Related
While studying this document on the Evolution of JPEG, i came across "The law of the eight" in Section 7.3 of the above document.
Despite the introduction of other block sizes from 1 to 16 with the SmartScale extension, beyond the fixed size 8 in the original JPEG standard, the fact remains that the block size of 8 will still be the default value, and that all other-size DCTs are scaled in reference to the standard 8x8 DCT.
The “Law of the Eight” explains, why the size 8 is the right default and reference value for the DCT size.
My question is
What exactly is this "law of the eight" ?
Historically, was a study performed that evaluated numerous images from a sample to arrive at the conclusion that 8x8 image block contains enough redundant data to support compression techniques using DCT? With very large image sizes like 8M(4Kx4K) fast becoming the norm in most digital images/videos, is this assumption still valid?
Another historic reason to limit the macro-block to 8x8 would have been the computationally prohibitive image-data size for larger macro-blocks. With modern super-scalar architectures (eg. CUDA) that restriction no longer applies.
Earlier similar questions exist - 1, 2 and 3. But none of them bother about any details/links/references to this mysterious fundamental "law of the eight".
1. References/excerpts/details of the original study will be highly appreciated as i would like to repeat it with a modern data-set with very large sized images to test the validity of 8x8 macro blocks being optimal.
2. In case a similar study has been recently carried-out, references to it are welcome too.
3. I do understand that SmartScale is controversial. Without any clear potential benefits 1, at best it is comparable with other backward-compliant extensions of the jpeg standard 2. My goal is to understand whether the original reasons behind choosing 8x8 as the DCT block-size (in jpeg image compression standard) are still relevant, hence i need to know what the law of the eight is.
My understanding is, the Law of the Eight is simply a humorous reference to the fact that the Baseline JPEG algorithm prescribed 8x8 as its only block size.
P.S. In other words, "the Law of the Eight" is a way to explain why "all other-size DCTs are scaled in reference to 8x8 DCT" by bringing in the historical perspective -- the lack of support for any other size in the original standard and its defacto implementations.
The next question to ask: why Eight? (Note that despite being a valid question, this is not the subject of the present discussion, which would still be relevant even if another value was picked historically, e.g. "Law of the Ten" or "Law of the Thirty Two".) The answer to that one is: because computational complexity of the problem grows as O(N^2) (unless FCT-class algorithms are employed, which grow slower as O(N log N) but are harder to implement on primitive hardware of embedded platforms, hence limited applicability), so larger block sizes quickly become impractical. Which is why 8x8 was chosen, as small enough to be practical on wide range of platforms but large enough to allow for not-too-coarse control of quantization levels for different frequencies.
Since the standard has clearly scratched an itch, a whole ecosphere soon grew around it, including implementations optimized for 8x8 as their sole supported block size. Once the ecosphere was in place, it became impossible to change the block size without breaking existing implementations. As that was highly undesirable, any tweaks to DCT/quantization parameters had to remain compatible with 8x8-only decoders. I believe this consideration must be what's referred to as the "Law of the Eight".
While not being an expert, I don't see how larger block sizes can help. First, dynamic range of values in one block will increase on average, requiring more bits to represent them. Second, relative quantization of frequencies ranging from "all" (represented by the block) to "pixel" has to stay the same (it is dictated by human perception bias after all), the quantization will get a bit smoother, that's all, and for the same compression level the potential quality increase will likely be unnoticeable.
How does the 68000 internally represent instructions.
I've read that there are different types of instructions: single effective operation word format instructions, brief and full extension word format instructions. The single effective operation word instruction seems to represent the instruction and the lower 6 bits of this instruction the addressing mode and register. Does this addressing mode and register tell you if there follows a brief or full extension word format instruction, which on his turn represents the operands for the instruction. Do you know a better manual than the 68000 programming reference manual.
Thanks in advance
The actual internal representation is a combination of "microcode" and "nanocode". The 68000 has 544 17-bit microcode words which dispaches to 366 68-bit nanocode words.
