DRAFT of 2020 May 28 Markku-Juhani O. Saarinen <[email protected]>
Many cryptographic functions require computation of Boolean functions from n bits to m bits. Sometimes these functions are large enough to have been traditionally implemented with table lookups, but timing side-channel considerations force their implementation in a bit-sliced manner.
The efficiency of bit-sliced logic is affected by three main factors:
-
The number of instructions.
-
Compressed (16-bit) and uncompressed (32-bit) instructions.
-
Logic depth for superscalar processing.
The truth table of any 2-input Boolean function has 22=4 bits. We write the truth table (TT) with the result of all-zero inputs on the right (at "bit 0"). All arithmetic instructions in RISC-V crypto extension have two source operands, even if they only use one.
| TT | C Expr | Assembler | Type |
|---|---|---|---|
0000 |
|
MV ZERO |
pseudo |
0001 |
|
OR + NOT |
n/a ("NOR") |
0010 |
|
ANDN |
ext (b) |
0011 |
|
NOT |
pseudo |
0100 |
|
ANDN |
ext (b) |
0110 |
|
XOR |
base (c) |
0101 |
|
NOT |
pseudo |
0111 |
|
AND + NOT |
n/a ("NAND") |
1000 |
|
AND |
base (c) |
1001 |
|
XNOR |
ext (b) |
1010 |
|
MV |
pseudo |
1011 |
|
ORN |
ext (b) |
1100 |
|
MV |
pseudo |
1101 |
|
ORN |
ext (b) |
1110 |
|
OR |
base (c) |
1111 |
|
NOT ZERO |
pseudo |
Pseudo:
In RISC-V the register X0 is permanently set to zero. Moving the contents of
a register to another is performed with an MV pseudoinstruction, which has
a compressed (16-bit rather than 32-bit) equivalent C.MV — both are
usually encoded as addition with zero). Single-operand one’s complement NOT
is implemented with immediate exclusive-or XORI rd, rs1, -1, which does not
have an equivalent form in the compressed instruction set. However one may
keep the value ~0=-1 in an auxiliary register, after which compressed
C.XOR always suffices.
Base:
The base set instruction set provides the basic commutative Boolean logic
operations AND, OR, XOR, each of which has compressed (half-size)
instruction counterparts C.AND, C.OR, X.XOR respectively. Note that
compressed forms can only be used if the destination operand is the same
as one of the input operands (corresponding to C operators
&=, |=, and ^=.)
Extension:
The Bitmanip and Crypto extensions add three counterparts to these
instructions by inverting the second operand: ANDN (&~), ORN (|~),
and XNOR (^~). Even though grouping the NOT (~) with the main
operation makes no difference in C syntax, we use this notation and hope
that compiler will understand this relationship and can use the correct
instructions. These operations have no compressed form. ANDN and ORN are
non-commutative, so these correspond to five two-input Boolean logic
functions.
Not available: Together the trivial (6), base (3), and expanded (5) make up 14 of the 16 possible functions. Two functions that are left out are NOR and NAND, which need to be constructed from two instructions.
Not really, at least for bigger bitsliced functions. To see this, consider a bitsliced circuit (directed acyclic graph) implementing an n-to-m bit Boolean function, where each gate has fan-in 2 and unlimited fan-out.
Out of the 16 Boolean functions, six can be eliminated trivially. To eliminate a NOT gate, modify all gates that it is driving (e.g. changing an AND into an ANDN, or vice versa).
Furthermore, any gate (except those directly driving the m output bits) can be changed into its inverted counterpart (fully inverted truth table) by modifying gates "downstream" in the circuit.
Therefore it is possible to replace all (two-instruction) "NOR" functions
~(x | y) with OR functions x | y and similarly "NAND"
functions ~(x & y) with simple AND function x & y by changing
all gates driven by them. Starting from input and proceeding in the acyclic
graph, we see that at most m additional output inverters are required
after this change is applied to a circuit; the structure and size of the
circuit remains the otherwise the same in this transformation.
In fact it may be preferable to replace XNOR functions x ^~ y with plain
XOR functions x ^ y since a compressed form (C.XOR) is available for it.
