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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 over ideal lattices and the sparse or low weight subset sum problem gentry s ph d thesis 15 provides additional details the gentry halevi implementation of gentry s original cryptosystem reported a timing of about 30 minutes per basic bit operation 16 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 17 which uses many of the tools of gentry s construction but which does not require ideal 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 and naccache in 2008 18 and also to one that was proposed by bram cohen in 1998 19 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 20 21 22 23 some of these works included also implementations of the resulting schemes second generation edit the homomorphic cryptosystems of this generation 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 24 building on techniques of brakerski vaikuntanathan 25 the ntru based scheme by lopez alt tromer and vaikuntanathan ltv 2012 26 the brakerski fan vercauteren bfv 2012 scheme 27 building on brakerski s scale invariant cryptosystem 28 the ntru based scheme by bos lauter loftus and naehrig blln 2013 29 building on ltv and brakerski s scale invariant cryptosystem 28 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 30 variant of the ntru computational problem this ntru variant was subsequently shown vulnerable to subfield lattice attacks 31 30 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 by craig gentry shai halevi and nigel smart resulted in cryptosystems with nearly optimal asymptotic complexity performing t displaystyle t operations on data encrypted with security parameter k displaystyle k has complexity of only t p o l y l o g k displaystyle t cdot mathrm polylog k 32 33 34 these optimizations build on the smart vercauteren techniques that enable packing of many plaintext values in a single ciphertext and operating on all these plaintext values in a simd fashion 35 many of the advances in these second generation cryptosystems were also ported to the cryptosystem over the integers 22 23 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 edit in 2013 craig gentry amit sahai and brent waters gsw proposed a new technique for building fhe schemes that avoids an expensive relinearization step in homomorphic multiplication 36 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 37 jacob alperin sheriff and chris peikert then described a very efficient bootstrapping technique based on this observation 38 these techniques were further improved to develop efficient ring variants of the gsw cryptosystem fhew 2014 39 and tfhe 2016 40 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 38 the efficiency of fhew was further improved by the tfhe scheme which implements a ring variant of the bootstrapping procedure 41 using a method similar to the one in fhew fourth generation edit in 2016 jung hee cheon andrey kim miran kim and yongsoo song ckks 42 proposed an approximate homomorphic encryption scheme that supports a special kind of fixed point arithmetic that is commonly referred to as block floating point arithmetic the ckks scheme includes an efficient rescaling operation that scales down an encrypted message after a multiplication for comparison such rescaling requires bootstrapping in the bgv and bfv schemes the rescaling operation makes ckks scheme the most efficient method for evaluating polynomial approximations and is the preferred approach for implementing privacy preserving machine learning applications the scheme introduces several approximation errors both nondeterministic and deterministic that require special handling in practice 43 a 2020 article by baiyu li and daniele micciancio discusses passive attacks against ckks suggesting that the standard ind cpa definition may not be sufficient in scenarios where decryption results are shared 44 the authors apply the attack to four modern homomorphic encryption libraries heaan seal helib and palisade and report that it is possible to recover the secret key from decryption results in several parameter