# 数学代写|密码学作业代写Cryptography代考|Other types of public-key cryptosystem

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## 数学代写|密码学作业代写Cryptography代考|Other types of public-key cryptosystem

There are several interesting extensions of the basic idea behind public-key encryption which have been gaining interest.

Identity-based encryption. The idea behind identity-based encryption is that the sender can specify an arbitrary identity string (for example, a user name or email address) as the public key of the recipient. This is in contrast to cryptosystems such as RSA and ElGamal, where the public keys cannot be so specifically selected. Identity-based encryption has some very interesting key management features (both positive and negative), so we will look at this idea in more detail in Section 11.4.2.

Attribute-based encryption. There are potential applications of encryption where it may not be possible (or desirable) to identify the intended recipients of a ciphertext. For example, we might want to encrypt a medical record in such a way that any qualified specialist can decrypt it. If using conventional public-key encryption, we would either have to encrypt the record using the public key of every such specialist (as well as being inefficient, we probably will not know the identities of all such specialists, and this list is likely to change over time), or deploy a common public key among all such specialists (which is unlikely to be a secure option in practice). Attribute-based encryption uses a combination of a set of attributes (features or properties such as ‘qualified radiologist’ or ‘clearance level 2’) and policies (rules defined on the attributes such as ‘must be a qualified radiologist who also has clearance level 2’) to determine who can decrypt a ciphertext. We will revisit this in Section 11.4.2.

Homomorphic encryption. In some applications, it is desirable to be able to perform computations on encrypted data, particularly when the data represents numerical values. For example, suppose $C_1$ is the encryption of $P_1, C_2$ is the encryption of $P_2$, and we would like to compute the ciphertext associated with $P_1$ multiplied by $P_2$ (all encryptions computed using the same key $K$ ). One obvious option is to decrypt the data first, perform the computation, and then re-encrypt the result. However, this solution requires that whoever does the decryption and computation is trusted, since they learn both the data and the result of the computation. This is not the case in, for example, cloud computing environments, where the cloud storage provider is usually a third party who is trusted to store encrypted data but not to know the plaintext data. It would be wonderful if the ciphertext associated with $P_1$ multiplied by $P_2$ could be computed directly from $C_1$ and $C_2$ (without first decrypting them), but this is sadly not the case for most traditional encryption schemes. Fully homomorphic encryption schemes allow any computation to be performed directly on the ciphertexts. There are only a few schemes of this type known, and they are extremely inefficient. Partially homomorphic encryption schemes allow certain specific computations (typically addition or multiplication) to be performed directly on the ciphertexts but not others. Interestingly, the ‘textbook’ version of RSA we presented in Section 5.2.2 is partially homomorphic. It supports multiplication since:
$$C_1 \times C_2=\left(P_1^e \times P_2^e\right) \quad \bmod n=\left(P_1 \times P_2\right)^e \bmod n,$$
hence $C_1 \times C_2$ is the encryption of $P_1 \times P_2$. Unfortunately, the more secure versions of RSA deployed in practice, such as RSA-OAEP (see Section 5.2.4), are not partially homomorphic. ElGamal is also partially homomorphic with respect to multiplication of plaintexts.

## 数学代写|密码学作业代写Cryptography代考|The future for public-key cryptosystems

As we have seen, the security of a (good) symmetric cryptosystem primarily relies on the difficulty of conducting an exhaustive key search. On the other hand, the security of a public-key cryptosystem is based on the difficulty of solving some underlying hard computational problem. As we discussed in Section 5.2.3, the security of RSA is based on the belief that factoring is hard, while, as we discussed in Section 5.3.3, the security of ElGamal is based on the belief that solving the discrete logarithm problem is hard. As suggested by Table 5.2, both of these problems are regarded as hard, and neither one is regarded as easier than the other.

There are considerable advantages to be gained from ensuring we retain efficient public-key cryptosystems based on different hard problems. If, one day, someone develops an efficient method for factoring large numbers, then the discrete logarithm problem might, at least in theory, still be secure. That said, there are some experts who hold the opinion that if a breakthrough occurs which results in one of these problems no longer being regarded as hard, then there may be similar implications for the other one. Only time will tell.

Of more concern is that both of these problems are no longer regarded as hard if an attacker has access to a quantum computer (see Section 4.7.2). It is known that a quantum computer can both factor and compute discrete logarithms in polynomial time. There is now considerable interest in developing public-key cryptosystems whose security is based on hard problems which cannot (apparently) be efficiently solved by a quantum computer. There are several candidate hard problems under investigation, and the development of algorithms based on them is sometimes referred to as post-quantum cryptography.

# 密码学代写

## 数学代写|密码学作业代写Cryptography代考|Other types of public-key cryptosystem

$$C_1 \times C_2=\left(P_1^e \times P_2^e\right) \quad \bmod n=\left(P_1 \times P_2\right)^e \bmod n,$$

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