Homomorphic Encryption

Homomorphic encryption is a form ofencryption that allowscomputation onciphertexts, generating an encrypted result which, when decrypted, matches the result of the operations as if they had been performed on theplaintext.

Homomorphic encryption can be used for privacy-preserving outsourcedstorage andcomputation. This allows data to be encrypted and out-sourced to commercial cloud environments for processing, all while encrypted. In highly regulated industries, such as health care, homomorphic encryption can be used to enable new services by removing privacy barriers inhibiting data sharing. For example,predictive analytics in health care can be hard to apply due tomedical data privacy concerns, but if the predictive analytics service provider can operate on encrypted data instead, these privacy concerns are diminished.

Description [ edit ]

Homomorphic encryption is a form ofencryption with an additional evaluation capability for computing over encrypted data without access to thesecret key. The result of such a computation remains encrypted. Homomorphic encryption can be viewed as an extension of eithersymmetric-key or public-key cryptography . Homomorphic refers tohomomorphism in algebra: the encryption and decryption functions can be thought 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.Some common types of homomorphic encryption are partially homomorphic, somewhat homomorphic, leveled fully homomorphic, and fully homomorphic encryption. The computations are represented as either Boolean or arithmetic circuits. 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 of bounded (pre-determined) depth. Fully homomorphic encryption (FHE) allows the evaluation of arbitrary circuits 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 inherentlymalleable. In terms of malleability, homomorphic encryption schemes have weaker security properties than non-homomorphic schemes.

History [ edit ]

Homomorphic encryption schemes have been developed using different approaches. Specifically, fully homomomorphic encryption schemes are often grouped into generations corresponding to the underlying approach.

Pre-FHE [ edit ]

The problem of constructing a fully homomorphic encryption scheme was first proposed in 1978, within a year of publishing of the RSA scheme.For more than 30 years, it was unclear whether a solution existed. During that period, partial results included the following schemes:

• RSA cryptosystem (unbounded number of modular multiplications);
• ElGamal cryptosystem (unbounded number of modular multiplications);
• Goldwasser–Micali cryptosystem (unbounded number ofexclusive or operations);
• Benaloh cryptosystem (unbounded number of modular additions);
• Paillier cryptosystem (unbounded number of modular additions);
• Sander-Young-Yung system (after more than 20 years solved the problem for logarithmic depth circuits);
• Boneh–Goh–Nissim cryptosystem (unlimited number of addition operations but at most one multiplication);
• Ishai-Paskin cryptosystem (polynomial-sizebranching programs).

First-Generation FHE [ edit ]

Craig Gentry, using lattice-based cryptography , described the first plausible construction for a fully homomorphic encryption scheme.Gentry's scheme supports both addition and multiplication operations on ciphertexts, from which it is possible to construct circuits for performing arbitrary computation. The construction starts from a somewhat homomorphic encryption scheme, which is limited to evaluating low-degree polynomials over encrypted data; it is limited because each ciphertext is noisy in some sense, and this noise grows as one adds and multiplies ciphertexts, until ultimately the noise makes the resulting ciphertext indecipherable. Gentry then shows how to slightly modify this scheme to make it bootstrappable , i.e., capable of evaluating its own decryption circuit and then at least one more operation. Finally, he shows that any bootstrappable somewhat homomorphic encryption scheme can be converted into a fully homomorphic encryption through a recursive self-embedding. For Gentry's "noisy" scheme, the bootstrapping procedure effectively "refreshes" the ciphertext by applying to it the decryption procedure homomorphically, thereby obtaining a new ciphertext that encrypts the same value as before but has lower noise. By "refreshing" the ciphertext periodically whenever the noise grows too large, it is possible to compute an arbitrary number of additions and multiplications without increasing the noise too much. Gentry based the security of his scheme on the assumed hardness of two problems: certain worst-case problems overideal lattices, and the sparse (or low-weight) subset sum problem. Gentry's Ph.D. thesisprovides additional details. The Gentry-Halevi implementation of Gentry's original cryptosystem reported timing of about 30 minutes per basic bit operation.Extensive design and implementation work in subsequent years have improved upon these early implementations by many orders of magnitude runtime performance.

