Lecture（Ⅱ）：Bits, Bytes, and Integers

p8-p10	Boolean Algebra ---And、Or、Not、Exclusive-Or General Boolean Algebras ---Operate on Bit Vectors & \| ^ ~ ---All of the Proper6es of Boolean Algebra Apply Example: Represen6ng & Manipulating Sets ---Representa6on ---Operations
p11	Bit-Level Operations in C ---Operations &, \|, ~, ^ Available in C ---Examples (Char data type)
p12-p13	Contrast: Logic Opera6ons in C ---Contrast to Logical Operators ---Examples (char data type) &&, \|\|, !
p14	Shift Operations ---Left Shift: x << y ---Right Shift: x >> y ---Undefined Behavior

3.Integers

(1)Representation: unsigned and signed

p16	Encoding Integers ---Unsigned、Two's Complement 注：有很多表示signed的方法，这门课我们只关注补码形式 ---C short 2 bytes long ---Sign Bit
p17	Two-complement Encoding Example (Cont.)
p18	Numeric Ranges ---Unsigned Values ---Two’s Complement Values ---Other Values
p19	Values for Different Word Sizes ---Observa6ons ---C Programming
p20	Unsigned & Signed Numeric Values ---Equivalence ---Uniqueness ---Can Invert Mappings

(2)Conversion, casting

p22	Mapping Between Signed & Unsigned ---Keep bit representations and reinterpret
p23-p24	Mapping Signed ↔ Unsigned
p26	Conversion Visualized ---2’s Comp. → Unsigned
p27	Signed vs. Unsigned in C ---Constants ---Casting
p28	Casting Surprises ---Expression Evalua6on
p29	Summary:Casting Signed ↔ Unsigned: Basic Rules --Bit paiern is maintained --But reinterpreted --Can have unexpected effects: adding or subtrac6ng 2w --Expression containing signed and unsigned int

(3)Expanding, truncating

p31

Sign Extension
---Task

---Rule

p32

Sign Extension Example
---Converting from smaller to larger integer data type

---C automatically performs sign extension

p33

Summary:Expanding, Truncating: Basic Rules
---Expanding (e.g., short int to int)

---Truncating (e.g., unsigned to unsigned short)

(4)Addition, negation, multiplication, shifting

p35	Unsigned Addition ---Standard Addition Function ---Implements Modular Arithmetic
p36	Visualizing (Mathematical) Integer Addi6on ---Integer Addition
p37	Visualizing Unsigned Addition ---Wraps Around 注：overflow溢出舍弃最高位
p38	Two’s Complement Addition ---TAdd and UAdd have Identical Bit-Level Behavior
p39	TAdd Overflow ---Functionality
p40	Visualizing 2’s Complement Addition ---Values ---Wraps Around
p41	Multiplication ---Goal: Computing Product of w-bit numbers x, y ---But, exact results can be bigger than w bits ---So, maintaining exact results…
p42	Unsigned Multiplication in C ---Standard Multiplication Function ---Implements Modular Arithmetic
p43	Signed Multiplication in C ---Standard Multiplication Func6on
p44	Power-of-2 Multiply with Shift ---Operation ---Examples
p45	Unsigned Power-of-2 Divide with Shift --Quotient of Unsigned by Power of 2

(5)Summary

p47	Arithmetic: Basic Rules --Addition: --Multiplication: 溢出分析
p48	Why Should I Use Unsigned? ---Don’t use without understanding implications
p49	Counting Down with Unsigned ---Proper way to use unsigned as loop index ---See Robert Seacord, Secure Coding in C and C++ ---Even better
p50	Why Should I Use Unsigned? (cont.) ---Do Use When Performing Modular Arithmetic ---Do Use When Using Bits to Represent Sets

4.Representations in memory, pointers, strings

p52	Byte-Oriented Memory Organization ---Programs refer to data by address ---Note: system provides private address spaces to each “process” 编程思想
p53	Machine Words ---Any given computer has a “Word Size”
p54	Word-Oriented Memory Organization ---Addresses Specify Byte Locations
p55	Example Data Representa6ons
p56	Byte Ordering ---So, how are the bytes within a multi-byte word ordered in memory? ---Conventions 大端（Big Endian0）、小端（Little Endian）
p57	Byte Ordering Example --Example
p58	Representing Integers
p59	Examining Data Representations --Code to Print Byte Representation of Data
p60	show_bytes Executition Example
p61	Representing Pointers
p62	Representing Strings ---Strings in C ---Compatibility
p63	Integer C Puzzles 后面的为Bonus

p65	Application of Boolean Algebra --Applied to Digital Systems by Claude Shannon
p66	Binary Number Property --w = 0: --Assume true for w-1:
p67	Code Security Example ---Similar to code found in FreeBSD’s implementation of getpeername ---There are legions of smart people trying to find vulnerabili6es in programs
p68	Typical Usage
p69	Malicious Usage
p70	Mathematical Properties ---Modular Addition Forms an Abelian Group
p71	Mathematical Properties of TAdd ---Isomorphic Group to unsigneds with UAdd ---Two’s Complement Under TAdd Forms a Group
p72	Characterizing TAdd ---Functionality
p73	Negation: Complement & Increment ---Claim: Following Holds for 2’s Complement ---Complement ---Complete Proof?
p74	Complement & Increment Examples
p75	Code Security Example #2 ---SUN XDR library
p76	XDR Code
p77	XDR Vulnerability ---What if: ---How can I make this function secure?
p78	Compiled Multiplication Code --C compiler automa6cally generates shift/add code when multoplying by constant
p79	Compiled Unsigned Division Code ---Uses logical shift for unsigned ---For Java Users
p80	Signed Power-of-2 Divide with Shift ---Quo6ent of Signed by Power of 2
p81	Correct Power-of-2 Divide ---Quotient of Negative Number by Power of 2
p82	Correct Power-of-2 Divide (Cont.)
p83	Compiled Signed Division Code
p84	Arithmetic: Basic Rules ---Unsigned ints, 2’s complement ints are isomorphic rings: isomorphism = cas6ng ---Left shift ---Right shift
p85	Properties of Unsigned Arithmetic ---Unsigned Multiplication with Addi6on Forms Commutative Ring
p86	Properties of Two’s Comp. Arithmetic ---Isomorphic Algebras ---Both Form Rings ---Comparison to (Mathema6cal) Integer Arithmetic
p87	Reading Byte-Reversed Listings ---Disassembly ---Example Fragment ---Deciphering Numbers