While this may not be what you wanted to know, this link may provide some insights:
http://www.easy68k.com/paulrsm/doc/dpbm68k1.htm
right, on m68000 indexed modes uses the brief extension. In "Address Register Indirect with Index (8-Bit Displacement) Mode" (d8, An, Xn), the BEW is filled with D/A (if Xn is a data or address register), Xn (the register number), W/L (to threat Xn contents as 16 or 32bits), scale to 0 (see note), and the 8-bit displacement.
on other hand, other modes, like the 16bit displacement, "Address with displacement" (d16,An) , the extension is only a word with the displacement.
the note is: brief extension word - m68k doesn't support the 2bits for scale so is set to 0; scale on BEW using the scale bits, and full extensions are only suported m68020,40,-> cpus. http://etd.dtu.dk/thesis/264182/bac10_19.pdf
I don't know very well about RAM and HDD architecture, or how electronics deals with chunks of memory, but this always triggered my curiosity:
Why did we choose to stop at 8 bits for the smallest element in a computer value ?
My question may look very dumb, because the answer are obvious, but I'm not very sure...
Is it because 2^3 allows it to fit perfectly when addressing memory ?
Are electronics especially designed to store chunk of 8 bits ? If yes, why not use wider words ?
It is because it divides 32, 64 and 128, so that processor words can be be given several of those words ?
Is it just convenient to have 256 value for such a tiny space ?
What do you think ?
My question is a little too metaphysical, but I want to make sure it's just an historical reason and not a technological or mathematical reason.
For the anecdote, I was also thinking about the ASCII standard, in which most of the first characters are useless with stuff like UTF-8, I'm also trying to think about some tinier and faster character encoding...
Historically, bytes haven't always been 8-bit in size (for that matter, computers don't have to be binary either, but non-binary computing has seen much less action in practice). It is for this reason that IETF and ISO standards often use the term octet - they don't use byte because they don't want to assume it means 8-bits when it doesn't.
Indeed, when byte was coined it was defined as a 1-6 bit unit. Byte-sizes in use throughout history include 7, 9, 36 and machines with variable-sized bytes.
8 was a mixture of commercial success, it being a convenient enough number for the people thinking about it (which would have fed into each other) and no doubt other reasons I'm completely ignorant of.
The ASCII standard you mention assumes a 7-bit byte, and was based on earlier 6-bit communication standards.
Edit: It may be worth adding to this, as some are insisting that those saying bytes are always octets, are confusing bytes with words.
An octet is a name given to a unit of 8 bits (from the Latin for eight). If you are using a computer (or at a higher abstraction level, a programming language) where bytes are 8-bit, then this is easy to do, otherwise you need some conversion code (or coversion in hardware). The concept of octet comes up more in networking standards than in local computing, because in being architecture-neutral it allows for the creation of standards that can be used in communicating between machines with different byte sizes, hence its use in IETF and ISO standards (incidentally, ISO/IEC 10646 uses octet where the Unicode Standard uses byte for what is essentially - with some minor extra restrictions on the latter part - the same standard, though the Unicode Standard does detail that they mean octet by byte even though bytes may be different sizes on different machines). The concept of octet exists precisely because 8-bit bytes are common (hence the choice of using them as the basis of such standards) but not universal (hence the need for another word to avoid ambiguity).
Historically, a byte was the size used to store a character, a matter which in turn builds on practices, standards and de-facto standards which pre-date computers used for telex and other communication methods, starting perhaps with Baudot in 1870 (I don't know of any earlier, but am open to corrections).
This is reflected by the fact that in C and C++ the unit for storing a byte is called char whose size in bits is defined by CHAR_BIT in the standard limits.h header. Different machines would use 5,6,7,8,9 or more bits to define a character. These days of course we define characters as 21-bit and use different encodings to store them in 8-, 16- or 32-bit units, (and non-Unicode authorised ways like UTF-7 for other sizes) but historically that was the way it was.
