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# 数学代写|密码学代写Cryptography Theory代考|Popularity of RSA

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## 数学代写|密码学Cryptography Theory代考|Popularity of RSA

Historically there is no doubt RSA has been by far the most popular public-key cryptosystem. There are several possible reasons for this:

Maturity. RSA was one of the first public-key cryptosystems to be proposed and was the first to gain widespread recognition. Thus, in many senses, RSA is the brand leader.

Less message expansion. ElGamal involves message expansion by default, which makes its use potentially undesirable. The ‘textbook’ version of RSA has no message expansion, and RSA-OAEP has limited message expansion.

Marketing. The use of RSA was marketed from an early stage by a commercial company. Indeed, it was at one stage subject to patent in certain parts of the world. ElGamal has not had such successful commercial backing. However, ECC does, and there are a number of patents on ECC primitives.

In comparison with most symmetric encryption algorithms, neither RSA nor any variants of ElGamal are particularly efficient. The main problem is that in each case encryption involves exponentiation. We saw in Section 3.2.3 that exponentiation has complexity $n^3$. This means it is easy to compute but is not as efficient as other more straightforward operations such as addition (complexity $n$ ) and multiplication (complexity $n^2$ ).

In this respect, RSA is more efficient for encryption than ElGamal variants, since it only requires one exponentiation (and by choosing the exponent $e$ to have a certain format, this can be made to be a faster-than-average exponentiation computation), whereas ElGamal variants need two. However, we already noted in Section 5.3 .4 that the computation of $C_1$ could be done in advance, and so some people argue there is very little difference in computational efficiency.

In contrast, decryption is slightly more efficient for ElGamal variants than for RSA. This is because the decryption exponentiation is typically performed with a smaller exponent than for RSA. If the exponent is carefully chosen, then, even with the additional ElGamal decryption costs of running the Extended Euclidean Algorithm, the result is typically a more efficient computation than an RSA decryption based on a much larger exponent.

There has been a lot of work invested in trying to speed up the exponentiation process in order to make RSA and ElGamal variants more efficient. A combination of clever engineering and mathematical expertise has led to faster implementations, but they are all slower than symmetric computations. For this reason, none of these public-key cryptosystems are normally used for bulk data encryption (see Section 5.5).

## 数学代写|密码学Cryptography Theory代考|Security issues

In order to compare different public-key cryptosystems, we first need to establish a means of relating the security of one public-key cryptosystem to another.
KEY LENGTHS OF PUBLIC-KEY CRYPTOSYSTEMS
Just as for symmetric cryptosystems, the length of a private (decryption) key is an important parameter of a public-key cryptosystem and is one which can be used to compare different public-key cryptosystems. A complicating factor is that keys in public-key cryptosystems are:

• First specified in terms of ‘numbers’; and
• Then converted into binary strings for implementation.
As a result, unlike in symmetric cryptosystems, the actual length in bits of a private key will vary, since a smaller ‘number’ will involve fewer bits when it is converted into binary. Thus, we tend to regard the ‘length’ of a private key as the maximum length the private key could possibly be.

In order to determine the (maximum) length of a private key, we have to consider the specifics of the public-key cryptosystem. For example, in RSA the decryption key $d$ is a number modulo $n$. This means the decryption key can be any number less than $n$. Hence, the maximum number of bits we need to represent an RSA private key is the smallest number $k$ such that:
$$2^k \geq n$$
This might sound a bit complicated since, given the modulus $n$, we would appear to have to perform some calculation before we can determine the length in bits of an RSA private key. However, the good news is that key length is of sufficient importance that we tend to approach this issue the other way around. In other words, public-key cryptosystems tend to be referred to directly in terms of their maximum private key lengths. When someone refers to 3072-bit RSA, they mean the modulus $n$ is 3072 bits long when written in binary, and thus the maximum private key length is also 3072 bits. This means the actual modulus $n$, when considered as a ‘number’, is much (much) bigger than 3072 . More precisely, the modulus $n$ will be a number in the range:
$$2^{3071} \leq n<2^{3072}$$
since these are the numbers which have 3072 bits when written in binary.

## 数学代写|密码学Cryptography Theory代考|Security issues

$$2^k \geq n$$

$$2^{3071} \leq n<2^{3072}$$

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