Today: Bits, Bytes, and Integers. Bits, Bytes, and Integers. For example, can count in binary. Everything is bits. Encoding Byte Values

Transcription

1 Today: Bits, Bytes, and Integers Representing information as bits Bits, Bytes, and Integers CSci : Machine Architectre and Organization September th-9th, Yor instrctor: Stephen McCamant Bit-level maniplations Integers Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Smmary Based on slides originally by: Randy Bryant, Dave O Hallaron Representations in memory, pointers, strings Everything is bits For example, can cont in binary Each bit is or By encoding/interpreting sets of bits in varios ways Compters determine what to do (instrctions) and represent and maniplate nmbers, sets, strings, etc Why bits? Electronic Implementation Easy to store with bistable elements Reliably transmitted on noisy and inaccrate wires Base Nmber Representation Represent 53 as Represent. as.[] Represent.53 X as. X 3.V.9V.V.V 3 Encoding Byte Vales Aside: ASCII table Byte bits Binary to Decimal: to 55 Hexadecimal to FF Base nmber representation Use characters to 9 and A to F Write FAD37B in C as xfad37b xfad37b A B C D 3 E F a b c d e f x_ \ ^A ^B ^C ^D ^E ^F ^G ^H \t \n ^K ^L ^M ^N ^O x_ ^P ^Q ^R ^S ^T ^U ^V ^W ^X ^Y ^Z ESC FS GS RS US x_ SPC! " # $ % & ' ( ) * +, -. / x3_ : ; < >? A B C D E F G H I J K L M N O x5_ P Q R S T U V W X Y Z [ \ ] ^ _ x_ ` a b c d e f g h I j k l m n o x7_ p q r s t v w x y z { ~ DEL 5

2 Example Data Representations C Data Type Typical 3-bit Typical -bit Today: Bits, Bytes, and Integers x- char short int long float doble long doble / pointer 7 Boolean Algebra Developed by George Boole in 9th Centry Or (math: ) A&B when both A and B A B when either A or B ~A when A 9 ^ ~ All of the properties of Boolean algebra apply Width w bit vector represents sbsets of {,, w aj if j A 753 Bit-Level Operations in C Representation & A^B when either A or B, bt not both Example: Representing & Maniplating Sets Operate on bit vectors Exclsive-Or xor (math: ) Operations applied bitwise Encode Tre as and False as Not (math: ) Representations in memory, pointers, strings General Boolean Algebras And (math: ) Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Smmary Algebraic representation of logic Representing information as bits Bit-level maniplations Integers Operations &,, ~, ^ Available in C Apply to any integral data type long, int, short, char, nsigned View argments as bit vectors Argments applied bit-wise {, 3, 5, Examples (Char data type) ~x xbe 753 {,,, Operations & ^ ~ Intersection Union Symmetric difference Complement ~ ~x xff ~ x9 & x55 x {, {,, 3,, 5, {, 3,, 5 {, 3, 5, 7 & x9 x55 x7d

3 Contrast: Logic Operations in C Shift Operations Contrast to Logical Operators Left Shift: x << y View as False Anything nonzero as Tre Always retrn or Early termination (AKA short-circit evalation ) Throw away extra bits on left Fill with s on right Watch ot for && vs. & (and vs. ) Examples (char data type) of the more common oopsies in!x one x!x C x programming Arith. >> Right Shift: x >> y Argment x Throw away extra bits on right Logical shift: fill with s on left Arithmetic shift: replicate most significant bit on left << 3 Log. >> Arith. >> Undefined Behavior Shift amont < or word size Signed shift into or ot of sign bit (i.e., arith. behavior not assred) (avoids nll pointer access) << 3 Log. >> Shift bit-vector x right y positions!!x x x9 && x55 x x9 x55 x p && *p Argment x Shift bit-vector x left y positions &&,,! 3 Interlde: ChimeIn Today: Bits, Bytes, and Integers Which of the following nmbers represented in hex is not a mltiple of? x73c xfd5f9e xca9 x97bd9 xac3f97 Representing information as bits Bit-level maniplations Integers Idea: it is enogh to look at the last digit Like divisibility by, and 5 for decimal x, x, x, and xc decimal are mltiples of xe is even bt not a mltiple of 5 Binary Nmber Property Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Smmary Representations in memory, pointers, strings Smmary Encoding Integers Claim Unsigned BU(X ) w- w w- + å i w w : x y w- + w w + w w+ BT (X ) xw Decimal Hex 3B D C 93 w w xi i i Sign Bit Binary Sign Bit For s complement, most significant bit indicates sign w Two s Complement i C short bytes long Assme tre for w-: xi short int x 53; short int y -53; w i i for nonnegative for negative 3