In languages which aim to be more consistent across machines, rather than reflecting the machine architecture, byte tends to be fixed in the language, and these days this generally means it is defined in the language as 8-bit. Given the point in history when they were made, and that most machines now have 8-bit bytes, the distinction is largely moot, though it's not impossible to implement a compiler, run-time, etc. for such languages on machines with different sized bytes, just not as easy.
A word is the "natural" size for a given computer. This is less clearly defined, because it affects a few overlapping concerns that would generally coïncide, but might not. Most registers on a machine will be this size, but some might not. The largest address size would typically be a word, though this may not be the case (the Z80 had an 8-bit byte and a 1-byte word, but allowed some doubling of registers to give some 16-bit support including 16-bit addressing).
Again we see here a difference between C and C++ where int is defined in terms of word-size and long being defined to take advantage of a processor which has a "long word" concept should such exist, though possibly being identical in a given case to int. The minimum and maximum values are again in the limits.h header. (Indeed, as time has gone on, int may be defined as smaller than the natural word-size, as a combination of consistency with what is common elsewhere, reduction in memory usage for an array of ints, and probably other concerns I don't know of).
Java and .NET languages take the approach of defining int and long as fixed across all architecutres, and making dealing with the differences an issue for the runtime (particularly the JITter) to deal with. Notably though, even in .NET the size of a pointer (in unsafe code) will vary depending on architecture to be the underlying word size, rather than a language-imposed word size.
Hence, octet, byte and word are all very independent of each other, despite the relationship of octet == byte and word being a whole number of bytes (and a whole binary-round number like 2, 4, 8 etc.) being common today.
Not all bytes are 8 bits. Some are 7, some 9, some other values entirely. The reason 8 is important is that, in most modern computers, it is the standard number of bits in a byte. As Nikola mentioned, a bit is the actual smallest unit (a single binary value, true or false).
As Will mentioned, this article http://en.wikipedia.org/wiki/Byte describes the byte and its variable-sized history in some more detail.
The general reasoning behind why 8, 256, and other numbers are important is that they are powers of 2, and computers run using a base-2 (binary) system of switches.
ASCII encoding required 7 bits, and EBCDIC required 8 bits. Extended ASCII codes (such as ANSI character sets) used the 8th bit to expand the character set with graphics, accented characters and other symbols.Some architectures made use of proprietary encodings; a good example of this is the DEC PDP-10, which had a 36 bit machine word. Some operating sytems on this architecture used packed encodings that stored 6 characters in a machine word for various purposes such as file names.
By the 1970s, the success of the D.G. Nova and DEC PDP-11, which were 16 bit architectures and IBM mainframes with 32 bit machine words was pushing the industry towards an 8 bit character by default. The 8 bit microprocessors of the late 1970s were developed in this environment and this became a de facto standard, particularly as off-the shelf peripheral ships such as UARTs, ROM chips and FDC chips were being built as 8 bit devices.
By the latter part of the 1970s the industry settled on 8 bits as a de facto standard and architectures such as the PDP-8 with its 12 bit machine word became somewhat marginalised (although the PDP-8 ISA and derivatives still appear in embedded sytem products). 16 and 32 bit microprocessor designs such as the Intel 80x86 and MC68K families followed.
Since computers work with binary numbers, all powers of two are important.
8bit numbers are able to represent 256 (2^8) distinct values, enough for all characters of English and quite a few extra ones. That made the numbers 8 and 256 quite important.
The fact that many CPUs (used to and still do) process data in 8bit helped a lot.
Other important powers of two you might have heard about are 1024 (2^10=1k) and 65536 (2^16=65k).
Computers are build upon digital electronics, and digital electronics works with states. One fragment can have 2 states, 1 or 0 (if the voltage is above some level then it is 1, if not then it is zero). To represent that behavior binary system was introduced (well not introduced but widely accepted).
So we come to the bit. Bit is the smallest fragment in binary system. It can take only 2 states, 1 or 0, and it represents the atomic fragment of the whole system.
