Bits, Bytes, and Integer

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1 19.3 Compter Architectre Spring 1 Bits, Bytes, and Integers Nmber Representations Bits, Bytes, and Integer Representing information as bits Bit-level maniplations Integers Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Smmary Acknowledgement: slides based on the cs:appe material Binary Representations Encoding Byte Vales Byte = bits 3.3V.V.5V.V 1 Binary to Decimal: 1 to 551 Hexadecimal 1 to 1 Base 1 nmber representation Use characters to 9 and A to F Write FA1D37B1 in C as xfa1d37b xfa1d37b A 1 11 B C 1 11 D E F Byte-Oriented Memory Organization Machine Words Programs Refer to Virtal Addresses Conceptally very large array of bytes Actally implemented with hierarchy of different memory types System provides address space private to particlar process Program being exected Program can clobber its own data, bt not that of others Compiler + Rn-Time System Control Allocation Where different program objects shold be stored All allocation within single virtal address space Machine Has Word Size Nominal size of integer-valed data Inclding addresses Most crrent machines se 3 bits ( bytes) words Limits addresses to GB Becoming too small for memory-intensive applications High-end systems se bits ( bytes) words Potential address space 1. X 1 19 bytes x- machines spport -bit addresses: 5 Terabytes Machines spport mltiple data formats Fractions or mltiples of word size Always integral nmber of bytes 5 1

2 19.3 Compter Architectre Spring 1 Word-Oriented Memory Organization 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 = 1?? -bit Words Addr =?? Addr =?? Bytes Addr Data Representations C Data Type Typical 3-bit Intel IA3 x- char short int long long long float doble long doble 1/1 1/1 pointer 7 Byte Ordering Byte Ordering Example How shold bytes within a mlti-byte word be ordered in memory? Conventions Big Endian: Sn, PPC Mac, Internet Least significant byte has highest address Little Endian: x Least significant byte has lowest address Big Endian Least significant byte has highest address Little Endian Least significant byte has lowest address Example Variable x has -byte representation x1357 Address given by &x is x1 Big Endian x1 x11 x1 x Little Endian x1 x11 x1 x Reading Byte-Reversed Listings Examining Data Representations 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: 1 c3 ab 1 add $x1ab,%ebx 3c: 3 bb cmpl $x,x(%ebx) Deciphering Nmbers Vale: x1ab Pad to 3 bits: x1ab Split into bytes: 1 ab Reverse: ab 1 Code to Print Byte Representation of Data Casting pointer to nsigned char * creates byte array void show_bytes(char *start, int len){ int 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 11 1

3 19.3 Compter Architectre Spring 1 show_bytes Exection Example int a = 1513; printf("int a = 1513;\n"); show_bytes((char *) &a, sizeof(int)); Reslt (Linx): int a = 1513; x11ffffcb xd x11ffffcb9 x3b x11ffffcba x x11ffffcbb x 13 Representing Integers int A = 1513; IA3, x- D 3B IA3, x- 93 C Sn 3B D int B = -1513; (= xffffc93) Sn C 93 1 Decimal: 1513 long int C = 1513; IA3 D 3B Binary: Hex: 3 B D x- D 3B Two s complement representation (Covered later) Sn 3B D Representing Pointers int B = -1513; int *P = &B; Sn IA3 x- different compilers & machines assign different locations to objects EF FB C D F BF C 9 EC 7F Representing Strings char S[] = "13"; Strings in C Represented by array of characters Linx/Alpha Sn Each character encoded in ASCII format Standard 7-bit encoding of character set Character has code x Digit i has code x3+i String shold be nll-terminated Final character = Compatibility Byte ordering not an isse 15 1 Bits, Bytes, and Integers Boolean Algebra Representing information as bits Bit-level maniplations Integers Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Smmary Developed by George Boole in 19th Centry Algebraic representation of logic Encode Tre as 1 and False as And Or A&B = 1 when both A=1 and B=1 A B = 1 when either A=1 or B=1 Not ~A = 1 when A= Exclsive-Or (Xor) A^B = 1 when either A=1 or B=1, bt not both