4 For-bit Example, nsigned For-bit Example, sign + magnitde Unsigned: 5 This approach can represent throgh 5 Sign + magnitde: Negate if -7 This approach can represent -7 throgh 7 Disadvantages: special cases, - 3 For-bit Example, two s complement For-bit Example Two s complement: - - Unsigned: 5 Two s complement: - This approach can represent - throgh 7-5 Encoding Integers Unsigned BU(X) w x i i i short int x 53; short int y -53; C short bytes long Sign Bit For s complement, most significant bit indicates sign for nonnegative for negative Two s Complement BT(X) x w w w x i i i Decimal Hex Binary x 53 3B D y -53 C 93 Sign Bit 7 Two-complement Encoding Example (Cont.) x 53: y -53: Weight Sm 53-53

5 Nmeric Ranges Unsigned Vales UMin UMax w Vales for W Two s Complement Vales TMin w TMax w Other Vales Mins Decimal Hex Binary UMax 5535 FF FF TMax 377 7F FF TMin FF FF Vales for Different Word Sizes W 3 UMax 55 5,535,9,97,95,,7,73,79,55,5 TMax 7 3,77,7,3,7 9,3,37,3,5,775,7 TMin - -3,7 -,7,3, -9,3,37,3,5,775, Observations TMin TMax + Asymmetric range UMax * TMax + C Programming #inclde <limits.h> Declares constants, e.g., ULONG_MAX LONG_MAX LONG_MIN Vales platform specific 9 3 Unsigned & Signed Nmeric Vales X BU(X) BT(X) Eqivalence Same encodings for nonnegative vales Uniqeness Every bit pattern represents niqe integer vale Each representable integer has niqe bit encoding Can Invert Mappings UB(x) BU - (x) Bit pattern for nsigned integer TB(x) BT - (x) Bit pattern for two s comp integer Today: Bits, Bytes, and Integers Representing information as bits Bit-level maniplations Integers Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Smmary Representations in memory, pointers, strings 3 3 Mapping Between Signed & Unsigned Two s Complement x Unsigned x TU TB X BU Maintain Same Bit Pattern UT UB X BT Maintain Same Bit Pattern Unsigned x Two s Complement x Mappings between nsigned and two s complement nmbers: Keep bit representations and reinterpret Mapping Signed Unsigned Bits Signed TU UT Unsigned

6 Mapping Signed Unsigned Bits Signed Unsigned 3 Relation between Signed & Unsigned Two s Complement 3 TB X x BU Maintain Same Bit Pattern w x x /- Large negative weight becomes Large positive weight 3 37 Signed vs. Unsigned in C Conversion Visalized Unsigned TU x s Comp. Unsigned UMax UMax Ordering Inversion Negative Big Positive Constants By defalt are considered to be signed integers Unsigned if have U as sffix U, 99759U TMax + TMax TMax s Complement Range Unsigned Range Casting Explicit casting between signed & nsigned same as UT and TU int tx, ty; nsigned x, y; tx (int) x; y (nsigned) ty; Implicit casting also occrs via assignments and procedre calls tx x; y ty; TMin 3 Casting and Comparison Srprises Casting and Comparison Srprises Expression Evalation Expression Evalation If there is a mix of nsigned and signed in single expression, If there is a mix of nsigned and signed in single expression, signed vales implicitly cast to nsigned Inclding comparison operations <, >,, <, > Examples for W 3: TMIN -,7,3,, TMAX,7,3,7 Inclding comparison operations <, >,, <, > Examples for W 3: TMIN -,7,3,, TMAX,7,3,7 Constant Constant Relation Evalation U U U U nsigned < signed > nsigned signed vales implicitly cast to nsigned Constant U 737U - - (nsigned) - (nsigned) Constant U U U U U 73U (int) 73U 73U (int) Relation Evalation nsigned < signed > > nsigned signed < nsigned > > signed nsigned < nsigned > signed