To make our lives easy the byte (8 bits) was introduced. To give u some analogy we don't express weight in grams, but that is the base measure of weight, but we use kilograms, because it is easier to use and to understand the use. One kilogram is the 1000 grams, and that can be expressed as 10 on the power of 3. So when we go back to the binary system and we use the same power we get 8 ( 2 on the power of 3 is 8). That was done because the use of only bits was overly complicated in every day computing.
That held on, so further in the future when we realized that 8 bytes was again too small and becoming complicated to use we added +1 on the power ( 2 on the power of 4 is 16), and then again 2^5 is 32, and so on and the 256 is just 2 on the power of 8.
So your answer is we follow the binary system because of architecture of computers, and we go up in the value of the power to represent get some values that we can simply handle every day, and that is how you got from a bit to an byte (8 bits) and so on!
(2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, and so on) (2^x, x=1,2,3,4,5,6,7,8,9,10 and so on)
The important number here is binary 0 or 1. All your other questions are related to this.
Claude Shannon and George Boole did the fundamental work on what we now call information theory and Boolean arithmetic. In short, this is the basis of how a digital switch, with only the ability to represent 0 OFF and 1 ON can represent more complex information, such as numbers, logic and a jpg photo. Binary is the basis of computers as we know them currently, but other number base computers or analog computers are completely possible.
In human decimal arithmetic, the powers of ten have significance. 10, 100, 1000, 10,000 each seem important and useful. Once you have a computer based on binary, there are powers of 2, likewise, that become important. 2^8 = 256 is enough for an alphabet, punctuation and control characters. (More importantly, 2^7 is enough for an alphabet, punctuation and control characters and 2^8 is enough room for those ASCII characters and a check bit.)
We normally count in base 10, a single digit can have one of ten different values. Computer technology is based on switches (microscopic) which can be either on or off. If one of these represents a digit, that digit can be either 1 or 0. This is base 2.
It follows from there that computers work with numbers that are built up as a series of 2 value digits.
1 digit,2 values
2 digits, 4 values
3 digits, 8 values etc.
When processors are designed, they have to pick a size that the processor will be optimized to work with. To the CPU, this is considered a "word". Earlier CPUs were based on word sizes of fourbits and soon after 8 bits (1 byte). Today, CPUs are mostly designed to operate on 32 bit and 64 bit words. But really, the two state "switch" are why all computer numbers tend to be powers of 2.
I believe the main reason has to do with the original design of the IBM PC. The Intel 8080 CPU was the first precursor to the 8086 which would later be used in the IBM PC. It had 8-bit registers. Thus, a whole ecosystem of applications was developed around the 8-bit metaphor. In order to retain backward compatibility, Intel designed all subsequent architectures to retain 8-bit registers. Thus, the 8086 and all x86 CPUs after that kept their 8-bit registers for backwards compatibility, even though they added new 16-bit and 32-bit registers over the years.
The other reason I can think of is 8 bits is perfect for fitting a basic Latin character set. You cannot fit it into 4 bits, but you can in 8. Thus, you get the whole 256-value ASCII charset. It is also the smallest power of 2 for which you have enough bits into which you can fit a character set. Of course, these days most character sets are actually 16-bit wide (i.e. Unicode).
Charles Petzold wrote an interesting book called Code that covers exactly this question. See chapter 15, Bytes and Hex.
Quotes from that chapter:
Eight bit values are inputs to the
adders, latches and data selectors,
and also outputs from these units.
Eight-bit values are also defined by
switches and displayed by lightbulbs,
The data path in these circuits is
thus said to be 8 bits wide. But
why 8 bits? Why not 6 or 7 or 9 or
10?
... there's really no reason why
it had to be built that way. Eight
bits just seemed at the time to be a
convenient amount, a nice biteful of
bits, if you will.
...For a while, a byte meant simply
the number of bits in a particular
data path. But by the mid-1960s. in
connection with the development of
IBM's System/360 (their large complex
of business computers), the word came
to mean a group of 8 bits.