In 2010, Marten van Dijk,Craig Gentry,Shai Halevi and Vinod Vaikuntanathan presented a second fully homomorphic encryption scheme,which uses many of the tools of Gentry's construction, but which does not requireideal lattices. Instead, they show that the somewhat homomorphic component of Gentry's ideal lattice-based scheme can be replaced with a very simple somewhat homomorphic scheme that uses integers. The scheme is therefore conceptually simpler than Gentry's ideal lattice scheme, but has similar properties with regards to homomorphic operations and efficiency. The somewhat homomorphic component in the work of van Dijk et al. is similar to an encryption scheme proposed by Levieil andNaccache in 2008,and also to one that was proposed byBram Cohen in 1998.Cohen's method is not even additively homomorphic, however. The Levieil–Naccache scheme supports only additions, but it can be modified to also support a small number of multiplications. Many refinements and optimizations of the scheme of van Dijk et al. were proposed in a sequence of works by Jean-Sébastien Coron, Tancrède Lepoint, Avradip Mandal,David Naccache, and Mehdi Tibouchi.Some of these works included also implementations of the resulting schemes.

Second-Generation FHE [ edit ]

The homomorphic cryptosystems in current use are derived from techniques that were developed starting in 2011-2012 by Zvika Brakerski,Craig Gentry, Vinod Vaikuntanathan, and others. These innovations led to the development of much more efficient somewhat and fully homomorphic cryptosystems. These include:

• The Brakerski-Gentry-Vaikuntanathan (BGV, 2011) scheme,building on techniques of Brakerski-Vaikuntanathan;
• TheNTRU-based scheme by Lopez-Alt, Tromer, and Vaikuntanathan (LTV, 2012);
• The Brakerski/Fan-Vercauteren (BFV, 2012) scheme,building on Brakerski's scale-invariant cryptosystem;
• TheNTRU-based scheme by Bos, Lauter, Loftus, and Naehrig (BLLN, 2013),building on LTV and Brakerski's scale-invariant cryptosystem;
• The Cheon-Kim-Kim-Song (CKKS, 2016) scheme.

The security of most of these schemes is based on the hardness of the (Ring) Learning With Errors (RLWE) problem, except for the LTV and BLLN schemes that rely on an overstretched variant of theNTRU computational problem. This NTRU variant was subsequently shown vulnerable to subfield lattice attacks,which is why these two schemes are no longer used in practice.

All the second-generation cryptosystems still follow the basic blueprint of Gentry's original construction, namely they first construct a somewhat homomorphic cryptosystem and then convert it to a fully homomorphic cryptosystem using bootstrapping.

A distinguishing characteristic of the second-generation cryptosystems is that they all feature a much slower growth of the noise during the homomorphic computations. Additional optimizations byCraig Gentry,Shai Halevi, andNigel Smart resulted in cryptosystems with nearly optimal asymptotic complexity: Performing operations on data encrypted with security parameter has complexity of only .These optimizations build on the Smart-Vercauteren techniques that enables packing of many plaintext values in a single ciphertext and operating on all these plaintext values in aSIMD fashion.Many of the advances in these second-generation cryptosystems were also ported to the cryptosystem over the integers.

Another distinguishing feature of second-generation schemes is that they are efficient enough for many applications even without invoking bootstrapping, instead operating in the leveled FHE mode.

Third-Generation FHE [ edit ]

In 2013,Craig Gentry,Amit Sahai, andBrent Waters (GSW) proposed a new technique for building FHE schemes that avoids an expensive "relinearization" step in homomorphic multiplication.Zvika Brakerski and Vinod Vaikuntanathan observed that for certain types of circuits, the GSW cryptosystem features an even slower growth rate of noise, and hence better efficiency and stronger security.Jacob Alperin-Sheriff and Chris Peikert then described a very efficient bootstrapping technique based on this observation.