It may also preferable to replace ORN (x |~ y, y |~ x) with its
inverse ANDN (y &~ x, x &~ y), for portability. ANDN (also called BIC,
"bit clear") is available on Intel and ARM platforms while ORN is not.
Any circuit can be transformed to use the four-gate set XOR, AND, OR, ANDN
with at most "m" additinal gates (NOT gates on outputs).
We list below optimized expressions for all 3-input Boolean functions; since the truth table is 23=8 bits, there are 28=256 such functions in all. There are several, sometimes contradictory optimality metrics for a function, and even with those there many equivalent expressions.
We used the following selection criteria in our exhaustive search. In order of importance, select expression with..
-
The smallest number of instructions (Len).
-
Smallest circuit depth (Dep, measured with longest topological path).
-
Minimum code size.
Code size is measured assuming that compressed instructions C.AND, C.OR, and C.XOR can be used and input variables can be overwritten. Note that this is among all code sequences with the same length and depth.
Two expressions are given; one for base instruction set (with compression) and another for expanded
SHA-2 Algorithms utilize two main nonlinear functions, Ch and Maj, defined in Section 4.1 of FIPS 180-4 as:
Ch(x,y,z) = (x & y) ^ (~x & z) // 4 insn: 2 * AND, 1 * XOR, 1 NOT Maj(x,y,z) = (x & y) ^ (x & z) ^ (y & z) // 5 insn: 3 * AND, 2 * XOR
One can see that AND, NOT sequence (~x & z) in Ch can be written using
a single ANDN instruction (z &~ x) from the extension, reducing the
instruction count to 3. Are further optimizations possible?
One can evaluate the truth tables bit-by-bit, or in "parallel" by setting
x=0xAA, y=0xCC, z=0xF0 and computing the 8-bit result.
This yields truth tables for Ch: 0xD8, 11011000 and Maj: 0xE8, 11101000.
It appears that the expression Ch = ((x & y) | (z &~ x)) is optimal if
&~ is a single instruction (ANDN). However the table also gives a
three-instruction sequence that uses just the base set:
Ch(x,y,z) = (z ^ (x & (y ^ z))). One may ask why is this not the "best"
for the extension too - it’s shorter! This is because of our prioritization
of depth over opcode length:
Extended: Base set:
((x & y) | (z &~ x)) (z ^ (x & (y ^ z)))
[Ch]
\
[Ch] (^)
\ / \
(|) z (&)
/ \ / \
(&) (&~) <-ANDN x (^)
/ \ / \ / \
x y z x y z
Len=3 Dep=2 Len=3 Dep=3
For Maj we obtain the same 4-instruction sequence for both base and
extended instruction set: Maj(x,y,z) = (x ^ ((x ^ y) & (x ^ z))). If the
input variables can be overwritten, this sequence could be implemented
(using pseudocode):
C.XOR z, z, x // z ^= x -- rs1=rd for compressed
C.XOR y, y, x // y ^= x
C.AND y, y, z // y &= z
C.XOR x, x, y // x ^= y -- the result is in x
With C extension compression this sequence is 4 * 2 = 8 bytes, the size of two "normal" instructions. This is probably why it was chosen from the set of all Len=3 and Dep=3 sequences.
In a typical SHA-2 implementation not all input variables can be overwritten. However the compressed form selection criteria has least priority (among otherwise equivalent candidates). It is difficult to predict the stack load/store sequence so giving the Compiler a form that allows the use of compressed instructions is generally preferable.
