number representation part 2 floating point representations little-endian vs. big-endian...
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Number Representation
Part 2Floating Point Representations
Little-Endian vs. Big-Endian Representations
Galois Field Representations
ECE 645: Lecture 2
Required Reading
Chapter 17, Floating-Point Representations
Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Design
J-P. Deschamps, G. Bioul, G. Sutter, Synthesis of Arithmetic Circuits: FPGA, ASIC and Embedded Systems,
Chapter 3.3, Real Numbers
Recommended Reading
Recommended Reading(to be covered at the next lecture)
Chapter 5, Basic Addition and Counting
Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Design
J-P. Deschamps, G. Bioul, G. Sutter, Synthesis of Arithmetic Circuits: FPGA, ASIC and Embedded Systems,
Chapter 4.1.1 Basic AlgorithmChapter 11.1 Basic AdderChapter 11.2 Carry-Chain Adder
The ANSI/IEEE standard floating-point number representation formats
Short (32-bit) format
Long (64-bit) format
Sign Exponent Significand
8 bits, bias = 127, –126 to 127
11 bits, bias = 1023, –1022 to 1023
52 bits for fractional part (plus hidden 1 in integer part)
23 bits for fractional part (plus hidden 1 in integer part)
Originally IEEE 754-1985.Superseded by IEEE 754-2008 Standard.
00 01 7F FE FF7E 800 1 127 254 255126 128
–126 0 +127–1 +1
Decimal codeHex code
Exponent value
f = 0: Representation of 0f 0: Representation of denormals, 0.f 2–126
f = 0: Representation of f 0: Representation of NaNs
Exponent encoding in 8 bits for the single/short (32-bit) ANSI/IEEE format
1.f 2e
Exponent Encoding
The IEEE 754-2008 standard includes five rounding modes:
Round to nearest, ties away from 0 (rtna)
Round to nearest, ties to even (rtne) [default rounding mode]
Round toward zero (inward)
Round toward + (upward)
Round toward – (downward)
Rounding Modes
Round to Nearest Number
Fig. 17.7 Rounding of a signed-magnitude value to the nearest number.
Rounding has a slight upward bias.
Consider rounding (xk–1xk–2 ... x1x0 . x–1x–2)two to an integer (yk–1yk–2 ... y1y0 . )two
The four possible cases, and their representation errors are:
x–1x–2 Round Error 00 down 0 01 down –0.25 10 up 0.5 11 up 0.25
With equal prob., mean = 0.125For certain calculations, the probability of getting a midpoint value can be much higher than 2–l
rtn(x)
–4
–3
–2
–1
x –4 –3 –2 –1 4 3 2 1
4
3
2
1
rtna(x)
Directed Rounding: Motivation
We may need result errors to be in a known direction
Example: in computing upper bounds, larger results are acceptable, but results that are smaller than correct values could invalidate the upper bound
This leads to the definition of directed rounding modesupward-directed rounding (round toward +) and downward-directed rounding (round toward –)(required features of IEEE floating-point standard)
Directed Rounding: Visualization
Fig. 17.12 Upward-directed rounding or rounding toward +.
Fig. 17.6 Truncation or chopping of a 2’s-complement number (same as downward-directed rounding).
up(x)
–4
–3
–2
–1
x –4 –3 –2 –1 4 3 2 1
4
3
2
1
chop(x) = down(x)
–4
–3
–2
–1
x –4 –3 –2 –1 4 3 2 1
4
3
2
1
Requirements for Arithmetic
Results of the 4 basic arithmetic operations (+, , , ) as well as square-rooting must match those obtained if all intermediate computations were infinitely precise
That is, a floating-point arithmetic operation should introduce no more imprecision than the error attributable to the final rounding of a result that has no exact representation (this is the best possible)
Example:(1 + 21) (1 + 223 )
Rounded result 1 + 21 + 222 Error = ½ ulp
Exact result 1 + 21 + 223 + 224
Little-Endian vs. Big-Endian Representation
A0 B1 C2 D3 E4 F5 67 8916
LSBMSB
MSB = A0
B1
C2D3
E4F5
67LSB = 89
Big-Endian Little-Endian
LSB = 89
0
MAX
67
F5E4
D3C2
B1MSB = A0
address
Little-Endian vs. Big-Endian Camps
Big-Endian Little-Endian
0
MAX
address
MSB
LSB
. . .