... One reason IBM gravitated toward
8-bit bytes was the ease in storing
numbers in a format known as BCD.
But as we'll see in the chapters ahead, quite by coincidence a byte is
ideal for storing text because most
written languages around the world
(with the exception of the ideographs
used in Chinese, Japanese and Korean)
can be represented with fewer than 256
characters.
Historical reasons, I suppose. 8 is a power of 2, 2^2 is 4 and 2^4 = 16 is far too little for most purposes, and 16 (the next power of two) bit hardware came much later.
But the main reason, I suspect, is the fact that they had 8 bit microprocessors, then 16 bit microprocessors, whose words could very well be represented as 2 octets, and so on. You know, historical cruft and backward compability etc.
Another, similarily pragmatic reason against "scaling down": If we'd, say, use 4 bits as one word, we would basically get only half the troughtput compared with 8 bit. Aside from overflowing much faster.
You can always squeeze e.g. 2 numbers in the range 0..15 in one octet... you just have to extract them by hand. But unless you have, like, gazillions of data sets to keep in memory side-by-side, this isn't worth the effort.
I read a while back that Quantum Computers can break most types of hashing and encryption in use today in a very short amount of time(I believe it was mere minutes). How is it possible? I've tried reading articles about it but I get lost at the a quantum bit can be 1, 0, or something else. Can someone explain how this relates to cracking such algorithms in plain English without all the fancy maths?
Preamble: Quantum computers are strange beasts that we really haven't yet tamed to the point of usefulness. The theory that underpins them is abstract and mathematical, so any discussion of how they can be more efficient than classical computers will inevitably be long and involved. You'll need at least an undergraduate understanding of linear algebra and quantum mechanics to understand the details, but I'll try to convey my limited understanding!
The basic premise of quantum computation is quantum superposition. The idea is that a quantum system (such as a quantum bit, or qubit, the quantum analogue of a normal bit) can, as you say, exist not only in the 0 and 1 states (called the computational basis states of the system), but also in any combination of the two (so that each has an amplitude associated with it). When the system is observed by someone, the qubit's state collapses into one of its basis states (you may have heard of the Schrödinger's cat thought experiment, which is related to this).
Because of this, a register of n qubits has 2^n basis states of its own (these are the states that you could observe the register being in; imagine a classical n-bit integer). Since the register can exist in a superposition of all these states at once, it is possible to apply a computation to all 2^n register states rather than just one of them. This is called quantum parallelism.
Because of this property of quantum computers, it may seem like they're a silver bullet that can solve any problem exponentially faster than a classical computer. But it's not that simple: the problem is that once you observe the result of your computation, it collapses (as I mentioned above) into the result of just one of the computations – and you lose all of the others.
The field of quantum computation/algorithms is all about trying to work around this problem by manipulating quantum phenomena to extract information in fewer operations than would be possible on a classical computer. It turns out that it's very difficult to contrive a "quantum algorithm" that is faster than any possible classical counterpart.
The example you ask about is that of quantum cryptanalysis. It's thought that quantum computers might be able to "break" certain encryption algorithms: specifically, the RSA algorithm, which relies on the difficulty of finding the prime factors of very large integers. The algorithm which allows for this is called Shor's algorithm, which can factor integers with polynomial time complexity. By contrast the best classical algorithm for the problem has (almost) exponential time complexity, and the problem is hence considered "intractable".
If you want a deeper understanding of this, get a few books on linear algebra and quantum mechanics and get comfortable. If you want some clarification, I'll see what I can do!
Aside: to better understand the idea of quantum superposition, think in terms of probabilities. Imagine you flip a coin and catch it on your hand, covered so that you can't see it. As a very tenuous analogy, the coin can be thought of as being in a superposition of the heads and tails "states": each one has a probability of 0.5 (and, naturally, since there are two states, these probabilities add up to 1). When you take your hand away and observe the coin directly, it collapses into either the heads state or the tails state, and so the probability of this state becomes 1, while the other becomes 0. One way to think about it, I suppose, is a set of scales that is balanced until observation, at which point it tips to one side as our knowledge of the system increases and one state becomes the "real" state.
