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main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 algorithm toggle algorithm subsection 1 1 key generation 1 2 encryption 1 3 decryption 1 4 homomorphic properties 1 5 background 1 6 semantic security 1 7 applications 1 7 1 electronic voting 1 7 2 electronic cash 1 7 3 electronic auction 1 7 4 threshold cryptosystem 2 see also 3 references toggle references subsection 3 1 notes 4 external links toggle the table of contents paillier cryptosystem 7 languages deutsch español français עברית 日本語 русский türkçe edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia algorithm for public key cryptography the paillier cryptosystem invented by and named after pascal paillier in 1999 is a probabilistic asymmetric algorithm for public key cryptography the problem of computing n th residue classes is believed to be computationally difficult the decisional composite residuosity assumption is the intractability hypothesis upon which this cryptosystem is based the scheme is an additive homomorphic cryptosystem this means that given only the public key and the encryption of m 1 displaystyle m_ 1 and m 2 displaystyle m_ 2 one can compute the encryption of m 1 m 2 displaystyle m_ 1 m_ 2 algorithm edit the scheme works as follows key generation edit choose two large prime numbers p displaystyle p and q displaystyle q randomly and independently of each other such that gcd p q p 1 q 1 1 displaystyle gcd pq p 1 q 1 1 this property is assured if both primes are of equal length 1 compute n p q displaystyle n pq and λ lcm p 1 q 1 displaystyle lambda operatorname lcm p 1 q 1 lcm means least common multiple select random integer g displaystyle g where g z n 2 displaystyle g in mathbb z _ n 2 ensure n displaystyle n divides the order of g displaystyle g by checking the existence of the following modular multiplicative inverse μ l g λ mod n 2 1 mod n displaystyle mu l g lambda bmod n 2 1 bmod n where function l displaystyle l is defined as l x x 1 n displaystyle l x frac x 1 n note that the notation a b displaystyle frac a b does not denote the modular multiplication of a displaystyle a times the modular multiplicative inverse of b displaystyle b but rather the quotient of a displaystyle a divided by b displaystyle b i e the largest integer value v 0 displaystyle v geq 0 to satisfy the relation a v b displaystyle a geq vb the public encryption key is n g displaystyle n g the private decryption key is λ μ displaystyle lambda mu if using p q of equivalent length a simpler variant of the above key generation steps would be to set g n 1 λ φ n displaystyle g n 1 lambda varphi n and μ φ n 1 mod n displaystyle mu varphi n 1 bmod n where φ n p 1 q 1 displaystyle varphi n p 1 q 1 1 the simpler variant is recommended for implementational purposes because in the general form the calculation time of μ displaystyle mu can be very high with sufficiently large primes p q encryption edit let m displaystyle m be a message to be encrypted where 0 m n displaystyle 0 leq m n select random r displaystyle r where 0 r n displaystyle 0 r n and gcd r n 1 displaystyle gcd r n 1 note if you find a value that has gcd r n 1 displaystyle gcd r n neq 1 you can use this to calculate the private key this is unlikely enough to ignore compute ciphertext as c g m r n mod n 2 displaystyle c g m cdot r n bmod n 2 decryption edit let c displaystyle c be the ciphertext to decrypt where c z n 2 displaystyle c in mathbb z _ n 2 compute the plaintext message as m l c λ mod n 2 μ mod n displaystyle m l c lambda bmod n 2 cdot mu bmod n as the original paper 2 points out decryption is essentially one exponentiation modulo n 2 displaystyle n 2 homomorphic properties edit a notable feature of the paillier cryptosystem is its homomorphic properties along with its non deterministic encryption see electronic voting in applications for usage as the encryption function is additively homomorphic the following identities can be described homomorphic addition of plaintexts the product of two ciphertexts will decrypt to the sum of their corresponding plaintexts d e m 1 r 1 e m 2 r 2 mod n 2 m 1 m 2 mod n displaystyle d e m_ 1 r_ 1 cdot e m_ 2 r_ 2 bmod n 2 m_ 1 m_ 2 bmod n the product of a ciphertext with a plaintext raising g displaystyle g will decrypt to the sum of the corresponding plaintexts d e m 1 r 1 g m 2 mod n 2 m 1 m 2 mod n displaystyle d e m_ 1 r_ 1 cdot g m_ 2 bmod n 2 m_ 1 m_ 2 bmod n homomorphic multiplication of plaintexts a ciphertext raised to the power of a plaintext will decrypt to the product of the two plaintexts