In computing, bit numbering is the convention used to identify the bit positions in a binary number or a container of such a value. The bit number starts with zero and is incremented by one for each subsequent bit position.
Least significant bit
In computing, the least significant bit (LSB) is the bit position in a binary integer giving the units value, that is, determining whether the number is even or odd. The LSB is sometimes referred to as the low-order bit or right-most bit, due to the convention in positional notation of writing less significant digits further to the right. It is analogous to the least significant digit of a decimal integer, which is the digit in the ones (right-most) position.^{[1]}
It is common to assign each bit a position number, ranging from zero to N-1, where N is the number of bits in the binary representation used. Normally, the bit number is simply the exponent for the corresponding bit weight in base-2 (such as in 2^{31}..2^{0}). A few CPU manufacturers have assigned bit numbers the opposite way (which is not the same as different endianness). In any case, the least significant bit itself remains unambiguous as the unit bit.
The least significant bits (plural) are the bits of the number closest to, and including, the LSB. The least significant bits have the useful property of changing rapidly if the number changes even slightly. For example, if 1 (binary 00000001) is added to 3 (binary 00000011), the result will be 4 (binary 00000100) and three of the least significant bits will change (011 to 100). By contrast, the three most significant bits (MSBs) stay unchanged (000 to 000). Because of this volatility, least significant bits are frequently employed in pseudorandom number generators, steganographic tools, hash functions and checksums.
Least significant bit in digital steganography
In digital steganography, sensitive messages may be concealed by manipulating and storing information in the least significant bits of an image or a sound file. In the context of an image, if a user were to manipulate the last two bits of a color in a pixel, the value of the color would change at most ±3 value places, which is likely to be indistinguishable by the human eye. The user may later recover this information by extracting the least significant bits of the manipulated pixels to recover the original message.
This allows the storage or transfer of digital information to remain concealed.
Least significant byte
LSB can also stand for least significant byte.^{[2]} The meaning is parallel to the above: it is the byte (or octet) in that position of a multi-byte number which has the least potential value. If the abbreviation's meaning least significant byte isn't obvious from context, it should be stated explicitly to avoid confusion with least significant bit.
To avoid this ambiguity, the less abbreviated terms "lsbit" or "lsbyte" may be used.
Most significant bit
In computing, the most significant bit (MSB, also called the high-order bit) is the bit position in a binary number having the greatest value. The MSB is sometimes referred to as the high-order bit or left-most bit due to the convention in positional notation of writing more significant digits further to the left.
The MSB can also correspond to the sign bit of a signed binary number. In one's and two's complement notation, "1" signifies a negative number and "0" signifies a positive number.
It is common to assign each bit a position number ranging from zero to N−1 where N is the number of bits in the binary representation used. Normally, this is simply the exponent for the corresponding bit weight in base-2 (such as in 2^{31}..2^{0}
). Although a few CPU manufacturers assign bit numbers the opposite way (which is not the same as different endianness), the MSB unambiguously remains the most significant bit. This may be one of the reasons why the term MSB is often used instead of a bit number, although the primary reason is probably that different number representations use different numbers of bits.
By extension, the most significant bits (plural) are the bits closest to, and including, the MSB.
Most significant byte
MSB can also stand for "most significant byte".^{[3]} The meaning is parallel to the above: it is the byte (or octet) in that position of a multi-byte number which has the greatest potential value.
To avoid this ambiguity, the less abbreviated terms "MSbit" or "MSbyte" are often used.^{[4]}^{[5]}^{[6]}
Unsigned integer example
This table illustrates an example of decimal value of 149 and the location of LSB. In this particular example, the position of unit value (decimal 1 or 0) is located in bit position 0 (n = 0). MSB stands for Most Significant Bit, while LSB stands for Least Significant Bit.
Binary (Decimal: 149) | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 |
---|---|---|---|---|---|---|---|---|
Bit weight for given bit position n ( 2^{n} ) | 2^{7} | 2^{6} | 2^{5} | 2^{4} | 2^{3} | 2^{2} | 2^{1} | 2^{0} |
Bit position label | MSB | LSB |
Position of LSB is independent of how the bit position is transmitted (Some system transmit MSB first, others transmit LSB first), which is a question more of a topic of Endianness.
Most- vs least-significant bit first
The expressions most significant bit first and least significant bit first are indications on the ordering of the sequence of the bits in the bytes sent over a wire in a transmission protocol or in a stream (e.g. an audio stream).
Most significant bit first means that the most significant bit will arrive first: hence e.g. the hexadecimal number 0x12
, 00010010
in binary representation, will arrive as the sequence 0 0 0 1 0 0 1 0
.