7 Brief annoncements Abot week left to go on HA Keep yor qestions coming, don t pt it off My office hors tomorrow will move to 3-pm Still in -5E Keller Smmary Casting Signed Unsigned: Basic Rles Bit pattern is maintained Bt reinterpreted Can have nexpected effects: adding or sbtracting w Expression containing signed and nsigned int int is cast to nsigned!! 3 Typical Usage Malicios Usage /* Declaration of library fnction memcpy */ void *memcpy(void *dest, void *src, size_t n); typedef nsigned long size_t; /* Kernel memory region holding ser-accessible data */ #define KSIZE char kbf[ksize]; /* Kernel memory region holding ser-accessible data */ #define KSIZE char kbf[ksize]; /* Copy at most maxlen bytes from kernel region to ser bffer */ int copy_from_kernel(void *ser_dest, int maxlen) { /* Byte cont len is minimm of bffer size and maxlen */ int len KSIZE < maxlen? KSIZE : maxlen; memcpy(ser_dest, kbf, len); retrn len; /* Copy at most maxlen bytes from kernel region to ser bffer */ int copy_from_kernel(void *ser_dest, int maxlen) { /* Byte cont len is minimm of bffer size and maxlen */ int len KSIZE < maxlen? KSIZE : maxlen; memcpy(ser_dest, kbf, len); retrn len; #define MSIZE 5 #define MSIZE 5 void getstff() { char mybf[msize]; copy_from_kernel(mybf, MSIZE); printf( %s\n, mybf); void getstff() { char mybf[msize]; copy_from_kernel(mybf, -MSIZE);... 5 Today: Bits, Bytes, and Integers Sign Extension Representing information as bits Bit-level maniplations Integers Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Smmary Representations in memory, pointers, strings Task: Given w-bit signed integer x Convert it to w+k-bit integer with same vale Rle: Make k copies of sign bit: X x w,, x w, x w, x w,, x k copies of MSB X w 7 X k w 7

8 Sign Extension Example Smmary: Expanding, Trncating: Basic Rles short int x 53; int ix (int) x; short int y -53; int iy (int) y; Decimal x ix y iy Hex 3B 3B C FF FF C Binary D D Unsigned: zeros added Signed: sign extension Both yield expected reslt Trncating (e.g., nsigned to nsigned short) Converting from smaller to larger integer data type C atomatically performs sign extension 9 UAddw(, v) Tre Sm: w+ bits Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Discard Carry: w bits Standard Addition Fnction Implements Modlar Arithmetic s 5 UAddw(, v) + v mod w 5 Visalizing Unsigned Addition Visalizing (Mathematical) Integer Addition Add(, v) -bit integers, v Compte tre sm Ignores carry otpt Representations in memory, pointers, strings Smmary Add(, v) Vales increase linearly with and v Forms planar srface +v +v Operands: w bits Integer Addition 5 Unsigned Addition Representing information as bits Bit-level maniplations Integers Unsigned/signed: bits are trncated Reslt reinterpreted Unsigned: mod operation Signed: similar to mod For small nmbers yields expected behavior Today: Bits, Bytes, and Integers Expanding (e.g., short int to int) Overflow Wraps Arond If tre sm w At most once Integer Addition UAdd(, v) 3 Tre Sm w+ Overflow v w 53 Modlar Sm v 5