LSB
MSB
. . .
Motorola 68xx, 680x0 Intel
IBM
Hewlett-PackardDEC VAX
Internet TCP/IP
Sun SuperSPARC
Bi-Endian
Motorola Power PC
Silicon Graphics MIPS
RS 232
AMD
Origin of the termsLittle-Endian vs. Big-Endian
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
Little-Endian vs. Big-Endian
Big-Endian Little-Endian
• easier to determine a sign of the number
• easier to compare two numbers
• easier to divide two numbers
• easier to print
• easier addition and multiplication of multiprecision numbers
Advantages and Disadvantages
Pointers (1)
89
67
F5E4
D3C2
B1
A0
Big-Endian Little-Endian
0
MAX
address
int * iptr;
(* iptr) = 8967; (* iptr) = 6789;
iptr+1
Pointers (2)
89
67
F5E4
D3C2
B1
A0
Big-Endian Little-Endian
0
MAX
address
long int * lptr;
(* lptr) = 8967F5E4; (* lptr) = E4F56789;
lptr + 1
Evariste Galois (1811-1832)
Studied the problem of finding algebraic solutions for the general
equations of the degree 5, e.g.,
f(x) = a5x5+ a4x4+ a3x3+ a2x2+ a1x+ a0 = 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.
Evariste Galois (1811-1832)
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 Prizein mathematics
Reviewer 2 Joseph Fourier – died shortly after receiving the manuscript
1831 Third submission to the French Academy of SciencesReviewer 3
Simeon-Denis Poisson – did not understand the manuscript and rejected it.
Evariste Galois (1811-1832)
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.
1843 Galois manuscript rediscovered by Joseph Liouville
1846 Galois manuscript published forthe first time in a mathematical journal
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.
{ set Zp={0, 1, 2, … , p-1}, + (mod p): addition modulo p, * (mod p): multiplication modulo p}
Examples of fieldsInfinite fields
Finite fields
{ R= set of real numbers, + addition of real numbers * multiplication of real numbers}
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
r – remainder (of a divided by n)
q = an = a div n
r = a - q n = a – an
n =
= a mod n
Integers coungruent 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
Rules of addition, subtraction and multiplicationmodulo 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
Laws of modular arithmetic
Modular addition
Modular multiplication
Regular addition
Regular multiplication
a+b = a+ciff
b=c
a+b a+c (mod n)iff
b c (mod n)
If a b = a c and a 0then b = c
If a b a c (mod n) and gcd (a, n) = 1then b c (mod n)
Modular Multiplication: Example
18 42 (mod 8) 6 3 6 7 (mod 8)
3 7 (mod 8)
x
6 x mod 8
0 1 2 3 4 5 6 7
0 6 4 2 0 6 4 2
x
5 x mod 8
0 1 2 3 4 5 6 7
0 5 2 7 4 1 6 3
Z[x] - polynomials with coefficients in Z,
Sets of polynomials
e.g., f(x) = -4 x3 + 254 x2 + 45 x + 7
Zn[x] - polynomials with coefficients in Zn
e.g., for n=15
f(x) = 3 x3 + 14 x2 + 4 x + 7
Z2[x] - polynomials with coefficients in Z2
e.g., f(x) = 1 x3 + 0 x2 + 1 x + 1 = x3 + x + 1
Finite sets of polynomials
Z2[x]/f(x) - polynomials with coefficients in Z2
of degree less than n=deg f(x)
Zp[x]/f(x) - polynomials with coefficients in Zp
of degree less than n=deg f(x)
e.g., for f(x) = x3 + x + 1
g7(x) = x2 + x + 1g6(x) = x2 + x g5(x) = x2 + 1g4(x) = x2
g3(x) = x + 1g2(x) = x g1(x) = 1g0(x) = 0
e.g., for f(x) = x3 + x + 1, and p=3
g0(x) = 0….gM-1(x) = 2x2 + 2x + 2
Total: 3n polynomials
