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Homomorphic Encryption

Wikipedia

Homomorphic encryption is a form of encryption with an additional evaluation capability for computing over encrypted data without access to the secret key. The result of such a computation remains encrypted. Homomorphic encryption can be viewed as an extension of public-key cryptography. Homomorphic refers to homomorphism in algebra: the encryption and decryption functions can be thought of as homomorphisms between plaintext and ciphertext spaces.

Homomorphic encryption includes multiple types of encryption schemes that can perform different classes of computations over encrypted data.^[3]^ The computations are represented as either Boolean or arithmetic circuits. Some common types of homomorphic encryption are partially homomorphic, somewhat homomorphic, leveled fully homomorphic, and fully homomorphic encryption:

  • Partially homomorphic encryption encompasses schemes that support the evaluation of circuits consisting of only one type of gate, e.g., addition or multiplication.
  • Somewhat homomorphic encryption schemes can evaluate two types of gates, but only for a subset of circuits.
  • Leveled fully homomorphic encryption supports the evaluation of arbitrary circuits composed of multiple types of gates of bounded (pre-determined) depth.
  • Fully homomorphic encryption (FHE) allows the evaluation of arbitrary circuits composed of multiple types of gates of unbounded depth and is the strongest notion of homomorphic encryption.

For the majority of homomorphic encryption schemes, the multiplicative depth of circuits is the main practical limitation in performing computations over encrypted data. Homomorphic encryption schemes are inherently malleable. In terms of malleability, homomorphic encryption schemes have weaker security properties than non-homomorphic schemes.

Types of Homomorphic Encryption - IEEE Digital Privacy

对加密的数据直接进行运算，不需要解密。

  • 为什么要对加密的数据进行运算，而不是对明文？

    一方不希望交出明文数据，另一方不希望或不能够让运算由前者进行。于是只能让前者交出加密数据，后者运算后交回。

  • 客户端加密，服务端运算的场景很好理解，但服务端加密，客户端运算的场景也存在吗？

    似乎基本只能当计数器用，比如客户端登录后服务端返回个加密后的初值，然后客户端运算 +1，后面每次请求时带上，调用成功时再 +1，服务端每次检查。比直接 +1 也就是更复杂了而已。

A 6-minute introduction to homomorphic encryption

Homomorphic Encryption Use Cases - IEEE Digital Privacy

Fully homomorphic encryption

Obfuscation

  • Homomorphic Indistinguishability Obfuscation and its Applications

    似乎确实有点用，如果不可区分混淆实现的话，就相当于不需要暴露程序也可以让别人执行，加上同态加密，就相当于程序和数据都不需要暴露就可以让别人执行，完全安全，虽然这有些不切实际。

Can homomorphic encryption offer program integrity and program obfuscation? - Cryptography Stack Exchange

  • Zero-knowledge proofs

Homomorphic encryption and program obfuscation - Cryptography Stack Exchange