9 Two s Complement Addition Mathematical Properties Modlar Addition Forms an Abelian Grop Tre Sm: w+ bits UAddw(, v) w Commtative UAddw(, v) UAddw(v, ) Associative UAddw(t, UAddw(, v)) UAddw(UAddw(t, ), v) is additive identity UAddw(, ) Every element has additive inverse Let UCompw ( ) w UAddw(, UCompw ( )) Discard Carry: w bits TAdd and UAdd have Identical Bit-Level Behavior Will give s t 55 5 NegOver Tre Sm w w PosOver bits s comp. integer TAddw(, v) Visalizing s Complement Addition Fnctionality Drop off MSB Treat remaining bits as int s, t,, v; s (int) ((nsigned) + (nsigned) v); t + v TAdd Overflow Tre sm reqires w+ Signed vs. nsigned addition in C: + v +v Operands: w bits Closed nder addition -bit two s comp. Range from - to +7 TAdd Reslt Vales Wraps Arond If sm w Becomes negative At most once If sm < w Becomes positive At most once w w NegOver TAdd(, v) v PosOver 5 Positive Overflow Characterizing TAdd Mathematical Properties of TAdd Isomorphic Grop to nsigneds with UAdd TAddw(, v) UT(UAddw(TU( ), TU(v))) TAdd(, v) > v < Fnctionality Tre sm reqires w+ bits Drop off MSB Treat remaining bits as s Since both have identical bit patterns < Negative Overflow comp. integer Two s Complement Under TAdd Forms a Grop Sign Bit Set Closed, Commtative, Associative, is additive identity Every element has additive inverse TComp w () TMinw TMinw TMinw w v w TAddw (,v) v v ww 59 v TMin w > TAdd(, v) > v < < > (NegOver) TMinw v TMax w TMax w v (PosOver) 9

10 Signed/Unsigned Overflow Differences Sign bit table, signed ordering Unsigned: Overflow if carry ot of last position Also jst called carry (C) Signed: Reslt wrong if inpt signs are the same bt otpt sign is different In CPUs, nqalified overflow sally means signed (O or V) UAdd(, v) v < v > TAdd(, v) Positive Overflow > < Carry Ot < v > Negative Overflow TAdd(, v) > < Sign Bit Set Sign bit table, nsigned ordering Negation: Complement & Increment Claim: Following Holds for s Complement ~x + -x Complement Observation: ~x + x - x ~x Complement & Increment Examples Mltiplication x 53 Decimal Hex Binary x 53 3B D ~x -5 C 9 ~x+ -53 C 93 y -53 C 93 x Decimal Hex Binary ~ - FF FF ~+ Goal: Compting Prodct of w-bit nmbers x, y Either signed or nsigned Bt, exact reslts can be bigger than w bits Unsigned: p to w bits Reslt range: x * y ( w ) w w+ + Two s complement min (negative): Up to w- bits Reslt range: x * y ( w )*( w ) w + w Two s complement max (positive): Up to w bits, bt only for (TMin w ) Reslt range: x * y ( w ) w So, maintaining exact reslts wold need to keep expanding word size with each prodct compted is done in software, if needed e.g., by arbitrary precision arithmetic packages 5

