- a/semester3/spca/parts/00_c/01_basics/06_integers.tex +++ b/semester3/spca/parts/00_c/01_basics/06_integers.tex @@ -4,6 +4,36 @@ As a reminder, integers are encoded as follows in big endian notation, with $x_i \item \bi{Unsigned}: $\displaystyle \sum_{i = 0}^{w - 1} x_i \cdot 2^i$ \item \bi{Signed}: $\displaystyle -x_{w - 1} \cdot 2^{w - 1} + \sum_{i = 0}^{w - 1} x_i \cdot 2^i$ (two's complement notation, with $x_{w - 1}$ being the sign-bit) \end{itemize} -The minimum number representable is $0$ and $-2^{w - 1}$, respectively, whereas the maximum number representable is $2^w - 1$ and $2^{w - 1} - 1$ +The minimum number representable is $0$ and $-2^{w - 1}$, respectively, whereas the maximum number representable is $2^w - 1$ and $2^{w - 1} - 1$. +\verb|limits.h| defines constants for the minimum and maximum values of different types, e.g. \verb|ULONG_MAX| or \verb|LONG_MAX| and \verb|LONG_MIN| We can use the shift operators to multiply and divide by two. Shift operations are usually \textit{much} cheaper than multiplication and division. +Left shift (\texttt{u << k} in \lC) always fills with zeros and throws away the extra bits on the left (equivalent to multiplication by $2^k$), +whereas right shift (\texttt{u >> k} in \lC) is implementation-defined, +either arithmetic (fill with most significant bit, division by $2^k$. This however rounds incorrectly, see below) +or logical shift (fill with zeros, unsigned division by $2^k$). + +Signed division using arithmetic right shifts has the issue of incorrect rounding when number is $< 0$. +Instead, we represent $s / 2^k = s + (2^k - 1) \texttt{ >> } k$ for $s < 0$ and $s / 2^k = s >> k$ for $s > 0$ + +\bi{In expressions, signed values are implicitly cast to unsigned} + +This can lead to all sorts of nasty exploits (e.g. provide $-1$ as the argument to \texttt{memcpy} and watch it burn, this was an actual exploit in FreeBSD) + +\fhlc{Cyan}{Addition \& Subtraction} + +A nice property of the two's complement notation is that addition and subtraction works exactly the same as in normal notation, due to over- and underflow. +This also obviously means that it implements modular arithmetic, i.e. +\mrmvspace +\begin{align*} + \texttt{Add}_w (u, v) = u + v \text{ mod } 2^w \ \text{ and } \ \texttt{Sub}_w (u, v) = u - v \text{ mod } 2^w +\end{align*} + +\mrmvspace +\fhlc{Cyan}{Multiplication \& Division} + +Unsigned multiplication with addition forms a commutative ring. +Again, it is doing modular arithmetic and +\begin{align*} + \texttt{UMult}_w (u, v) = u \cdot v \text{ mod } 2^w +\end{align*} diff --git a/semester3/spca/parts/00_c/01_basics/07_pointers.tex b/semester3/spca/parts/00_c/01_basics/07_pointers.tex index e69de29..ad65d30 100644 --- a/semester3/spca/parts/00_c/01_basics/07_pointers.tex +++ b/semester3/spca/parts/00_c/01_basics/07_pointers.tex @@ -0,0 +1 @@ +\subsubsection{Pointers} diff --git a/semester3/spca/spca-summary.pdf b/semester3/spca/spca-summary.pdf index 9777a2ef522268857f6b411ace2cdfa6114e0c54..238fc539433e3f15a846cafed346eb70a8fb1286 100644 GIT binary patch delta 8 PcmbR6PU5%LhH6m&6hs63 literal 407312 zcmce+V~{A@vNhVaZQHhO+qPHR?$x$!+xBYPwrzKRd*64yi2LH5hZIvv;zEq8AdfcPG$h_{*M=;eSqDDEhx={^u!1D0*2#XD7?QEExZ?_-p>}>t7a1 zroZf6oQzF>5wQL>q69@RV`^vaY(c=tO7O4auT?B7gBEj@`(EpKiVq*U9Iq^3L{xeRn{r8;s 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