configurations the authors also propose mitigation strategies for these attacks and include a responsible disclosure in the paper suggesting that the homomorphic encryption libraries already implemented mitigations for the attacks before the article became publicly available further information on the mitigation strategies implemented in the homomorphic encryption libraries has also been published 45 46 partially homomorphic cryptosystems edit in the following examples the notation e x displaystyle mathcal e x is used to denote the encryption of the message x displaystyle x unpadded rsa if the rsa public key has modulus n displaystyle n and encryption exponent e displaystyle e then the encryption of a message m displaystyle m is given by e m m e mod n displaystyle mathcal e m m e bmod n the homomorphic property is then e m 1 e m 2 m 1 e m 2 e mod n m 1 m 2 e mod n e m 1 m 2 displaystyle begin aligned mathcal e m_ 1 cdot mathcal e m_ 2 m_ 1 e m_ 2 e bmod n 6pt m_ 1 m_ 2 e bmod n 6pt mathcal e m_ 1 cdot m_ 2 end aligned elgamal in the elgamal cryptosystem in a cyclic group g displaystyle g of order q displaystyle q with generator g displaystyle g if the public key is g q g h displaystyle g q g h where h g x displaystyle h g x and x displaystyle x is the secret key then the encryption of a message m displaystyle m is e m g r m h r displaystyle mathcal e m g r m cdot h r for some random r 0 q 1 displaystyle r in 0 ldots q 1 the homomorphic property is then e m 1 e m 2 g r 1 m 1 h r 1 g r 2 m 2 h r 2 g r 1 r 2 m 1 m 2 h r 1 r 2 e m 1 m 2 displaystyle begin aligned mathcal e m_ 1 cdot mathcal e m_ 2 g r_ 1 m_ 1 cdot h r_ 1 g r_ 2 m_ 2 cdot h r_ 2 6pt g r_ 1 r_ 2 m_ 1 cdot m_ 2 h r_ 1 r_ 2 6pt mathcal e m_ 1 cdot m_ 2 end aligned goldwasser micali in the goldwasser micali cryptosystem if the public key is the modulus n displaystyle n and quadratic non residue x displaystyle x then the encryption of a bit b displaystyle b is e b x b r 2 mod n displaystyle mathcal e b x b r 2 bmod n for some random r 0 n 1 displaystyle r in 0 ldots n 1 the homomorphic property is then e b 1 e b 2 x b 1 r 1 2 x b 2 r 2 2 mod n x b 1 b 2 r 1 r 2 2 mod n e b 1 b 2 displaystyle begin aligned mathcal e b_ 1 cdot mathcal e b_ 2 x b_ 1 r_ 1 2 x b_ 2 r_ 2 2 bmod n 6pt x b_ 1 b_ 2 r_ 1 r_ 2 2 bmod n 6pt mathcal e b_ 1 oplus b_ 2 end aligned where displaystyle oplus denotes addition modulo 2 i e exclusive or benaloh in the benaloh cryptosystem if the public key is the modulus n displaystyle n and the base g displaystyle g with a blocksize of c displaystyle c then the encryption of a message m displaystyle m is e m g m r c mod n displaystyle mathcal e m g m r c bmod n for some random r 0 n 1 displaystyle r in 0 ldots n 1 the homomorphic property is then e m 1 e m 2 g m 1 r 1 c g m 2 r 2 c mod n g m 1 m 2 r 1 r 2 c mod n e m 1 m 2 mod c displaystyle begin aligned mathcal e m_ 1 cdot mathcal e m_ 2 g m_ 1 r_ 1 c g m_ 2 r_ 2 c bmod n 6pt g m_ 1 m_ 2 r_ 1 r_ 2 c bmod n 6pt mathcal e m_ 1 m_ 2 bmod c end aligned paillier in the paillier cryptosystem if the public key is the modulus n displaystyle n and the base g displaystyle g then the encryption of a message m displaystyle m is e m g m r n mod n 2 displaystyle mathcal e m g m r n bmod n 2 for some random r 0 n 1 displaystyle r in 0 ldots n 1 the homomorphic property is then e m 1 e m 2 g m 1 r 1 n g m 2 r 2 n mod n 2 g m 1 m 2 r 1 r 2 n mod n 2 e m 1 m 2 displaystyle begin aligned mathcal e m_ 1 cdot mathcal e m_ 2 g m_ 1 r_ 1 n g m_ 2 r_ 2 n bmod n 2 6pt g m_ 1 m_ 2 r_ 1 r_ 2 n bmod n 2 6pt mathcal e m_ 1 m_ 2 end aligned other partially homomorphic cryptosystems okamoto uchiyama cryptosystem naccache stern cryptosystem damgård jurik cryptosystem sander young yung encryption scheme 47 boneh goh nissim cryptosystem 48 ishai paskin cryptosystem joye libert cryptosystem 49 castagnos laguillaumie cryptosystem 50 implementations edit there are several open source implementations of partially somewhat and fully homomorphic encryption schemes second generation and fourth generation fhe scheme implementations typically operate in the leveled