11 Unsigned Mltiplication in C Signed Mltiplication in C Operands: w bits Tre Prodct: *w bits v Discard w bits: w bits * v UMlt w (, v) Operands: w bits Tre Prodct: *w bits v Discard w bits: w bits * v TMlt w (, v) Standard Mltiplication Fnction Ignores high order w bits Implements Modlar Arithmetic UMlt w (, v) v mod w Standard Mltiplication Fnction Ignores high order w bits Some of which are different for signed vs. nsigned mltiplication Lower bits are the same 7 Code Secrity Example # XDR Code SUN XDR library Widely sed library for transferring data between machines void* copy_elements(void *ele_src[], int ele_cnt, size_t ele_size); ele_src malloc(ele_cnt * ele_size) void* copy_elements(void *ele_src[], int ele_cnt, size_t ele_size) { /* * Allocate bffer for ele_cnt objects, each of ele_size bytes * and copy from locations designated by ele_src */ void *reslt malloc(ele_cnt * ele_size); if (reslt NULL) /* malloc failed */ retrn NULL; void *next reslt; int i; for (i ; i < ele_cnt; i++) { /* Copy object i to destination */ memcpy(next, ele_src[i], ele_size); /* Move pointer to next memory region */ next + ele_size; retrn reslt; 9 7 XDR Vlnerability Power-of- Mltiply with Shift malloc(ele_cnt * ele_size) What if: ele_cnt + ele_size 9 Allocation?? Chime in: (Qestion 57) ( + ) How can I make this fnction secre? Operation << k gives * k Both signed and nsigned Operands: w bits Tre Prodct: w+k bits k Discard k bits: w bits UMlt w (, k ) TMlt w (, k ) Examples << 3 * ( << 5) ( << 3) * Most machines shift and add faster than mltiply * k k 7 7

12 Compiled Mltiplication Code Backgrond: Ronding in Math C Fnction long ml(long x) { retrn x*; Compiled Arithmetic Operations leaq (%rax,%rax,), %rax salq $, %rax Explanation t <- x+x* retrn t << ; C compiler atomatically generates shift/add code when mltiplying by constant How to rond to the nearest integer? Cannot have both: rond(x + k) rond(x) + k (k integer), translation invariance rond(-x) -rond(x) negation invariance x, read floor : always rond down (to - ):.,.7, x, read ceiling : always rond p (to + ):.,.7, -. - C integer operators mostly se rond to zero, which is like floor for positive and ceiling for negative 73 7 Division in C Undefined behavior Integer division /: ronds towards Choice (settled in C99) is historical, via FORTRAN and most CPUs Division by zero: ndefined, sally fatal Unsigned division: no overflow possible Signed division: overflow almost impossible Exception: TMin/- is n-representable, and so ndefined On x this too is a defalt-fatal exception Many things yo shold not do are officially called ndefined by the C langage standard Meaning: compiler can do anything it wants Examples: Accessing beyond the ends of an array Dividing by zero Overflow in signed operations Shifts of negative vales Things yo do in this section of the corse! Bad interaction with improving compiler optimizers Gap between standard and lenient practical compilers not yet resolved 75 7 Unsigned Power-of- Divide with Shift Compiled Unsigned Division Code Qotient of Unsigned by Power of >> k gives / k Uses logical shift k Operands: / k Division: / k. Reslt: / k Binary Point C Fnction nsigned long div (nsigned long x) { retrn x/; Compiled Arithmetic Operations shrq $3, %rax Explanation # Logical shift retrn x >> 3; Division Compted Hex Binary x B D x >> D B x >> B x >> B Uses logical shift for nsigned For Java Users Logical shift written as >>> 77 7