Of course, we don't think of the coin as a quantum system: for all practical purposes, the coin has a definite state, even if we can't see it. For genuine quantum systems, however (such as an individual particle trapped in a box), we can't think about it in this way. Under the conventional interpretation of quantum mechanics, the particle fundamentally has no definite position, but exists in all possible positions at once. Only upon observation is its position constrained in space (though only to a limited degree; cf. uncertainty principle), and even this is purely random and determined only by probability.
By the way, quantum systems are not restricted to having just two observable states (those that do are called two-level systems). Some have a large but finite number, some have a countably infinite number (such as a "particle in a box" or a harmonic oscillator), and some even have an uncountably infinite number (such as a free particle's position, which isn't constrained to individual points in space).
It's highly theoretical at this point. Quantum Bits might offer the capability to break encryption, but clearly it's not at that point yet.
At the Quantum Level, the laws that govern behavior are different than in the macro level.
To answer your question, you first need to understand how encryption works.
At a basic level, encryption is the result of multiplying two extremely large prime numbers together. This super large result is divisible by 1, itself, and these two prime numbers.
One way to break encryption is to brute force guess the two prime numbers, by doing prime number factorization.
This attack is slow, and is thwarted by picking larger and larger prime numbers. YOu hear of key sizes of 40bits,56bits,128bits and now 256,512bits and beyond. Those sizes correspond to the size of the number.
The brute force algorithm (in simplified terms) might look like
for(int i = 3; i < int64.max; i++)
{
if( key / i is integral)
{
//we have a prime factor
}
}
So you want to brute force try prime numbers; well that is going to take awhile with a single computer. So you might try grouping a bunch of computers together to divide and conquer. That works, but is still slow for very large keysizes.
How a quantum bit address this is that they are both 0 and 1 at the same time. So say you have 3 quantum bits (no small feat mind you).
With 3 qbits, your program can have the values of 0-7 simulatanously
(000,001,010,011 etc)
, which includes prime numbers 3,5,7 at the same time.
so using the simple algorithm above, instead of increasing i by 1 each time, you can just divide once, and check
0,1,2,3,4,5,6,7
all at the same time.
Of course quantum bits aren't to that point yet; there is still lots of work to be done in the field; but this should give you an idea that if we could program using quanta, how we might go about cracking encryption.
The Wikipedia article does a very good job of explaining this.
In short, if you have N bits, your quantum computer can be in 2^N states at the same time. Similar conceptually to having 2^N CPU's processing with traditional bits (though not exactly the same).
A quantum computer can implement Shor's algorithm which can quickly perform prime factorization. Encryption systems are build on the assumption that large primes can not be factored in a reasonable amount of time on a classical computer.
Almost all our public-key encryptions (ex. RSA) are based solely on math, relying on the difficulty of factorization or discrete-logarithms. Both of these will be efficiently broken using quantum computers (though even after a bachelors in CS and Math, and having taken several classes on quantum mechanics, I still don't understand the algorithm).
However, hashing algorithms (Ex. SHA2) and symmetric-key encryptions (ex. AES), which are based mostly on diffusion and confusion, are still secure.
In the most basic terms, a normal no quantum computer works by operating on bits (sates of on or off) uesing boolean logic. You do this very fast for lots and lots of bits and you can solve any problem in a class of problems that are computable.
However they are "speed limits" namely something called computational complexity.This in lay mans terms means that for a given algorithm you know that the time it takes to run an algorithm (and the memory space required to run the algorithm) has a minimum bound. For example a algorithm that is O(n^2) means that for a data size of n it will require n^2 time to run.
