1's complement and 2's complement Representation

一的补码(one's complement) 指的是正数=原码,负数=反码
而二的补码(two's complement) 指的就是通常所指的补码

IP checksum definition

The IP checksum is the 16 bit one's complement of the one's complement sum of all 16 bit words in the header.

One question many people may ask is "What is the 1's complement sum ?". This is because all computers utilize the 2's complement representation and the 1's complement is not used. The following gives a short introduction.

2's complement fixed point integers (8-bit)

Binary	Decimal	Hex
0000 0000	0	00
0000 0001	1	01
0000 0010	2	02
0000 0011	3	03
1111 1111	-1	FF
1111 1110	-2	FE
1111 1101	-3	FD

Let's add two intergers:
-3 + 5 = 2
FD + 05 = 01 02
Discarding the carry (01) gives the correct result.

1's complement fixed point integers (8-bit)

Binary	Decimal	Hex
0000 0000	0	00
0000 0001	1	01
0000 0010	2	02
0000 0011	3	03
1111 1111	-0	FF
1111 1110	-1	FE
1111 1101	-2	FD
1111 1100	-3	FC

Add the same numbers:
-3 + 5 = 2
FC + 05 = 01 01
Adding the carry (01) to the LSB (01) gives the correct result:
01 + 01 = 02

So, the 1's complement sum is done by summing the numbers and adding the carry (or carries) to the result..

Simple Internet checksum example

Suppose we have an 8-bit, 2's complement, machine and send the packet

FE 05 00

where 00 is the checksum field.

Let's calculate and verify the Internet checksum.

FE + 05 = 01 03

This is the result of the normal (2's complement) addition. The 1's complement sum requires the addition of the carry to the 8-bit word (even though we will not get the same result)

03 + 01 = 04

so the 1's complement sum of FE + 05 is 04.

The 1's complement of the 1's complement sum (Internet checksum) will be

~04 = FB

and the packet will be sent as

FE 05 FB

Now, at the receiving end we add all the received bytes, including the checksum (again using the 2's complement representation)

FE + 05 + FB = 01 FE

The 1's complement sum is

FE + 01 = FF = -0

which checks that the transmission was OK (see below).

A more complex example (32-bit machine)

As shown in RFC 1071, the checksum calculation is done in the following way:

(1) Adjacent octets to be checksummed are paired to form 16-bit integers, and the 1's complement sum of these 16-bit integers is formed.

(2) To generate a checksum, the checksum field itself is cleared, the 16-bit 1's complement sum is computed over the octets concerned, and the 1's complement of this sum is placed in the checksum field.

(3) To check a checksum, the 1's complement sum is computed over the same set of octets, including the checksum field. If the result is all 1 bits (-0 in 1's complement arithmetic), the check succeeds.

Packet

01 00 F2 03 F4 F5 F6 F7 00 00

(00 00 is the checksum field)

Form the 16-bit words

0100 F203 F4F5 F6F7

Calculate 2's complement sum

0100 + F203 + F4F5 + F6F7 = 0002 DEEF (store the sum in a 32-bit word)

Add the carries (0002) to get the 16-bit 1's complement sum

DEEF + 002 = DEF1

Calculate 1's complement of the 1's complement sum

~DEF1 = 210E

We send the packet including the checksum 21 0E

01 00 F2 03 F4 F5 F6 F7 21 0E

At the receiving

0100 + F203 + F4F5 + F6F7 + 210E = 0002 FFFD
FFFD + 0002 = FFFF

which checks OK.

Comments
It may look awkword to use a 1's complement addition on 2's complement machines. This method however has its own benefits.

Probably the most important is that it is endian independent. Little Endian computers store hex numbers with the LSB last (Intel processors for example). Big Endian computers put the LSB first (IBM mainframes for example). When carry is added to the LSB to form the 1's complement sum (see the example) it doesn't matter if we add 03 + 01 or 01 + 03. The result is the same.

Other benefits include the easiness of checking the transmission and the checksum calculation plus a variety of ways to speed up the calculation by updating only IP fields that have changed.

The IP Header Checksum is computed on the header fields only.
Before starting the calculation, the checksum fields (octets 11 and 12)
are made equal to zero.

In the example code,
u16 buff[] is an array containing all octets in the header with octets 11 and 12 equal to zero.
u16 len_ip_header is the length (number of octets) of the header.

/*
**************************************************************************
Function: ip_sum_calc
Description: Calculate the 16 bit IP sum.
***************************************************************************
*/
typedef unsigned short u16;
typedef unsigned long u32;

u16 ip_sum_calc(u16 len_ip_header, u16 buff[])
{
u16 word16;
u32 sum=0;
u16 i;

// make 16 bit words out of every two adjacent 8 bit words in the packet
// and add them up
for (i=0;i<len_ip_header;i=i+2){
word16 =((buff[i]<<8)&0xFF00)+(buff[i+1]&0xFF);
sum = sum + (u32) word16;
}

// take only 16 bits out of the 32 bit sum and add up the carries
while (sum>>16)
sum = (sum & 0xFFFF)+(sum >> 16);

// one's complement the result
sum = ~sum;

return ((u16) sum);
}

1's complement and 2's complement Representation相关推荐

What’s difference between 1’s Complement and 2’s Complement?
1's complement of a binary number is another binary number obtained by toggling all bits in it, i.e. ...
ones' complement 和 two's complement
ones' complement 译为"一的补码",或译为"反码". two's complement 译为"二的补码",或译 ...
one's-complement 反码, two's-complement 补码, one's complement sum, two's complement sum
1.one's-complement: 反码,高位为符号位: two's-complement: 补码,高位为符号位: 2. one's complement sum 反码加法,需要加上进位: two ...
[Swift]LeetCode1009. 十进制整数的补码 | Complement of Base 10 Integer
★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★★ ➤微信公众号:山青咏芝(shanqingyongzhi) ➤博客园地址:山青咏芝(https://www.cnblog ...
整数二进制补码的数学原理(two's complement)
转载自整数二进制补码的数学原理(two's complement) ================================================================== ...
计算机导论简答芯片,吉大计算机 - 计算机导论简答题 (2011级)
C1 2. Q: What model is the basis for today's computers? A: Von Neumann Model. 5. Q: What are the sub ...
二进制文件签名_二进制数的签名表示
二进制文件签名 Prerequisite: Number systems 先决条件: 数字系统 Until now, we have only talked about positive number ...
软件工程计算机导论试题及答案,2010级计算机学院计算机导论试题A卷.doc
2010级计算机学院计算机导论试题A卷,计算机导论试题,计算机导论考试试题,护理学导论试题及答案,软件工程导论期末试题,社区护理导论试题,软件工程导论试题,药学导论试题,公共政策导论试题,语言学导论期 ...
华南理工大学计算机科学概论试卷,华南理工大学计算机概论试卷b.doc
华南理工大学计算机概论试卷b = 1 \* ROMAN I. Multiple-Choice Questions (1.5*20=30 points) 1. What is a unique char ...

1's complement and 2's complement Representation

1's complement and 2's complement Representation相关推荐

最新文章

热门文章