So, based on these considerations, and prioritizing length over depth, we recommend:
Ch(x,y,z) = (z ^ (x & (y ^ z))) // 3 insn: 1 * AND, 2 * XOR Maj(x,y,z) = (x ^ ((x ^ y) & (x ^ z))) // 4 insn: 1 * AND, 3 * XOR
| Numb | TT | Len | Dep | C expr (expanded) | Len | Dep | C expr (base set) |
|---|---|---|---|---|---|---|---|
0x00 |
00000000 |
0 |
0 |
|
0 |
0 |
|
0x01 |
00000001 |
3 |
2 |
|
3 |
3 |
|
0x02 |
00000010 |
2 |
2 |
|
3 |
3 |
|
0x03 |
00000011 |
2 |
2 |
|
2 |
2 |
|
0x04 |
00000100 |
2 |
2 |
|
3 |
3 |
|
0x05 |
00000101 |
2 |
2 |
|
2 |
2 |
|
0x06 |
00000110 |
2 |
2 |
|
3 |
2 |
|
0x07 |
00000111 |
3 |
2 |
|
3 |
3 |
|
0x08 |
00001000 |
2 |
2 |
|
3 |
2 |
|
0x09 |
00001001 |
2 |
2 |
|
3 |
3 |
|
0x0A |
00001010 |
1 |
1 |
|
2 |
2 |
|
0x0B |
00001011 |
2 |
2 |
|
4 |
4 |
|
0x0C |
00001100 |
1 |
1 |
|
2 |
2 |
|
0x0D |
00001101 |
2 |
2 |
|
4 |
3 |
|
0x0E |
00001110 |
2 |
2 |
|
3 |
2 |
|
0x0F |
00001111 |
1 |
1 |
|
1 |
1 |
|
0x10 |
00010000 |
2 |
2 |
|
3 |
3 |
|
0x11 |
00010001 |
2 |
2 |
|
2 |
2 |
|
0x12 |
00010010 |
2 |
2 |
|
3 |
2 |
|
0x13 |
00010011 |
3 |
3 |
|
3 |
3 |
|
0x14 |
00010100 |
2 |
2 |
|
3 |
2 |
|
0x15 |
00010101 |
3 |
2 |
|
3 |
3 |
|
0x16 |
00010110 |
4 |
3 |
|
4 |
3 |
|
0x17 |
00010111 |
4 |
3 |
|
5 |
4 |
|
0x18 |
00011000 |
3 |
2 |
|
3 |
2 |
|
0x19 |
00011001 |
3 |
2 |
|
4 |
3 |
|
0x1A |
00011010 |
3 |
2 |
|
3 |
3 |
|
0x1B |
00011011 |
3 |
2 |
|
4 |
3 |
|
0x1C |
00011100 |
3 |
2 |
|
3 |
3 |
|
0x1D |
00011101 |
3 |
2 |
|
4 |
3 |
|
0x1E |
00011110 |
2 |
2 |
|
2 |
2 |
|
0x1F |
00011111 |
3 |
2 |
|
3 |
3 |
|
0x20 |
00100000 |
2 |
2 |
|
3 |
3 |
|
0x21 |
00100001 |
2 |
2 |
|
3 |
3 |
|
0x22 |
00100010 |
1 |
1 |
|
2 |
2 |
|
0x23 |
00100011 |
2 |
2 |
|
4 |
3 |
|
0x24 |
00100100 |
3 |
2 |
|
3 |
2 |
|
0x25 |
00100101 |
3 |
2 |
|
4 |
3 |
|
0x26 |
00100110 |
3 |
2 |
|
3 |
3 |
|
0x27 |
00100111 |
3 |
2 |
|
4 |
3 |
|
0x28 |
00101000 |
2 |
2 |
|
2 |
2 |
|
0x29 |
00101001 |
4 |
3 |
|
5 |
3 |
|
0x2A |
00101010 |
2 |
2 |
|
3 |
3 |
|
0x2B |
00101011 |
4 |
3 |
|
5 |
3 |
|
0x2C |
00101100 |