4 19.3 Compter Architectre Spring 1 Application of Boolean Algebra General Boolean Algebras Applied to Digital Systems by Clade Shannon 1937 MIT Master s Thesis Reason abot networks of relay switches Encode closed switch as 1, open switch as A&~B Connection when A ~B A&~B ~A&B Operate on Bit Vectors Operations applied bitwise 1111 & All of the Properties of Boolean Algebra Apply 1111 ^ ~ ~A B ~A&B = A^B 19 Representing & Maniplating Sets Representation Width w bit vector represents sbsets of {,, w 1 a j = 1 if j A 1111 {, 3, 5, {,,, 7531 Operations & Intersection 11 {, Union {,, 3,, 5, ^ Symmetric difference 1111 {, 3,, 5 ~ Complement 1111 { 1, 3, 5, 7 Bit-Level Operations in C Operations &,, ~, ^ available in C Apply to any integral data type long, int, short, char, nsigned View argments as bit vectors Argments applied bit-wise Examples (Char data type) ~x1 xbe ~ ~x x ~ x9 & x55 x & x9 x55 x7d Contrast: Logic Operations in C Shift Operations Contrast to Logical Operators &&,,! View as False Anything nonzero as Tre Always retrn or 1 Early termination Examples (char data type)!x1 x!x x1!!x1 x1 x9 && x55 x1 x9 x55 x1 p && *p (avoids nll pointer access) Left Shift: x << y Shift bit-vector x left y positions Throw away extra bits on left Fill with s on right Right Shift: x >> y Shift bit-vector x right y positions Throw away extra bits on right Logical shift Fill with s on left Arithmetic shift Replicate most significant bit on right Undefined Behavior Shift amont < or word size Argment x 111 << 3 1 Log. >> 11 Arith. >> 11 Argment x 111 << 3 1 Log. >> 11 Arith. >>

5 19.3 Compter Architectre Spring 1 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 Encoding Integers BU(X) w 1 x i i i C short bytes long short int x = 1513; short int y = -1513; Two s Complement BT(X) x w 1 w 1 w x i i i Decimal Hex Binary x B D y C Sign Bit Sign Bit For s complement, most significant bit indicates sign for nonnegative 1 for negative 5 Encoding Example (Cont.) x = 1513: y = -1513: Weight Sm Encoding Example Visalized Two s Complement BU(X) w 1 x i i BT(X) x w 1 w 1 w x i i i i 3 = 3 = = = 1 = 1 = = 1 = [1] [1] [11] [11] [111] [111] [1111] [1111] 7 Nmeric Ranges Vales UMin = UMax = w Two s Complement Vales TMin = w 1 1 TMax = w Other Vales Mins 1 Vales for W = Decimal Hex Binary UMax TMax 377 7F TMin Vales for Different Word Sizes W 1 3 UMax 55 5,535,9,97,95 1,,7,73,79,551,15 TMax 17 3,77,17,3,7 9,3,37,3,5,775,7 TMin -1-3,7 -,17,3, -9,3,37,3,5,775, Observations TMin = TMax + 1 Asymmetric range UMax = * TMax + 1 C Programming #inclde <limits.h> Declares constants, e.g., ULONG_MAX LONG_MAX LONG_MIN Vales platform specific 9 3 5

6 19.3 Compter Architectre Spring 1 & 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 -1 (x) Bit pattern for nsigned integer TB(x) = BT -1 (x) Bit pattern for two s comp integer An (Easier) Way To Look At It 11 BU(X) w 1 x i i i Two s Complement BT(X) x w 1 w 1 w x i i i UMax TMin TMax Bits, Bytes, and Integers Mapping Between Signed & Representing information as bits Bit-level maniplations Integers Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Smmary Two s Complement TU x TB X BU Maintain Same Bit Pattern UT x UB X BT x Two s Complement x Maintain Same Bit Pattern Mappings between nsigned and two s complement nmbers: keep bit representations and reinterpret 33 3 Mapping Signed Mapping Signed Bits Signed Bits Signed TU UT = +/