13 Signed Power-of- Divide with Shift Correct Power-of- Divide Qotient of Signed by Power of x >> k gives x / k Uses arithmetic shift Ronds wrong direction when < / k x / k Want x / k (Rond Toward ) Compte as (x+k-)/ k k Binary Point RondDown(x / k) Division: Reslt: x Operands: Qotient of Negative Nmber by Power of Case : No ronding Dividend:. In C: (x + (<<k)-) >> k Biases dividend toward k +k y y >> y >> y >> Division Compted Hex C 93 E 9 FC 9 FF C Binary Divisor: Binary Point. Biasing has no effect 79 / k / k Compiled Signed Division Code Correct Power-of- Divide (Cont.) C Fnction Case : Ronding Dividend: x +k k long idiv(long x) { retrn x/; Incremented by Divisor: / k x / k testq %rax, %rax js L L3: sarq $3, %rax ret L: addq $7, %rax jmp L3 Binary Point. Incremented by Remainder operator Uses arithmetic shift for int For Java Users Today: Bits, Bytes, and Integers Written as % in C x % y is the remainder after division x / y E.g., x % is the lowest digit of non-negative x Behavior for negative vales matches / s ronding toward zero Representing information as bits Bit-level maniplations Integers I.e. sign of remainder matches sign of dividend (Some other langages have other conventions: sign of reslt eqals sign of divisor, sometimes distingished as modlo, or always positive) b*(a / b) + (a % b) a if x < x + 7; # Arithmetic shift retrn x >> 3; Arith. shift written as >> Biasing adds to final reslt Explanation Compiled Arithmetic Operations 3 Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Smmary Representations in memory, pointers, strings 3

14 Arithmetic: Basic Rles Arithmetic: Basic Rles Addition: Unsigned/signed: Normal addition followed by trncate, same operation on bit level Unsigned: addition mod w Mathematical addition + possible sbtraction of w Signed: modified addition mod w (reslt in proper range) Mathematical addition + possible addition or sbtraction of w Unsigned ints, s complement ints are isomorphic rings: isomorphism casting Left shift Unsigned/signed: mltiplication by k Always logical shift Mltiplication: Unsigned/signed: Normal mltiplication followed by trncate, same operation on bit level Unsigned: mltiplication mod w Signed: modified mltiplication mod w (reslt in proper range) Right shift Unsigned: logical shift, div (division + rond to zero) by k Signed: arithmetic shift Positive nmbers: div (division + rond to zero) by k Negative nmbers: div (division + rond away from zero) by k Use biasing to fix 5 Properties of Unsigned Arithmetic Unsigned Mltiplication with Addition Forms Commtative Ring Addition is commtative grop Closed nder mltiplication UMlt w (, v) w Mltiplication Commtative UMlt w (, v) UMlt w (v, ) Mltiplication is Associative UMlt w (t, UMlt w (, v)) UMlt w (UMlt w (t, ), v) is mltiplicative identity UMlt w (, ) Mltiplication distribtes over addtion UMlt w (t, UAdd w (, v)) UAdd w (UMlt w (t, ), UMlt w (t, v)) 7 Properties of Two s Comp. Arithmetic Isomorphic Algebras Unsigned mltiplication and addition Trncating to w bits Two s complement mltiplication and addition Trncating to w bits Both Form Rings Isomorphic to ring of integers mod w Comparison to (Mathematical) Integer Arithmetic Both are rings Integers obey ordering properties, e.g., > + v > v >, v > v > These properties are not obeyed by two s comp. arithmetic TMax + TMin 53 * 3-3 (-bit words) Why Shold I Use Unsigned? Conting Down with Unsigned Don t se withot nderstanding implications Easy to make mistakes nsigned i; for (i cnt-; i > ; i--) a[i] + a[i+]; Can be very sbtle #define DELTA sizeof(int) int i; for (i CNT; i-delta > ; i- DELTA)... 9 Proper way to se nsigned as loop index nsigned i; for (i cnt-; i < cnt; i--) a[i] + a[i+]; See Robert Seacord, Secre Coding in C and C++ C Standard garantees that nsigned addition will behave like modlar arithmetic UMax Even better size_t i; for (i cnt-; i < cnt; i--) a[i] + a[i+]; Data type size_t defined as nsigned vale with length word size Code will work even if cnt UMax What if cnt is signed and <? 9