3 |
2 |
|
3 |
2 |
|
0x2D |
00101101 |
2 |
2 |
|
3 |
3 |
|
0x2E |
00101110 |
3 |
2 |
|
3 |
2 |
|
0x2F |
00101111 |
2 |
2 |
|
4 |
3 |
|
0x30 |
00110000 |
1 |
1 |
|
2 |
2 |
|
0x31 |
00110001 |
2 |
2 |
|
4 |
3 |
|
0x32 |
00110010 |
2 |
2 |
|
3 |
3 |
|
0x33 |
00110011 |
1 |
1 |
|
1 |
1 |
|
0x34 |
00110100 |
3 |
2 |
|
3 |
3 |
|
0x35 |
00110101 |
3 |
3 |
|
4 |
3 |
|
0x36 |
00110110 |
2 |
2 |
|
2 |
2 |
|
0x37 |
00110111 |
3 |
3 |
|
3 |
3 |
|
0x38 |
00111000 |
3 |
2 |
|
3 |
2 |
|
0x39 |
00111001 |
2 |
2 |
|
3 |
3 |
|
0x3A |
00111010 |
3 |
3 |
|
3 |
3 |
|
0x3B |
00111011 |
2 |
2 |
|
4 |
4 |
|
0x3C |
00111100 |
1 |
1 |
|
1 |
1 |
|
0x3D |
00111101 |
3 |
2 |
|
4 |
3 |
|
0x3E |
00111110 |
3 |
2 |
|
4 |
3 |
|
0x3F |
00111111 |
2 |
2 |
|
2 |
2 |
|
0x40 |
01000000 |
2 |
2 |
|
3 |
2 |
|
0x41 |
01000001 |
2 |
2 |
|
3 |
3 |
|
0x42 |
01000010 |
3 |
2 |
|
3 |
2 |
|
0x43 |
01000011 |
3 |
2 |
|
4 |
3 |
|
0x44 |
01000100 |
1 |
1 |
|
2 |
2 |
|
0x45 |
01000101 |
2 |
2 |
|
4 |
3 |
|
0x46 |
01000110 |
3 |
2 |
|
3 |
3 |
|
0x47 |
01000111 |
3 |
2 |
|
4 |
4 |
|
0x48 |
01001000 |
2 |
2 |
|
2 |
2 |
|
0x49 |
01001001 |
4 |
3 |
|
5 |
4 |
|
0x4A |
01001010 |
3 |
2 |
|
3 |
2 |
|
0x4B |
01001011 |
2 |
2 |
|
3 |
3 |
|
0x4C |
01001100 |
2 |
2 |
|
3 |
3 |
|
0x4D |
01001101 |
4 |
3 |
|
5 |
4 |
|
0x4E |
01001110 |
3 |
2 |
|
3 |
2 |
|
0x4F |
01001111 |
2 |
2 |
|
4 |
3 |
|
0x50 |
01010000 |
1 |
1 |
|
2 |
2 |
|
0x51 |
01010001 |
2 |
2 |
|
4 |
3 |
|
0x52 |
01010010 |
3 |
2 |
|
3 |
3 |
|
0x53 |
01010011 |
3 |
3 |
|
4 |
4 |
|
0x54 |
01010100 |
2 |
2 |
|
3 |
2 |
|
0x55 |
01010101 |
1 |
1 |
|
1 |
1 |
|
0x56 |
01010110 |
2 |
2 |
|
2 |
2 |
|
0x57 |
01010111 |
3 |
2 |
|
3 |
3 |
|
0x58 |
01011000 |
3 |
2 |
|
3 |
2 |
|
0x59 |
01011001 |
2 |
2 |
|
3 |
3 |
|
0x5A |
01011010 |
1 |
1 |
|
1 |
1 |
|
0x5B |
01011011 |
3 |
2 |
|
4 |
3 |
|
0x5C |
01011100 |
3 |
3 |
|
3 |
3 |
|
0x5D |
01011101 |
2 |
2 |
|
4 |
3 |
|
0x5E |
01011110 |
3 |
2 |
|
4 |
3 |
|
0x5F |
01011111 |
2 |
2 |
|
2 |
2 |
|
0x60 |
01100000 |
2 |
2 |
|
2 |
2 |
|
0x61 |