7 19.3 Compter Architectre Spring 1 Relation between Signed & Conversion Visalized Two s Complement x TU TB X BU x s Comp. Ordering Inversion Negative Big Positive UMax UMax 1 w 1 x x Large negative weight becomes Large positive weight Maintain Same Bit Pattern x x x x w x s Complement Range TMax 1 TMin TMax + 1 TMax Range 37 3 Conversion Visalized () UMax Signed vs. in C s Comp. Ordering Inversion Negative Big Positive s Comp. Ordering Inversion Big Positive Negative Conversion Nmber <= TMax: == s Comp Otherwise: = s Comp + w +/- 1 == -1 1 TMin TMax Constants By defalt are considered to be signed integers if have U as sffix U, 99759U 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; 39 Casting Srprises Expression Evalation 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 = -,17,3,, TMAX =,17,3,7 Constant 1 Constant Relation Evalation U == nsigned -1-1 < signed -1-1 U > nsigned > signed 1737U 1737U < nsigned > signed (nsigned) > nsigned U < nsigned (int) 173U > signed Code Secrity Example /* Kernel memory region holding ser-accessible data */ #define KSIZE 1 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; /* memcpy definition */ void *memcpy(void *dest, const void *src, size_t n); Similar to code fond in FreeBSD s implementation of getpeername There are legions of smart people trying to find vlnerabilities in programs 1 7

8 19.3 Compter Architectre Spring 1 Carnegie Mello Carnegie Mello Typical Usage Malicios Usage /* Kernel memory region holding ser-accessible data */ #define KSIZE 1 char kbf[ksize]; /* Kernel memory region holding ser-accessible data */ #define KSIZE 1 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; /* memcpy definition */ void *memcpy(void *dest, const void *src, size_t n); /* memcpy definition */ void *memcpy(void *dest, const void *src, size_t n); #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);... 3 Understanding What Can Go Wrong /* Kernel memory region holding ser-accessible data */ #define KSIZE 1 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; /* memcpy definition */ void *memcpy(void *dest, const void *src, size_t n); Carnegie Mello Smmary Casting Signed : 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!! #define MSIZE 5 void getstff() { char mybf[msize]; copy_from_kernel(mybf, -MSIZE);... determine type of size_t: $ echo "#inclde <stdio.h>" \ gcc -E - \ grep -E "typedef.* size_t;" 5 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 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 1,, x w 1, x w 1, x w,, x k copies of MSB w X X 7 k w

9 19.3 Compter Architectre Spring 1 Sign Extension Example Smmary: Expanding, Trncating: Basic Rles short int x = 1513; int ix = (int) x; short int y = -1513; int iy = (int) y; Decimal Hex Binary x B D ix B D y C iy C Converting from smaller to larger integer data type C atomatically performs sign extension Expanding (e.g., short int to int) : zeros added Signed: sign extension Both yield expected reslt Trncating (e.g., nsigned to nsigned short) /signed: bits are trncated Reslt reinterpreted : mod operation Signed: similar to mod For small nmbers yields expected behavior 9 5 Bits, Bytes, and Integers Negation: Complement & Increment Representing information as bits Bit-level maniplations Integers Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Smmary Claim: Following Holds for s Complement ~x + 1 == -x Complement Observation: ~x + x == == -1 + x ~x Complement & Increment Examples Addition x = 1513 Decimal Hex Binary x B D ~x -151 C ~x C y C Operands: w bits + v Tre Sm: w+1 bits + v Discard Carry: w bits UAdd w (, v) x = Decimal Hex Binary ~ ~+1 Standard Addition Fnction Ignores carry otpt Implements Modlar Arithmetic s = UAdd w (, v) = + v mod w UAdd w (,v) v v w v w v w

