Lecture 2: Number Representations (2)

Transcription

1 Lecture 2: Number Representations (2) ECE 645 Computer Arithmetic 1/29/08 ECE 645 Computer Arithmetic Lecture Roadmap Number systems (cont'd) Floating point number system representations Residue number number systems Galois Field number systems Endian-ness Big endian Little endian 2

2 Required Reading B. Parhami, Computer Arithmetic: Algorithms and Hardware Design Chapter 17, Floating-Point Representations ( , 17.6) Chapter 4, Residue Number Systems (4.1, 4.3) Note errata at: 3 Floating Point Number Systems ECE 645 Computer Arithmetic

3 Number Systems Modern number systems used in digital arithmetic can be broadly classified as: Fixed-point number representation systems Integers I = {-N,., N} Rational numbers of the form x = a/2 f, a is an integer, f is a positive integer Floating-point number representation systems x * b E, where x is a rational number, b the integer base, and E the exponent There are also special number representation systems: Residue number systems Galois Field number systems 5 Real Numbers 6

4 Fixed Point Numbers 7 Floating Point Numbers Floating point range of values is [-max,-min] and [min,max] max = largest significand x blargest exponent min = smallest significand x bsmallest exponent 8

5 Floating Point Format 9 Spacing of Floating Point Numbers 10

6 ANSI/IEEE Short and Double Precision 11 Fig The ANSI/IEEE Standard Floating-Point Number Representation Formats 8 bits, bias = 127, 126 to 127 Short (32-bit) format 23 bits for fractional part (plus hidden 1 in integer part) IEEE 754 Standard (now being revised to yield IEEE 754R) Sign Exponent 11 bits, bias = 1023, 1022 to 1023 Significand 52 bits for fractional part (plus hidden 1 in integer part) Long (64-bit) format 12

7 Table 17.1 ANSI/IEEE Standard Floating Point Number Representation Formats 13 Exponent Encoding Exponent encoding in 8 bits for the single/short (32-bit) ANSI/IEEE format Decimal code Hex code Exponent value E 7F 80 FE FF f = 0: Representation of ±0 f 0: Representation of denormals, 0.f Exponent encoding in 11 bits for the double/long (64-bit) format is similar Overflow region Sparser f = 0: Representation of ± f 0: Representation of NaNs Negative numbers Positive numbers max FLP min ±0 min + FLP + max + + Midway example 1.f 2 e Denser Underflow example Underflow regions Denser Typical example Sparser Overflow region Overflow example 14

8 Exceptions 15 Special Operands 16

9 Denormals in the IEEE Single-Precision Format Denormals defined as numbers without a hidden 1 and with the smallest possible exponent Provided to make the effect of underflow less abrupt Requires extra hardware overhead 17 Other Features of IEEE standard 18

10 Floating-Point Rounding Modes 19 Fixed-Point and Logarithmic Number Systems Fixed-point can be roughly viewed as case of floating-point with exponent equal to 0 (making exponent field unecessary) Logarithmic number systems is the other extreme (assuming the significant field is always 1) 20

11 Logarithmic Number System Sign-and-logarithm number system: Limiting case of FLP representation x = ± b e 1 e = log b x We usually call b the logarithm base, not exponent base Using an integer-valued e wouldn t be very useful, so we consider e to be a fixed-point number Sign ± Fixed-point exponent e Implied radix point Fig Logarithmic number representation with sign and fixed-point exponent. 21 Residue Number Systems ECE 645 Computer Arithmetic

12 Divisibility Rules and Operations Modulo 23 Divisibility a b a divides b a is a divisor of b a b iff c Z such that b = c a a b a does not divide b a is not a divisor of b 24

13 True or False? Quotient and Remainder Given integers a and n, n>0! q, r Z such that a = q n + r and 0 r < n q quotient q = a n = a div n r remainder (of a divided by n) r = a - q n = a a n n = = a mod n 26

14 Examples 32 mod 5 = -32 mod 5 = 27 Integers Congruent modulo n Two integers a and b are congruent modulo n (equivalent modulo n) written a b iff a mod n = b mod n or a = b + kn, k Z or n a - b 28