However this kind of goes out the window when we have qbits (quantum bits) when you are doing operations on qbits that can have "in between" values. algorithms that would have very high computational complexity (like factoring huge numbers, the key to cracking many encryption algorithms) can be done in much much lower computational complexity. This is the reason that quantum computing will be able to crack encrypted streams orders of magnitude quicker then normal computers.
First of all, quantum computing is still barely out of the theoretical stage. Lots of research is going on and a few experimental quantum cells and circuits, but a "quantum computer" does not yet exist.
Second, read the wikipedia article: http://en.wikipedia.org/wiki/Quantum_computer
In particular, "In general a quantum computer with n qubits can be in an arbitrary superposition of up to 2^n different states simultaneously (this compares to a normal computer that can only be in one of these 2^n states at any one time). "
What makes cryptography secure is the use of encryption keys that are very long numbers that would take a very, very long time to factor into their constituent primes, and the keys are sufficiently long enough that brute-force attempts to try every possible key value would also take too long to complete.
Since quantum computing can (theoretically) represent a lot of states in a small number of qubit cells, and operate on all of those states simultaneously, it seems there is the potential to use quantum computing to perform brute-force try-all-possible-key-values in a very short amount of time.
If such a thing is possible, it could be the end of cryptography as we know it.
quantum computers etc all lies. I dont believe these science fiction magazines.
in fact rsa system is based on two prime numbers and their multipilation.
p1,p2 is huge primes p1xp2=N modulus.
rsa system is
like that
choose a prime number..maybe small its E public key
(p1-1)*(p2-1)=R
find a D number that makes E*D=1 mod(R)
we are sharing (E,N) data as public key publicly
we are securely saving (D,N) as private.
To solve this Rsa system cracker need to find prime factors of N.
*mass of the Universe is closer to 10^53 kg*
electron mass is 9.10938291 × 10^-31 kilograms
if we divide universe to electrons we can create 10^84 electrons.
electrons has slower speeds than light. its move frequency can be 10^26
if anybody produces electron size parallel rsa prime factor finders from all universe mass.
all universe can handle (10^84)*(10^26)= 10^110 numbers/per second.
rsa has limitles bits of alternative prime numbers. maybe 4096 bits
4096 bit rsa has 10^600 possible prime numbers to brute force.
so your universe mass quantum solver need to make tests during 10^500 years.
rsa vs universe mass quantum computer
1 - 0
maybe quantum computer can break 64/128 bits passwords. because 128 bit password has 10^39 possible brute force nodes.
This circuit is a good start to understand how qubit parallelism works. The 2-qubits-input is on the left side. Top qubit is x and bottom qubit ist y. The y qubit is 0 at the input, just like a normal bit. The x qubit on the other hand is in superposition at the input. y (+) f(x) stands here for addition modulo 2, just meaning 1+1=0, 0+1=1+0=1. But the interesting part is, since the x-qubit is in superposition, f(x) is f(0) and f(1) at the same time and we can perform the evaluation of the f function for all states simultaneously without using any (time consuming) loops. Having enough quibits we can branch this into endlessly complicating curcuits.
Even more bizarr imo. is the Grover's algorithm. As input we get here an unsorted array of integers with arraylength = n. What is the expected runtime of an algorithm, that finds the min value of this array? Well classically we have at least to check every 1..n element of the array resulting in an expected runtime of n. Not so for quantum computers, on a quantum computer we can solve this in expected runtime of maximum root(n), this means we don't even have to check every element to find the guaranteed solution...
For a computer working with a 64 bit processor, the largest number that it can handle would be 264 = 18,446,744,073,709,551,616. How does programming languages, say Java or be it C, C++ handle arithmetic of numbers higher than this value. Any register cannot hold it as a single piece. How was this issue tackled?
There are lots of specialized techniques for doing calculations on numbers larger than the register size. Some of them are outlined in this wikipedia article on arbitrary precision arithmetic
Low level languages, like C and C++, leave large number calculations to the library of your choice. One notable one is the GNU Multi-Precision library. High level languages like Python, and others, integrate this into the core of the language, so normal numbers and very large numbers are identical to the programmer.