01100001 |
4 |
3 |
|
5 |
3 |
|
0x62 |
01100010 |
3 |
2 |
|
3 |
2 |
|
0x63 |
01100011 |
2 |
2 |
|
3 |
3 |
|
0x64 |
01100100 |
3 |
2 |
|
3 |
2 |
|
0x65 |
01100101 |
2 |
2 |
|
3 |
3 |
|
0x66 |
01100110 |
1 |
1 |
|
1 |
1 |
|
0x67 |
01100111 |
3 |
2 |
|
4 |
3 |
|
0x68 |
01101000 |
4 |
3 |
|
4 |
3 |
|
0x69 |
01101001 |
2 |
2 |
|
3 |
2 |
|
0x6A |
01101010 |
2 |
2 |
|
2 |
2 |
|
0x6B |
01101011 |
4 |
3 |
|
5 |
3 |
|
0x6C |
01101100 |
2 |
2 |
|
2 |
2 |
|
0x6D |
01101101 |
4 |
3 |
|
5 |
3 |
|
0x6E |
01101110 |
3 |
2 |
|
4 |
3 |
|
0x6F |
01101111 |
2 |
2 |
|
3 |
2 |
|
0x70 |
01110000 |
2 |
2 |
|
3 |
3 |
|
0x71 |
01110001 |
4 |
3 |
|
5 |
4 |
|
0x72 |
01110010 |
3 |
2 |
|
3 |
2 |
|
0x73 |
01110011 |
2 |
2 |
|
4 |
3 |
|
0x74 |
01110100 |
3 |
2 |
|
3 |
2 |
|
0x75 |
01110101 |
2 |
2 |
|
4 |
3 |
|
0x76 |
01110110 |
3 |
2 |
|
4 |
3 |
|
0x77 |
01110111 |
2 |
2 |
|
2 |
2 |
|
0x78 |
01111000 |
2 |
2 |
|
2 |
2 |
|
0x79 |
01111001 |
4 |
3 |
|
5 |
3 |
|
0x7A |
01111010 |
3 |
2 |
|
4 |
3 |
|
0x7B |
01111011 |
2 |
2 |
|
3 |
2 |
|
0x7C |
01111100 |
3 |
2 |
|
4 |
3 |
|
0x7D |
01111101 |
2 |
2 |
|
3 |
2 |
|
0x7E |
01111110 |
3 |
2 |
|
3 |
2 |
|
0x7F |
01111111 |
3 |
2 |
|
3 |
3 |
|
0x80 |
10000000 |
2 |
2 |
|
2 |
2 |
|
0x81 |
10000001 |
3 |
2 |
|
4 |
3 |
|
0x82 |
10000010 |
2 |
2 |
|
3 |
3 |
|
0x83 |
10000011 |
3 |
2 |
|
4 |
3 |
|
0x84 |
10000100 |
2 |
2 |
|
3 |
3 |
|
0x85 |
10000101 |
3 |
2 |
|
4 |
3 |
|
0x86 |
10000110 |
4 |
3 |
|
4 |
3 |
|
0x87 |
10000111 |
2 |
2 |
|
3 |
2 |
|
0x88 |
10001000 |
1 |
1 |
|
1 |
1 |
|
0x89 |
10001001 |
3 |
2 |
|
4 |
4 |
|
0x8A |
10001010 |
2 |
2 |
|
3 |
3 |
|
0x8B |
10001011 |
3 |
2 |
|
4 |
3 |
|
0x8C |
10001100 |
2 |
2 |
|
3 |
3 |
|
0x8D |
10001101 |
3 |
2 |
|
4 |
3 |
|
0x8E |
10001110 |
4 |
3 |
|
4 |
3 |
|
0x8F |
10001111 |
2 |
2 |
|
3 |
2 |
|
0x90 |
10010000 |
2 |
2 |
|
3 |
3 |
|
0x91 |
10010001 |
3 |
2 |
|
4 |
3 |
|
0x92 |
10010010 |
4 |
3 |
|
4 |
3 |
|
0x93 |
10010011 |
2 |
2 |
|
3 |
2 |
|
0x94 |
10010100 |
4 |
3 |
|
4 |
4 |
|
0x95 |
10010101 |
2 |
2 |
|
3 |
2 |
|
0x96 |