15 Addition, Subtraction, Multiplication modulo n a + b mod n = ((a mod n) + (b mod n)) mod n a - b mod n = ((a mod n) - (b mod n)) mod n a b mod n = ((a mod n) (b mod n)) mod n 29 Examples 9 13 mod 5 = mod 26 = 30

16 Chinese Remainder Theorem: Intro 31 Chinese Remainder Theorem Let and N = n 1 n 2 n 3 for any i, j gcd(n i, n j ) = 1... n M Then, any number 0 A N-1 RANGE OF NUMBER can be represented uniquely by A (a 1 = A mod n 1, a 2 = A mod n 2,, a M = A mod n M ) A can be reconstructed from (a 1, a 2,, a M ) using equation M A = i=1 (a i N i N i -1 mod n i ) mod N where N i = N n i = = n 1 n 2... n i-1 n i+1... n M 32

17 Multiplicative Inverse Modulo n The multiplicative inverse of a modulo n is an integer [!!!] x such that a x 1 (mod n) The multiplicative inverse of a modulo n is denoted by a -1 mod n (in some books a or a * ). According to this notation: a a -1 1 (mod n) 33 Examples 3-1 mod 11 = 9-1 mod 26 = 5-1 mod 26 = 15-1 mod 13 = 34

18 Residue Number System Arithmetic Range of number in RNS( ) is 0 to 8*7*5*3 = 840 or (-420 to +419) or any other interval of 840 consecutive integers ( ) represents -1 or 839 or 35 RNS Hardware Arithmetic 36

19 RSA as a trapdoor one way function PUBLIC KEY M C = f(m) = M e mod N C M = f -1 (C) = C d mod N PRIVATE KEY N = P Q P, Q - large prime numbers e d 1 mod ((P-1)(Q-1)) 37 Chinese Remainder Theorem for N=P Q N = P Q gcd(p, Q) = 1 greatest common divisor M (M p = M mod P, M Q = M mod Q) M = M P N P N P -1 mod P + M Q N Q N Q -1 mod Q mod N = M P Q ((Q -1 ) mod P) + M Q P ((P -1 ) mod Q) mod N = = M P R Q + M Q R P mod N 38

20 Fast Modular Exponentiation Using Chinese Remainder Theorem M d = C mod N C P = C mod P d P = d mod (P-1) C Q = C mod Q d Q = d mod (Q-1) d Q mod d P M P = C P mod P M Q = C Q Q where M = M P R Q + M Q R P mod N R P = (P -1 mod Q) P = P Q-1 mod N R Q = (Q -1 mod P) Q= Q P-1 mod N 39 Time of Exponentiation without and with Chinese Remainder Theorem SOFTWARE Without CRT HARDWARE t EXP (k) = c s k 3 With CRT k t EXP-CRT (k) 2 c s ( ) 3 = t EXP (k) Without CRT t EXP (k) = c h k 2 With CRT k 1 t EXP-CRT (k) c h ( ) 2 = t EXP (k)

21 Galois Field Number Systems ECE 645 Computer Arithmetic Polynomial Representation of the Galois Field Elements 42

22 Evariste Galois ( ) 43 Evariste Galois ( ) Studied the problem of finding algebraic solutions for the general equations of the degree 5, e.g., f(x) = a 5 x 5 + a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x+ a 0 = 0 Answered definitely the question which specific equations of a given degree have algebraic solutions On the way, he developed group theory, one of the most important branches of modern mathematics. 44

23 Evariste Galois ( ) 1829 Galois submits his results for the first time to the French Academy of Sciences Reviewer 1 Augustin-Luis Cauchy forgot or lost the communication 1830 Galois submits the revised version of his manuscript, hoping to enter the competition for the Grand Prize in mathematics Reviewer 2 Joseph Fourier died shortly after receiving the manuscript 1831 Third submission to the French Academy of Sciences Reviewer 3 Simeon-Denis Poisson did not understand the manuscript and rejected it. 45 Evariste Galois ( ) May 1832 Galois provoked into a duel The night before the duel he writes a letter to his friend containing the summary of his discoveries. The letter ends with a plea: Eventually there will be, I hope, some people who will find it profitable to decipher this mess. May 30, 1832 Galois is grievously wounded in the duel and dies in the hospital the following day Galois manuscript rediscovered by Joseph Liouville 1846 Galois manuscript published for the first time in a mathematical journal 46