fhe mode though bootstrapping is still available in some libraries and support efficient simd like packing of data they are typically used to compute on encrypted integers or real complex numbers 51 third generation fhe scheme implementations often bootstrap after each operation but have limited support for packing they were initially used to compute boolean circuits over encrypted bits but have been extended to support integer arithmetics and univariate function evaluation 52 the choice of using a second generation vs third generation vs fourth generation scheme depends on the input data types and the desired computation phe swhe libraries library language rsa elgamal exponential elgamal elliptic curve elgamal paillier damgård jurik okamoto uchiyama benaloh naccache stern goldwasser micali sander young yung boneh goh nissim ishai paskin joye libert castagnos laguillaumie notes lightphe 53 python go java typescript yes yes yes yes yes yes yes yes yes yes yes yes no no no fhe libraries name developer bgv 24 ckks 42 bfv 27 fhew 39 ckks bootstrapping 54 tfhe 40 description helib 55 ibm yes yes no no no no bgv scheme with the ghs optimizations microsoft seal 56 microsoft yes yes yes no no no openfhe duality technologies samsung advanced institute of technology kr intel mit university of california san diego and others yes yes yes yes yes yes successor to palisade palisade 57 new jersey institute of technology duality technologies raytheon bbn technologies mit university of california san diego and others yes yes yes yes no yes general purpose lattice cryptography library predecessor of openfhe heaan 58 cryptolab no yes no no yes no fhew 39 leo ducas and daniele micciancio no no no yes no no tfhe 40 ilaria chillotti nicolas gama mariya georgieva and malika izabachene no no no no no yes fv nfllib 59 cryptoexperts no no yes no no no nufhe 60 nucypher no no no no no yes provides a gpu implementation of tfhe redcufhe 61 twc group no no no no no yes a multi gpu implementation of tfhe lattigo 62 epfl lds tune insight yes yes yes no yes 63 no implementation in go along with their distributed variants 64 enabling secure multi party computation tfhe rs 65 zama no no no no no yes rust implementation of tfhe extended supporting boolean integer operation and univariate function evaluation via programmable bootstrapping 66 liberate fhe 67 desilo no yes no no no no a multi gpu implementation of ckks fhe frameworks name developer fhew 39 tfhe helib seal palisade lattigo description concrete 68 zama no yes no no no no tfhe extended compiler with a python frontend 69 e3 70 moma lab at nyu abu dhabi yes yes yes yes yes no sheep 71 alan turing institute no yes yes yes yes no t2 72 twc group no yes yes yes yes yes helm 73 twc group no yes no no no no juliet 74 twc group no yes no no no no peev 75 twc group no no no yes no no verifiable encrypted computations based on rinocchio zkp and bgv homomorphic encryption standardization edit in 2017 researchers from ibm microsoft intel the nist and others formed the open homomorphic encryption standardization consortium which maintains a community security homomorphic encryption standard 76 77 78 related methods edit secure multi party computation is another method for the same goal performing computations while keeping the inputs private see secure multi party computation homomorphic for a detailed comparison see also edit client side encryption type of encryption confidential computing privacy enhancing computing technique format preserving encryption method in cryptography homomorphic secret sharing homomorphic signatures for network coding network coding concept private biometrics private set intersection secure multiparty computation cryptographic technique searchable symmetric encryption system allowing searching of encrypted documents verifiable computing using a fully homomorphic scheme references edit technology massachusetts institute of researchers develop innovative method for secure operations on encrypted data without decryption techxplore com retrieved 2025 04 01 combining machine learning and homomorphic encryption in...