15 Why Shold I Use Unsigned? (cont.) Today: Bits, Bytes, and Integers Do Use When Performing Modlar Arithmetic Mltiprecision arithmetic Do Use When Using Bits to Represent Sets Logical right shift, no sign extension Representing information as bits Bit-level maniplations Integers Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Smmary Representations in memory, pointers, strings 9 9 Byte-Oriented Memory Organization Machine Words Any given compter has a Word Size Nominal size of integer-valed data and of addresses Programs refer to data by address Conceptally, envision it as a very large array of bytes In reality, it s not, bt can think of it that way An address is like an index into that array and, a pointer variable stores an address Note: system provides private address spaces to each process Think of a process as a program being exected So, a program can clobber its own data, bt not that of others Until recently, most machines sed 3 bits ( bytes) as word size Limits addresses to GB ( 3 bytes) Increasingly, machines have -bit word size Potentially, cold have EB (exabytes) of addressable memory That s. X Machines still spport mltiple data formats Fractions or mltiples of word size Always integral nmber of bytes 93 9 Word-Oriented Memory Organization Example Data Representations Addresses Specify Byte Locations Address of first byte in word Addresses of sccessive words differ by (3-bit) or (-bit) 3-bit Words Addr?? Addr?? Addr?? Addr?? -bit Words Addr?? Addr?? Bytes Addr C Data Type Typical 3-bit Typical -bit x- char short int long float doble long doble / pointer 9 5

16 Byte Ordering Byte Ordering Example So, how are the bytes within a mlti-byte word ordered in memory? Conventions Big Endian: Sn, PPC Mac, Internet Example Variable x has -byte vale of x357 Address given by &x is x Least significant byte has highest address Little Endian: x, ARM processors rnning Android, ios, and Windows Least significant byte has lowest address Big Endian x x x x3 Little Endian x x x x Decimal: 53 Representing Integers Binary: Hex: Examining Data Representations 3 B D int A 53; IA3, x- D 3B Sn 3B D IA3 x- D 3B D 3B int B -53; IA3, x- 93 C FF FF Sn FF FF C 93 Code to Print Byte Representation of Data Casting pointer to nsigned char * allows treatment as a byte array long int C 53; Sn typedef nsigned char *pointer; 3B D void show_bytes(pointer start, size_t len){ size_t i; for (i ; i < len; i++) printf( %p\tx%.x\n",start+i, start[i]); printf("\n"); Printf directives: %p: Print pointer %x: Print hexadecimal Two s complement representation 99 show_bytes Exection Example Representing Pointers int B -53; int *P &B; int a 53; printf("int a 53;\n"); show_bytes((pointer) &a, sizeof(int)); Sn IA3 x- EF AC 3C FF B Reslt (Linx x-): FB F5 FE int a 53; x7fffb7f7dbc x7fffb7f7dbd x7fffb7f7dbe x7fffb7f7dbf C FF d 3b FD 7F Different compilers & machines assign different locations to objects Even get different reslts each time rn program

17 Representing Strings Reading Byte-Reversed Listings char S[] "3"; Strings in C Represented by array of characters Each character encoded in ASCII format IA3 Sn Standard 7-bit encoding of character set 3 3 Character has code x3 3 3 Digit i has code x3+i 3 3 String shold be nll-terminated 3 3 Final character Compatibility Byte ordering not an isse Disassembly Text representation of binary machine code Generated by program that reads the machine code Example Fragment Address Instrction Code Assembly Rendition 35: 5b pop %ebx 3: c3 ab add $xab,%ebx 3c: 3 bb cmpl $x,x(%ebx) Deciphering Nmbers Vale: xab Pad to 3 bits: xab Split into bytes: ab Reverse: ab 3 Integer C Pzzles Bons: More Integer C Pzzles. x < ((x*) < ). x > - 3. x > && y > x + y > Initialization. (x -x)>>3 - int x foo(); int y bar(); nsigned x x; nsigned y y; Initialization int x foo(); int y bar(); nsigned x x; nsigned y y; x < ((x*) < ) x > x & 7 7 (x<<3) < x > - x > y -x < -y x * x > x > && y > x + y > x > -x < x < -x > (x -x)>>3 - x >> 3 x/ x >> 3 x/ x & (x-)! 5 7