These techniques were further improved to develop efficient ring variants of the GSW cryptosystem: FHEW (2014)and TFHE (2016).The FHEW scheme was the first to show that by refreshing the ciphertexts after every single operation, it is possible to reduce the bootstrapping time to a fraction of a second. FHEW introduced a new method to compute Boolean gates on encrypted data that greatly simplifies bootstrapping, and implemented a variant of the bootstrapping procedure.The efficiency of FHEW was further improved by the TFHE scheme, which implements a ring variant of the bootstrapping procedureusing a method similar to the one in FHEW.

Fully Homomorphic Encryption [ edit ]

A cryptosystem that supports arbitrary computation on ciphertexts is known as fully homomorphic encryption (FHE). Such a scheme enables the construction of programs for any desirable functionality, which can be run on encrypted inputs to produce an encryption of the result. Since such a program need never decrypt its inputs, it can be run by an untrusted party without revealing its inputs and internal state. Fully homomorphic cryptosystems have great practical implications in the outsourcing of private computations, for instance, in the context ofcloud computing.

Implementations [ edit ]

There are no actively developed implementations of first-generation schemes. There are several open-source implementations of second- and third-generation fully homomorphic encryption schemes. Second-generation FHE implementations typically operate in the leveled FHE mode (though bootstrapping is still available in some libraries) and support efficientSIMD-like packing of data; they are typically used to compute on encrypted integers or real/complex numbers. Third-generation FHE implementations often bootstrap after each Boolean gate operation but have limited support for packing and efficient arithmetic computations; they are typically used to compute Boolean circuits over encrypted bits. The choice of using a second-generation vs. third-generation scheme depends on the input data types and the desired computation.

Second-generation FHE implementations [ edit ]

• HElibbyIBM implements the BGV scheme with the GHS optimizations, and the CKKS scheme;
• Microsoft SEALimplements the BFVand the CKKSencryption schemes;
• PALISADEby a consortium of defense contractors and academics, including New Jersey Institute of Technology , Duality Technologies , Raytheon BBN Technologies , MIT , University of California, San Diego and others. PALISADE is a general-purpose lattice cryptography library implementing the BFV, BGV, and other schemes;

Third-generation FHE implementations [ edit ]

• FHEWby Leo Ducas and Daniele Micciancio implements the FHEW scheme.
• TFHEby Ilaria Chillotti, Nicolas Gama, Mariya Georgieva and Malika Izabachene implements the TFHE scheme.

FHE Frameworks [ edit ]

• E3by MoMA Lab at NYU Abu Dhabi implements TFHE, FHEW, HElib and SEAL.

Standardization [ edit ]

A community standard for homomorphic encryption is maintained by the HomomorphicEncryption.org group, an open industry/government/academia consortium founded in 2017. The current standard document includes specifications of secure parameters for RLWE.

An up-to-date list of homomorphic encryption implementations is also maintained by the consortium.

Partially homomorphic cryptosystems [ edit ]

In the following examples, the notation is used to denote the encryption of the message .

Unpadded RSA

If theRSA public key is modulus and exponent , then the encryption of a message is given by . The homomorphic property is then

ElGamal

In theElGamal cryptosystem, in a cyclic group of order with generator , if the public key is , where , and is the secret key, then the encryption of a message is , for some random . The homomorphic property is then

Goldwasser–Micali

In the Goldwasser–Micali cryptosystem , if the public key is the modulus m and quadratic non-residue x , then the encryption of a bit b is , for some random . The homomorphic property is then

where denotes addition modulo 2, (i.e.exclusive-or).

Benaloh

In theBenaloh cryptosystem, if the public key is the modulus m and the base g with a blocksize of c , then the encryption of a message x is , for some random . The homomorphic property is then

Paillier

In the Paillier cryptosystem , if the public key is the modulus m and the base g , then the encryption of a message x is , for some random . The homomorphic property is then

Other partially homomorphic cryptosystems

• Okamoto–Uchiyama cryptosystem
• Naccache–Stern cryptosystem
• Damgård–Jurik cryptosystem
• Sander–Young–Yung encryption scheme
• Boneh–Goh–Nissim cryptosystem
• Ishai–Paskin cryptosystem
• Castagnos–Laguillaumie cryptosystem

See also [ edit ]