Least significant bit first means that the least significant bit will arrive first: hence e.g. the same hexadecimal number 0x12
, again 00010010
in binary representation, will arrive as the (reversed) sequence 0 1 0 0 1 0 0 0
.
LSB 0 bit numbering
When the bit numbering starts at zero for the least significant bit (LSB) the numbering scheme is called "LSB 0".^{[7]} This bit numbering method has the advantage that for any unsigned number the value of the number can be calculated by using exponentiation with the bit number and a base of 2. The value of an unsigned binary integer is therefore
where b_{i} denotes the value of the bit with number i, and N denotes the number of bits in total.
MSB 0 bit numbering
Similarly, when the bit numbering starts at zero for the most significant bit (MSB) the numbering scheme is called "MSB 0".
The value of an unsigned binary integer is therefore
Other
ALGOL 68's elem operator is effectively "MSB 1 bit numbering" as the bits are numbered from left to right, with the first bit (bits elem 1) being the "most significant bit", and the expression (bits elem bits width) giving the "least significant bit". Similarly, when bits are coerced (typecast) to an array of Boolean ([ ]bool bits), the first element of this array (bits[lwb bits]) is again the "most significant bit".
For MSB 1 numbering, the value of an unsigned binary integer is
PL/I numbers BIT strings starting with 1 for the leftmost bit.
The Fortran BTEST function uses LSB 0 numbering.
Usage
Little-endian CPUs usually employ "LSB 0" bit numbering, however both bit numbering conventions can be seen in big-endian machines. Some architectures like SPARC and Motorola 68000 use "LSB 0" bit numbering, while S/390, PowerPC and PA-RISC use "MSB 0".^{[8]}
The recommended style for Request for Comments (RfC) documents is "MSB 0" bit numbering.^{[9]}^{[10]}
Bit numbering is usually transparent to the software, but some programming languages like Ada and hardware description languages like VHDL and verilog allow specifying the appropriate bit order for data type representation.^{[11]}
See also
- ARINC 429
- Binary numeral system
- Signed number representations
- Two's complement
- Endianness
- Binary logarithm
- Unit in the last place (ULP)
- Find first set
- MAC address: Bit-reversed notation
References
- ^ "IBM SNA Formats Bit Ordering is Opposite of Intel Convention". Microsoft. 2014-02-23.
- ^ Buczynski, Don (2002-09-05). "MSB/LSB Tutorial".
- ^ Rouse, Margaret (September 2005). "most significant bit or byte". TechTarget.
- ^ II, Patrick J. Sweeney (2010-03-11). RFID For Dummies. John Wiley & Sons. ISBN 9781118054475.
- ^ "Amulet Bitmap Format". web.mit.edu. Retrieved 2018-02-25.
- ^ "Memory MAYHEM! Memory, Byte Ordering and Alignment". www.cs.umd.edu. Retrieved 2018-02-25.
- ^ Langdon, Glen G. (1982). Computer Design. Computeach Press Inc. p. 52. ISBN 0-9607864-0-6.
- ^ David V. James (June 1990). "Multiplexed buses: the endian wars continue". IEEE Micro. 10 (3): 9–21. doi:10.1109/40.56322. ISSN 0272-1732. S2CID 24291134.
their first bits and nibbles came from ALU chips, which map zero (0) to the least significant bit. (...) some (otherwise) big-endian designers insist on using the little-endian notation to describe bits and the big-endian notation to describe bytes. (...) Note that IBM (on the S/360 and 370) and Hewlett-Packard (on the PA-RISC processor) consistently map zero to the MSB
- ^
Scott, Gregor (June 1998). "RFC 2360 - Guide for Internet Standards Writers". Internet Engineering Task Force (IETF). p. 11. Retrieved 2010-02-14.
The preferred form for packet diagrams is a sequence of long words in network byte order, with each word horizontal on the page and bit numbering at the top
- ^
"RFC 1166 - INTERNET NUMBERS". Internet Engineering Task Force (IETF). July 1990. p. 1. Retrieved 2014-06-11.
Whenever an octet represents a numeric quantity the left most bit in the diagram is the high order or most significant bit
- ^ Norman H. Cohen (January 1994). "Endian-independent record representation clauses" (PDF). Ada Letters. ACM SIGAda. XIV (1): 27–29. doi:10.1145/181492.181493. ISSN 1094-3641. S2CID 31612852. Retrieved 2008-12-20.
External links
- Bit Numbers
- Bit numbering for different CPUs:
- Motorola 68000 ("Bit manipulation" and "Reversed Bit-Numbering" sections)
- IBM Cell Broadband Processors ("Byte ordering and bit numbering")