10010110 |
2 |
2 |
|
2 |
2 |
|
0x97 |
10010111 |
4 |
3 |
|
5 |
3 |
|
0x98 |
10011000 |
3 |
2 |
|
4 |
3 |
|
0x99 |
10011001 |
1 |
1 |
|
2 |
2 |
|
0x9A |
10011010 |
2 |
2 |
|
3 |
2 |
|
0x9B |
10011011 |
3 |
2 |
|
4 |
3 |
|
0x9C |
10011100 |
2 |
2 |
|
3 |
2 |
|
0x9D |
10011101 |
3 |
2 |
|
4 |
3 |
|
0x9E |
10011110 |
4 |
3 |
|
4 |
3 |
|
0x9F |
10011111 |
2 |
2 |
|
3 |
3 |
|
0xA0 |
10100000 |
1 |
1 |
|
1 |
1 |
|
0xA1 |
10100001 |
3 |
2 |
|
4 |
4 |
|
0xA2 |
10100010 |
2 |
2 |
|
3 |
3 |
|
0xA3 |
10100011 |
3 |
3 |
|
4 |
3 |
|
0xA4 |
10100100 |
3 |
2 |
|
4 |
3 |
|
0xA5 |
10100101 |
1 |
1 |
|
2 |
2 |
|
0xA6 |
10100110 |
2 |
2 |
|
3 |
2 |
|
0xA7 |
10100111 |
3 |
2 |
|
4 |
3 |
|
0xA8 |
10101000 |
2 |
2 |
|
2 |
2 |
|
0xA9 |
10101001 |
2 |
2 |
|
3 |
2 |
|
0xAA |
10101010 |
0 |
0 |
|
0 |
0 |
|
0xAB |
10101011 |
2 |
2 |
|
3 |
3 |
|
0xAC |
10101100 |
3 |
3 |
|
3 |
3 |
|
0xAD |
10101101 |
3 |
2 |
|
4 |
3 |
|
0xAE |
10101110 |
2 |
2 |
|
3 |
3 |
|
0xAF |
10101111 |
1 |
1 |
|
2 |
2 |
|
0xB0 |
10110000 |
2 |
2 |
|
3 |
3 |
|
0xB1 |
10110001 |
3 |
2 |
|
4 |
3 |
|
0xB2 |
10110010 |
4 |
3 |
|
4 |
3 |
|
0xB3 |
10110011 |
2 |
2 |
|
3 |
2 |
|
0xB4 |
10110100 |
2 |
2 |
|
3 |
2 |
|
0xB5 |
10110101 |
3 |
2 |
|
4 |
3 |
|
0xB6 |
10110110 |
4 |
3 |
|
4 |
3 |
|
0xB7 |
10110111 |
2 |
2 |
|
3 |
3 |
|
0xB8 |
10111000 |
3 |
2 |
|
3 |
3 |
|
0xB9 |
10111001 |
3 |
2 |
|
4 |
3 |
|
0xBA |
10111010 |
2 |
2 |
|
3 |
3 |
|
0xBB |
10111011 |
1 |
1 |
|
2 |
2 |
|
0xBC |
10111100 |
3 |
2 |
|
3 |
2 |
|
0xBD |
10111101 |
3 |
2 |
|
4 |
3 |
|
0xBE |
10111110 |
2 |
2 |
|
2 |
2 |
|
0xBF |
10111111 |
2 |
2 |
|
3 |
3 |
|
0xC0 |
11000000 |
1 |
1 |
|
1 |
1 |
|
0xC1 |
11000001 |
3 |
2 |
|
4 |
4 |
|
0xC2 |
11000010 |
3 |
2 |
|
4 |
3 |
|
0xC3 |
11000011 |
1 |
1 |
|
2 |
2 |
|
0xC4 |
11000100 |
2 |
2 |
|
3 |
3 |
|
0xC5 |
11000101 |
3 |
3 |
|
4 |
4 |
|
0xC6 |
11000110 |
2 |
2 |
|
3 |
2 |
|
0xC7 |
11000111 |
3 |
2 |
|
4 |
3 |
|
0xC8 |
11001000 |
2 |
2 |
|
2 |
2 |
|
0xC9 |
11001001 |
2 |
2 |
|
3 |
2 |
|
0xCA |
11001010 |
3 |
3 |
|
3 |
3 |
|