10 19.3 Compter Architectre Spring 1 Visalizing (Mathematical) Integer Addition Visalizing Addition Integer Addition -bit integers, v Compte tre sm Add (, v) Vales increase linearly with and v Forms planar srface Add (, v) Integer Addition v Wraps Arond If tre sm w At most once Tre Sm w+1 Overflow w Modlar Sm Overflow UAdd (, v) v 55 5 Mathematical Properties Modlar Addition Forms an Abelian Grop Closed nder addition UAdd w (, v) w 1 Commtative UAdd w (, v) = UAdd w (v, ) Associative UAdd w (t, UAdd w (, v)) = UAdd w (UAdd w (t, ), v) is additive identity UAdd w (, ) = Every element has additive inverse Let UComp w ( ) = w UAdd w (, UComp w ( )) = Two s Complement Addition Operands: w bits + v Tre Sm: w+1 bits + v Discard Carry: w bits TAdd w (, v) TAdd and UAdd have Identical Bit-Level Behavior Signed vs. nsigned addition in C: int s, t,, v; s = (int) ((nsigned) + (nsigned) v); t = + v Will give s == t 57 5 TAdd Overflow Visalizing s Complement Addition Fnctionality Tre sm reqires w+1 bits Drop off MSB Treat remaining bits as s comp. integer w 1 w 1 w 1 1 w Tre Sm PosOver NegOver TAdd Reslt Vales -bit two s comp. Range from - to +7 Wraps Arond If sm w 1 Becomes negative At most once If sm < w 1 Becomes positive At most once NegOver TAdd (, v) v - PosOver 59 1

11 19.3 Compter Architectre Spring 1 Characterizing TAdd Mathematical Properties of TAdd Fnctionality Tre sm reqires w+1 bits Drop off MSB Treat remaining bits as s c omp. integer Positive Overflow TAdd(, v) > v < < > Negative Overflow Isomorphic Grop to nsigneds with UAdd TAdd w (, v) = UT(UAdd w (TU( ), TU(v))) Since both have identical bit patterns Two s Complement Under TAdd Forms a Grop Closed, Commtative, Associative, is additive identity Every element has additive inverse TAdd w (,v) v w 1 w v v w 1 w v TMin (NegOver) w TMin w v TMax w TMax w v (PosOver) TComp w () TMin w TMin w TMin w 1 Mltiplication Mltiplication in C Compting Exact Prodct of w-bit nmbers x, y Either signed or nsigned Ranges : x * y ( w 1) = w w Up to w bits Operands: w bits Tre Prodct: *w bits Discard w bits: w bits v * v UMlt w (, v) Two s complement min: x * y ( w 1 )*( w 1 1) = w + w 1 Up to w 1 bits Two s complement max: x * y ( w 1 ) = w Standard Mltiplication Fnction Ignores high order w bits Up to w bits, bt only for (TMin w ) Maintaining Exact Reslts Wold need to keep expanding word size with each prodct compted Done in software by arbitrary precision arithmetic packages Implements Modlar Arithmetic UMlt w (, v) = v mod w 3 Signed Mltiplication in C Operands: w bits Tre Prodct: *w bits Discard w bits: w bits v Standard Mltiplication Fnction Ignores high order w bits Some of which are different for signed vs. nsigned mltiplication Lower bits are the same * v TMlt w (, v) Power-of- Mltiply with Shift Operation << k gives * k Both signed and nsigned Operands: w bits Tre Prodct: w+k bits Examples k Discard k bits: w bits UMlt w (, k ) TMlt w (, k ) << 3 == * << 5 - << 3 == * Most machines shift and add faster than mltiply Compiler generates this code atomatically * k 1 k 5 11

