I have seen many examples of encryption up to about 256-bit. But how does one programmatically scale the logic upwards in a language such as PHP or Java to say 1024-bit or even 4096-bit and higher?
Are there any websites that explain the workings of a very large encryption on a modern 64-bit computer? For example what kind of numbers (Long Int, Double, etc.) would one use and how can one even extend that range and still keep it computable to the computer?
Would this scaling's security be compromised by the limitations of the processing power available, i.e. 64-bit processor?
I understand that time is a factor in very large encryptions, but in this case time doesn't matter that much.
Dear Saleh,
As we know, using something like 3AES should work to scale up the security. But seriously, why would you even need 512-bits of security in a symmetric algorithm? According to the analysis on wikipedia it would take 50,955,671,114,250,100,000,000,000,000,000,000,000,000,000,000,000,000 years to bruteforce 256 bit AES (given a few assumptions on speed). I doubt you have any data that needs to be secured for that long.
As far as whether the CPU limits the security strength, it doesn't. There are libraries out there which provide "infinite" precision numbers (i.e., gmplib for c/c++, BigInteger and BigDecimal for Java, etc). For more on computing with large numbers, see chapter 14 of the Handbook of Applied Cryptography (HAC). Internally, these libraries use things like long long int or even plain int stored in an array. HAC has the algorithms.
There are methods to use extra keying material and increase the security of a block cipher (I assume you are talking about that). In general multiple re-encrypting the encryption works but not always as expected. 3DES double the security of DES, and so 3AES-256 will double the strength. Other methods exist like XDES we can have X-AES-256, and so on. These ideas are in the cryptographic literature and Wikipedia covers these methods. (Performance is an issue and always need to be taken into account, most importantly in real time operation).
I hope this answers some of the question (it is not completely defined as a question, so I tried).
To encrypt and decrypt would not take much time if you have the key, you would not need to worry about if that can be computed unless you use millions of encryptions per second or encrypt with million million digits long gygabytes of data. But if you try to bruteforce it without the key, then it takes from few hours to ........zillion years to guess it even with supercomputer. As these digits increase, the amount of computing needed to brute force encryption with them increases much much more, that is why 256 bit is said to take more time than our universe existence to break it even with supercomputer. And that is probably why it will be difficult to find more complicated encryption, that not many people need, because you need larger numbers and you may need to generate them yourself. It is not difficult, you may just need a longer than 256 digits in binary prime number key for yourself and for your target. Then you use these numbers with a very simple formula and your data that can be executed with very small and simple code. But as you said you may need much larger numbers than maybe even data type Long, then you could use MatLab, it allows you to use crazy large numbers. (More about that encryption formula is nicely explained here : https://www.youtube.com/watch?v=vgTtHV04xRI). If you need such extreme lengths to do something, then try thinking whether you doing this for fun or for something you need, because there might be no need to encrypt anything so hard that it would take more than thousand years to decrypt with supercomputers.
Thank you, Peter, for correction. I will edit the answer to not be completely wrong.
Let's sort out a possible misunderstanding: n-bit security means that any attacker needs at least 2^n operations to break the system in some specified sense.
AES-128 is believed to provide about 126-bit security.
RSA-1024 is estimated to provide about 80-bit security.
The length of the key is related to the security level, but it's not the same. The question asks for 1024-bit security or more which seems to think of RSA.
However, these statements are not mathematically sharp as up to date no one can prove things about "all attackers", for example Cook's hypothesis P \neq NP is still open. Have a look at https://www.cryptopp.com/wiki/Security_level.
As mentioned, 3AES-256 would be twice as secure as AES-256, so you can scale (and iterate that if need by). Unfortunately, the ratio between key length and security level gets worse with each tripling. X-AES does *not* provide better security against linear or differential attacks, despite a longer key. Peter's proposal is better here but amounts to the construction of new ciphers which have to be analyzed first.
Moreover, better hardware -so far- can only gain constant factors. So it does not matter that much whether you are using a 64-bit architecture or something else.
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