You assume the wrong thing. The biggest number it can handle in a single register is a 64-bits number. However, with some smart programming techniques, you could just combined a few dozens of those 64-bits numbers in a row to generate a huge 6400 bit number and use that to do more calculations. It's just not as fast as having the number fit in one register.
Even the old 8 and 16 bits processors used this trick, where they would just let the number overflow to other registers. It makes the math more complex but it doesn't put an end to the possibilities.
However, such high-precision math is extremely unusual. Even if you want to calculate the whole national debt of the USA and store the outcome in Zimbabwean Dollars, a 64-bits integer would still be big enough, I think. It's definitely big enough to contain the amount of my savings account, though.
Programming languages that handle truly massive numbers use custom number primitives that go beyond normal operations optimized for 32, 64, or 128 bit CPUs. These numbers are especially useful in computer security and mathematical research.
The GNU Multiple Precision Library is probably the most complete example of these approaches.
You can handle larger numbers by using arrays. Try this out in your web browser. Type the following code in the JavaScript console of your web browser:
The point at which JavaScript fails
console.log(9999999999999998 + 1)
// expected 9999999999999999
// actual 10000000000000000 oops!
JavaScript does not handle plain integers above 9999999999999998. But writing your own number primitive is to make this calculation work is simple enough. Here is an example using a custom number adder class in JavaScript.
Passing the test using a custom number class
// Require a custom number primative class
const {Num} = require('./bases')
// Create a massive number that JavaScript will not add to (correctly)
const num = new Num(9999999999999998, 10)
// Add to the massive number
num.add(1)
// The result is correct (where plain JavaScript Math would fail)
console.log(num.val) // 9999999999999999
How it Works
You can look in the code at class Num { ... } to see details of what is happening; but here is a basic outline of the logic in use:
Classes:
The Num class contains an array of single Digit classes.
The Digit class contains the value of a single digit, and the logic to handle the Carry flag
Steps:
The chosen number is turned into a string
Each digit is turned into a Digit class and stored in the Num class as an array of digits
When the Num is incremented, it gets carried to the first Digit in the array (the right-most number)
If the Digit value plus the Carry flag are equal to the Base, then the next Digit to the left is called to be incremented, and the current number is reset to 0
... Repeat all the way to the left-most digit of the array
Logistically it is very similar to what is happening at the machine level, but here it is unbounded. You can read more about about how digits are
carried here; this can be applied to numbers of any base.
Ada actually supports this natively, but only for its typeless constants ("named numbers"). For actual variables, you need to go find an arbitrary-length package. See Arbitrary length integer in Ada
More-or-less the same way that you do. In school, you memorized single-digit addition, multiplication, subtraction, and division. Then, you learned how to do multiple-digit problems as a sequence of single-digit problems.
If you wanted to, you could multiply two twenty-digit numbers together using nothing more than knowledge of a simple algorithm, and the single-digit times tables.
In general, the language itself doesn't handle high-precision, high-accuracy large number arithmetic. It's far more likely that a library is written that uses alternate numerical methods to perform the desired operations.
For example (I'm just making this up right now), such a library might emulate the actual techniques that you might use to perform that large number arithmetic by hand. Such libraries are generally much slower than using the built-in arithmetic, but occasionally the additional precision and accuracy is called for.
As a thought experiment, imagine the numbers stored as a string. With functions to add, multiply, etc these arbitrarily long numbers.
In reality these numbers are probably stored in a more space efficient manner.
Think of one machine-size number as a digit and apply the algorithm for multi-digit multiplication from primary school. Then you don't need to keep the whole numbers in registers, just the digits as they are worked on.
Most languages store them as array of integers. If you add/subtract two to of these big numbers the library adds/subtracts all integer elements in the array separately and handles the carries/borrows.
It's like manual addition/subtraction in school because this is how it works internally.
Some languages use real text strings instead of integer arrays which is less efficient but simpler to transform into text representation.