Finite Fields = Galois Fields
GF(p) GF(2m)
Polynomial basisrepresentation
Normal basisrepresentation
Fast in hardware
Arithmetic operations
presentin many libraries
Fast squaring
GF(pm)p – primepm – number of elements in the field
Most significantspecial cases
Elements of the Galois Field GF(2m)
Binary representation (used for storing and processing in computer systems):
Polynomial representation(used for the definition of basic arithmetic operations):
A = (am-1, am-2, …, a2, a1, a0) ai {0, 1}
A(x) = aixi = am-1xm-1 + am-2xm-2 + …+ a2x2 + a1x+a0
multiplication+ addition modulo 2 (XOR)
i=0
m-1
Addition and Multiplicationin the Galois Field GF(2m)
Inputs
A = (am-1, am-2, …, a2, a1, a0)B = (bm-1, bm-2, …, b2, b1, b0)
ai , bi {0, 1}
Output
C = (cm-1, cm-2, …, c2, c1, c0) ci {0, 1}
Addition
A A(x)B B(x)C C(x) = A(x) + B(x) = = (am-1+bm-1)xm-1 + (am-2+bm-2)xm-2+ …+ + (a2+b2)x2 + (a1+b1)x + (a0+b0) = = cm-1xm-1 + cm-2xm-2 + …+ c2x2 + c1x+c0
Addition in the Galois Field GF(2m)
multiplication+ addition modulo 2 (XOR)
ci = ai + bi = ai XOR bi
C = A XOR B
Multiplication
A A(x)B B(x)C C(x) = A(x) B(x) mod P(X) = cm-1xm-1 + cm-2xm-2 + …+ c2x2 + c1x+c0
Multiplication in the Galois Field GF(2m)
P(x) - irreducible polynomial of the degree m
P(x) = pmxm + pm-1xm-1 + …+ p2x2 + p1x+p0
AES MixColumns Operation
a0,0 a0,1 a0,2 a0,3
a1,0 a1,1 a1,2 a1,3
a2,0 a2,1 a2,2 a2,3
a3,0 a3,1 a3,2 a3,3
b0,0 b0,1 a0,2 b0,3
b1,0 b1,1 a1,2 b1,3
b2,0 b2,1 a2,2 b2,3
b3,0 b3,1 a3,2 b3,3
a1,j
a0,j
a2,j
a3,j
b1,j
b0,j
b2,j
b3,j
2 3 1 1 1 2 3 11 1 2 33 1 1 2
All operations in the Galois Field GF(28)
AES InvMixColumns Operations
a0,0 a0,1 a0,2 a0,3
a1,0 a1,1 a1,2 a1,3
a2,0 a2,1 a2,2 a2,3
a3,0 a3,1 a3,2 a3,3
b0,0 b0,1 a0,2 b0,3
b1,0 b1,1 a1,2 b1,3
b2,0 b2,1 a2,2 b2,3
b3,0 b3,1 a3,2 b3,3
a1,j
a0,j
a2,j
a3,j
b1,j
b0,j
b2,j
b3,j
E B D 9 9 E B DD 9 E BB D 9 E
All operations in the Galois Field GF(28)
Multiplication by a constantin the Galois Field GF(28)
Hardware8 8
MUL GF(28)
X
Y
C = const
x0 x3 x7
y0
. . .
x0 x3 x7
y7
x4
8
Hardware implementation - MixColumns
architecture mul_03 of mul_03 isbegin
output(7) <= input(7) xor input(6);output(6) <= input(6) xor input(5);output(5) <= input(5) xor input(4);output(4) <= input(4) xor input(3) xor input(7);output(3) <= input(3) xor input(2) xor input(7);output(2) <= input(2) xor input(1);output(1) <= input(1) xor input(0) xor input(7);output(0) <= input(0) xor input(7);
end mul_03;
b0 <= a0_02 xor a1_03 xor a2 xor a3;
Hardware implementation - InvMixColumns
architecture mul_0E of mul_0E isbegin output(7) <= input(7) xor input(6) xor input(5) xor input(4); output(6) <= input(6) xor input(5) xor input(4) xor input(3) xor input(7); output(5) <= input(5) xor input(4) xor input(3) xor input(2) xor input(6); output(4) <= input(4) xor input(3) xor input(2) xor input(1) xor input(5); output(3) <= input(3) xor input(2) xor input(1) xor input(0) xor input(6) xor input(5); output(2) <= input(2) xor input(1) xor input(0) xor input(6); output(1) <= input(1) xor input(0) xor input(5); output(0) <= input(0) xor input(7) xor input(6) xor input(5);end mul_0E;
b0 <= a0_0E xor a1_0B xor a2_0D xor a3_09;
Conclusion: In hardware, InvMixColumns slower than MixColumns
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