0xCB |
11001011 |
3 |
2 |
|
4 |
3 |
|
0xCC |
11001100 |
0 |
0 |
|
0 |
0 |
|
0xCD |
11001101 |
2 |
2 |
|
3 |
3 |
|
0xCE |
11001110 |
2 |
2 |
|
3 |
3 |
|
0xCF |
11001111 |
1 |
1 |
|
2 |
2 |
|
0xD0 |
11010000 |
2 |
2 |
|
3 |
3 |
|
0xD1 |
11010001 |
3 |
2 |
|
4 |
4 |
|
0xD2 |
11010010 |
2 |
2 |
|
3 |
2 |
|
0xD3 |
11010011 |
3 |
2 |
|
4 |
3 |
|
0xD4 |
11010100 |
4 |
3 |
|
4 |
3 |
|
0xD5 |
11010101 |
2 |
2 |
|
3 |
2 |
|
0xD6 |
11010110 |
4 |
3 |
|
4 |
3 |
|
0xD7 |
11010111 |
2 |
2 |
|
3 |
3 |
|
0xD8 |
11011000 |
3 |
2 |
|
3 |
3 |
|
0xD9 |
11011001 |
3 |
2 |
|
4 |
3 |
|
0xDA |
11011010 |
3 |
2 |
|
3 |
2 |
|
0xDB |
11011011 |
3 |
2 |
|
4 |
3 |
|
0xDC |
11011100 |
2 |
2 |
|
3 |
3 |
|
0xDD |
11011101 |
1 |
1 |
|
2 |
2 |
|
0xDE |
11011110 |
2 |
2 |
|
2 |
2 |
|
0xDF |
11011111 |
2 |
2 |
|
3 |
3 |
|
0xE0 |
11100000 |
2 |
2 |
|
2 |
2 |
|
0xE1 |
11100001 |
2 |
2 |
|
3 |
2 |
|
0xE2 |
11100010 |
3 |
2 |
|
3 |
3 |
|
0xE3 |
11100011 |
3 |
2 |
|
4 |
3 |
|
0xE4 |
11100100 |
3 |
2 |
|
3 |
3 |
|
0xE5 |
11100101 |
3 |
2 |
|
4 |
3 |
|
0xE6 |
11100110 |
3 |
2 |
|
3 |
2 |
|
0xE7 |
11100111 |
3 |
2 |
|
4 |
3 |
|
0xE8 |
11101000 |
4 |
3 |
|
4 |
3 |
|
0xE9 |
11101001 |
4 |
3 |
|
5 |
3 |
|
0xEA |
11101010 |
2 |
2 |
|
2 |
2 |
|
0xEB |
11101011 |
2 |
2 |
|
3 |
3 |
|
0xEC |
11101100 |
2 |
2 |
|
2 |
2 |
|
0xED |
11101101 |
2 |
2 |
|
3 |
3 |
|
0xEE |
11101110 |
1 |
1 |
|
1 |
1 |
|
0xEF |
11101111 |
2 |
2 |
|
3 |
2 |
|
0xF0 |
11110000 |
0 |
0 |
|
0 |
0 |
|
0xF1 |
11110001 |
2 |
2 |
|
3 |
3 |
|
0xF2 |
11110010 |
2 |
2 |
|
3 |
3 |
|
0xF3 |
11110011 |
1 |
1 |
|
2 |
2 |
|
0xF4 |
11110100 |
2 |
2 |
|
3 |
3 |
|
0xF5 |
11110101 |
1 |
1 |
|
2 |
2 |
|
0xF6 |
11110110 |
2 |
2 |
|
2 |
2 |
|
0xF7 |
11110111 |
2 |
2 |
|
3 |
3 |
|
0xF8 |
11111000 |
2 |
2 |
|
2 |
2 |
|
0xF9 |
11111001 |
2 |
2 |
|
3 |
3 |
|
0xFA |
11111010 |
1 |
1 |
|
1 |
1 |
|
0xFB |
11111011 |
2 |
2 |
|
3 |
2 |
|
0xFC |
11111100 |
1 |
1 |
|
1 |
1 |
|
0xFD |
11111101 |
2 |
2 |
|
3 |
2 |
|
0xFE |
11111110 |
2 |
2 |
|
2 |
2 |
|
0xFF |
11111111 |
0 |
1 |
|
0 |
1 |
|