24 47 Definition of Field Set F, and two operations typically denoted by (but not necessarily equivalent to) + and * Set F, and definitions of these two operations must fulfill special conditions. 48

25 Examples of Fields Infinite fields { R= set of real numbers, + addition of real numbers * multiplication of real numbers } Finite fields { set Zp={0, 1, 2,, p-1}, + (mod p): addition modulo p, * (mod p): multiplication modulo p } 49 Finite Fields = Galois Fields GF(p m ) p prime p m number of elements in the field Arithmetic operations present in many libraries GF(p) GF(2 m ) Polynomial basis representation Most significant special cases Normal basis representation Fast in hardware Fast squaring 50

26 Elements of the Galois Field GF(2 m ) Binary representation (used for storing and processing in computer systems): A = (a m-1, a m-2,, a 2, a 1, a 0 ) a i {0, 1} Polynomial representation (used for the definition of basic arithmetic operations): m-1 A(x) = a i x i = a m-1 x m-1 + a m-2 x m a 2 x 2 + a 1 x+a 0 i=0 multiplication + addition modulo 2 (XOR) 51 Addition and Multiplication in the Galois Field GF(2 m ) Inputs A = (a m-1, a m-2,, a 2, a 1, a 0 ) B = (b m-1, b m-2,, b 2, b 1, b 0 ) a i, b i {0, 1} Output C = (c m-1, c m-2,, c 2, c 1, c 0 ) c i {0, 1} 52

27 Addition in the Galois Field GF(2 m ) Addition A A(x) B B(x) C C(x) = A(x) + B(x) = = (a m-1 +b m-1 ) x m-1 + (a m-2 +b m-2 ) x m (a 2 +b 2 ) x 2 + (a 1 +b 1 ) x + (a 0 +b 0 ) = = c m-1 x m-1 + c m-2 x m c 2 x 2 + c 1 x+c 0 multiplication + addition modulo 2 (XOR) c i = a i + b i = a i XOR b i C = A XOR B 53 Multiplication in the Galois Field GF(2 m ) Multiplication A A(x) B B(x) C C(x) = A(x) B(x) mod P(X) = c m-1 x m-1 + c m-2 x m c 2 x 2 + c 1 x+c 0 P(x) - irreducible polynomial of the degree m P(x) = p m x m + p m-1 x m p 2 x 2 + p 1 x+p 0 54

28 Applications Galois fields/finite fields used in cryptography and coding theory 55 Little-Endian and Big-Endian Representation of Integers ECE 645 Computer Arithmetic

29 Little-Endian vs Big-Endian Representation A0 B1 C2 D3 E4 F LSB MSB Big-Endian Little-Endian MSB = A0 B1 C2 D3 E4 F5 67 LSB = 89 0 address MAX LSB = F5 E4 D3 C2 B1 MSB = A0 57 Little-Endian vs Big-Endian Camps MSB... LSB Big-Endian 0 address MAX LSB... MSB Little-Endian Motorola 68xx, 680x0 IBM Hewlett-Packard Sun SuperSPARC Internet TCP/IP Bi-Endian Motorola Power PC Silicon Graphics MIPS Intel AMD DEC VAX RS

30 Little Endian versus Big Endian: Origin Jonathan Swift, Gulliver s Travels A law requiring all citizens of Lilliput to break their soft-eggs at the little ends only A civil war breaking between the Little Endians and the Big-Endians, resulting in the Big Endians taking refuge on a nearby island, the kingdom of Blefuscu Satire over holy wars between Protestant Church of England and the Catholic Church of France 59 60

31 Comparing Little-Endian and Big-Endian Big-Endian Little-Endian easier to determine a sign of the number easier addition and multiplication of multiprecision numbers easier to compare two numbers easier to divide two numbers easier to print 61 Pointers (1) Big-Endian Little-Endian 0 address MAX F5 E4 D3 C2 B1 A0 int * iptr; (* iptr) = 8967; (* iptr) = 6789; iptr+1 62

32 Pointers (2) Big-Endian Little-Endian 0 address MAX F5 E4 D3 C2 B1 A0 long int * lptr; (* lptr) = 8967F5E4; (* lptr) = E4F56789; lptr