Homomorphic Cryptography & Its Applications Aaram Yun, UNIST FIF presentation, 2015 1
Cryptography 2
Cryptography » Bike lock of Internet » Provides many important building blocks for security » For example, encryption or authentication 3
Very secure 4
Using data Eventually, crypto-protected data should be used somewhere 5
Using data However, this usually means removing cryptographic protection Often makes it hard to apply crypto in designing secure systems 6
Homomorphic cryptography » Computation using cryptographically protected data » While maintaining security » Many potential applications, for example, secure cloud computing 7
RSA Encryption » » » » 'Homomorphism' in mathematics » From and , can be computed by anybody 8
Homomorphic encryption » E.g., from , compute a ciphertext , such that, » ' can be homomorphically evaluated using ciphertexts' » What kind of function can be computed like this? » Any function, if you don't care about security » Just use the identity function as Enc() 9
It remains to be seen whether it is possible to have a [homomorphic encryption] with a large set of operations which is highly secure. — Rivest, Adleman, and Dertouzos, 1978 10
Fully homomorphic encryption » Craig Gentry, "Fully Homomorphic Encryption Using Ideal Lattices" , STOC 2009 » Any polynomial-time algorithm can be homomorphically evaluated 11
Homomorphic encryption » ' Somewhat homomorphic': can handle low-degree polynomial functions mod » ' Fully homomorphic': can handle arbitrary polynomial-sized polynomial functions mod » When , it can handle practically any boolean functions, so, any practical algorithms 12
'Gentry's blueprint' 13
Somewhat homomorphic encryption (SHE) » Each ciphertext has its 'noise level' » Freshly encrypted ciphertext has low loise level » Adding ciphertexts: noises are added » Multiplying ciphertexts: noises are multiplied 14
Somewhat homomorphic encryption (SHE) » If the noise level is above certain threshold, correct decryption is not guaranteed » Addition is okay: 1-bit increment » Multiplication is the problem: noise level is doubled » Relatively efficient, but only 'somewhat homomorphic' 15
Refreshing the ciphertext » There is a way to reduce the noise level of a 'noisy', but yet correctly decryptable ciphertext » Idea: decrypt to get the plaintext , then encrypt again to obtain the refreshed ciphertext » Problem: can be done only by the owner of the decryption key 16
Bootstrapping » Gentry's solution: can be done if SHE can homomorphically evaluate its own decryption circuit » This is called 'bootstrapping' » 'SHE + bootstrapping' gives you a FHE 17
Efficiency of HE » Slower, with larger ciphertext expansion » Still, if your application requires only SHE, it is relatively efficient » FHE is the problem 18
Efficiency of FHE » First FHEs were s..l..o..w.. » About 6-30 minutes to refresh one bit! » Many improvements made afterwards » Better noise management » Parallelization 19
AES evaluation » Can evaluate AES homomorphically » About 180 blocks in 1050 sec. (Eurocrypt 2015) 20
In comparision... » Native evaluation of AES: » About 18 cycles/ block » Roughly 10,000,000,000 blocks during the same time » About 10 8 times faster 21
Efficiency of FHE » Still a long way to go » Perhaps completely different paradigm needed for drastic speedup » Everything still within Gentry's blueprint » I am optimistic... » If bootstrapping is not needed, already somewhat usable 22
Homomorphic signature » Each block of data is signed: » When computing using , anyone can homomorphically evaluate the signature for using corresponding signatures » Given and , anyone can check if » Even when you don't have 23
Applications 24
Private cloud computing » Storage is outsourced » No local copy » Computation is outsourced » Only receive the output » Data is sensitive, private » E.g., medical, or genomic data 25
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FHE is not always needed » Some of the previous conditions are not needed » Conventional encryption is enough » Your application may not require evaluating any circuit » SHE would be enough » Also, conventionally encrypted data can be turned into FHE encrypted data when necessary » Homomorphically evaluate the decryption circuit, later 27
Veri fj able cloud computing » Storage is outsourced » No local copy » Computation is outsourced » Only receive the output » You want guarantee that the output is correct 28
Secure cloud computing » Right now not very realistic economically, for general computation » In the future, when FHE becomes practically practical, not theoretically practical, ... » That's what we are doing: ' Future Internet' » After Snowden's revelation about NSA surveillance, perhaps secure cloud makes more sense already 29
Questions » Imagine very practical homomorphic crypto in the future » What other new Internet applications can we build out of it? » Can we use it even in building services on lower layers? » What can we do with SHE right now? 30
Conclusions » In homomorphic cryptography, it is possible to provide rich functionality without sacrificing security » This is a promising, relatively new area with big dreams and many applications » Still a long way to go, but progresses are being actively made 31
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