12 19.3 Compter Architectre Spring 1 Compiled Mltiplication Code Power-of- Divide with Shift C Fnction Qotient of by Power of int ml1(int x) { retrn x*1; Compiled Arithmetic Operations leal (%eax,%eax,), %eax sall $, %eax Explanation t <- x+x* retrn t << ; >> k gives / k Uses logical shift Operands: Division: Reslt: / k / k / k k 1. Binary Point C compiler atomatically generates shift/add code when mltiplying by a constant Division Compted Hex Binary x B D x >> D B x >> B x >> B Compiled Division Code Signed Power-of- Divide with Shift C Fnction nsigned div(nsigned x) { retrn x/; Compiled Arithmetic Operations shrl $3, %eax Uses logical shift for nsigned For Java Users Logical shift written as >>> Explanation # Logical shift retrn x >> 3; Qotient of Signed by Power of x >> k gives x / k Uses arithmetic shift Ronds wrong direction when < k x Binary Point Operands: / k 1 Division: Reslt: x / k RondDown(x / k ). Division Compted Hex Binary y C y >> E y >> FC y >> C Correct Power-of- Divide Qotient of Negative Nmber by Power of Want x / k (Rond Toward ) Compte as (x+ k -1)/ k In C: (x + (1<<k)-1) >> k Biases dividend toward Case 1: No ronding Dividend: Divisor: / Biasing has no effect k / k k 1 + k Binary Point Correct Power-of- Divide (Cont.) Case : Ronding Dividend: Divisor: / x k x / k Biasing adds 1 to final reslt k 1 + k Incremented by Incremented by 1 Binary Point

13 19.3 Compter Architectre Spring 1 Compiled Signed Division Code C Fnction int idiv(int x) { retrn x/; Compiled Arithmetic Operations testl %eax, %eax js L L3: sarl $3, %eax ret L: addl $7, %eax jmp L3 Explanation if x < x += 7; # Arithmetic shift retrn x >> 3; Uses arithmetic shift for int For Java Users Arith. shift written as >> Arithmetic: Basic Rles Addition: /signed: Normal addition followed by trncate, same operation on bit level : 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 Mltiplication: /signed: Normal mltiplication followed by trncate, same operation on bit level : mltiplication mod w Signed: modified mltiplication mod w (reslt in proper range) 73 7 Arithmetic: Basic Rles Bits, Bytes, and Integers ints, s complement ints are isomorphic rings: isomorphism = casting Left shift /signed: mltiplication by k Always logical shift Right shift : 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 Representing information as bits Bit-level maniplations Integers Representation: nsigned and signed Conversion, casting Expanding, trncating Addition, negation, mltiplication, shifting Smmary 75 7 Properties of Arithmetic Mltiplication with Addition Forms Commtative Ring Addition is commtative grop Closed nder mltiplication UMlt w (, v) w 1 Mltiplication Commtative UMlt w (, v) = UMlt w (v, ) Mltiplication is Associative UMlt w (t, UMlt w (, v)) = UMlt w (UMlt w (t, ), v) 1 is mltiplicative identity UMlt w (, 1) = Mltiplication distribtes over addtion UMlt w (t, UAdd w (, v)) = UAdd w (UMlt w (t, ), UMlt w (t, v)) Properties of Two s Comp. Arithmetic Isomorphic Algebras 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 + 1 == TMin 1513 * 3 == -13 (1-bit words)

14 19.3 Compter Architectre Spring 1 Why Shold I Use? Don t Use Jst Becase Nmber Nonnegative Easy to make mistakes nsigned i; for (i = cnt-; i >= ; i--) a[i] += a[i+1]; Can be very sbtle #define DELTA sizeof(int) int i; for (i = CNT; i-delta >= ; i-= DELTA)... Do Use When Performing Modlar Arithmetic Mltiprecision arithmetic Do Use When Using Bits to Represent Sets Logical right shift, no sign extension 79 1