d e m 1 r 1 m 2 mod n 2 m 1 m 2 mod n displaystyle d e m_ 1 r_ 1 m_ 2 bmod n 2 m_ 1 m_ 2 bmod n d e m 2 r 2 m 1 mod n 2 m 1 m 2 mod n displaystyle d e m_ 2 r_ 2 m_ 1 bmod n 2 m_ 1 m_ 2 bmod n more generally a ciphertext raised to a constant k will decrypt to the product of the plaintext and the constant d e m 1 r 1 k mod n 2 k m 1 mod n displaystyle d e m_ 1 r_ 1 k bmod n 2 km_ 1 bmod n however given the paillier encryptions of two messages there is no known way to compute an encryption of the product of these messages without knowing the private key background edit paillier cryptosystem exploits the fact that certain discrete logarithms can be computed easily for example by binomial theorem 1 n x k 0 x x k n k 1 n x x 2 n 2 higher powers of n displaystyle 1 n x sum _ k 0 x x choose k n k 1 nx x choose 2 n 2 text higher powers of n this indicates that 1 n x 1 n x mod n 2 displaystyle 1 n x equiv 1 nx pmod n 2 therefore if y 1 n x mod n 2 displaystyle y 1 n x bmod n 2 then x y 1 n mod n displaystyle x equiv frac y 1 n pmod n thus l 1 n x mod n 2 x mod n displaystyle l 1 n x bmod n 2 equiv x pmod n where function l displaystyle l is defined as l u u 1 n displaystyle l u frac u 1 n quotient of integer division and x z n displaystyle x in mathbb z _ n semantic security edit the original cryptosystem as shown above does provide semantic security against chosen plaintext attacks ind cpa the ability to successfully distinguish the challenge ciphertext essentially amounts to the ability to decide composite residuosity the so called decisional composite residuosity assumption dcra is believed to be intractable because of the aforementioned homomorphic properties however the system is malleable and therefore does not enjoy the highest level of semantic security protection against adaptive chosen ciphertext attacks ind cca2 usually in cryptography the notion of malleability is not seen as an advantage but under certain applications such as secure electronic voting and threshold cryptosystems this property may indeed be necessary paillier and pointcheval however went on to propose an improved cryptosystem that incorporates the combined hashing of message m with random r similar in intent to the cramer shoup cryptosystem the hashing prevents an attacker given only c from being able to change m in a meaningful way through this adaptation the improved scheme can be shown to be ind cca2 secure in the random oracle model applications edit electronic voting edit semantic security is not the only consideration there are situations under which malleability may be desirable secure electronic voting systems can utilize the above homomorphic properties consider a simple binary for or against vote let m voters cast a vote of either 1 for or 0 against each voter encrypts their choice before casting their vote the election official takes the product of the m encrypted votes and then decrypts the result and obtains the value n which is the sum of all the votes the election official then knows that n people voted for and m n people voted against the role of the random r ensures that two equivalent votes will encrypt to the same value only with negligible likelihood hence ensuring voter privacy electronic cash edit another feature named in paper is the notion of self blinding this is the ability to change one ciphertext into another without changing the content of its decryption this has application to the development of ecash an effort originally spearheaded by david chaum imagine paying for an item online without the vendor needing to know your credit card number and hence your identity the goal in both electronic cash and electronic voting is to ensure the e coin likewise e vote is valid while at the same time not disclosing the identity of the person with whom it is currently associated electronic auction edit the paillier cryptosystem plays a crucial role in enhancing the security of electronic auctions it prevents fraudulent activities such as dishonest auctioneers and collusion between bidders and auctioneers who manipulate bids by ensuring the confidentiality of actual bidding values while revealing auction results the pailler cryptosystem successfully promotes fair practices 3 threshold cryptosystem edit the homomorphic property of paillier cryptosystem is sometimes used to build threshold ecdsa signature 4 see also edit the naccache stern cryptosystem and the okamoto uchiyama cryptosystem are historical antecedents of paillier the damgård jurik cryptosystem is a generalization of paillier references edit paillier pascal 1999 public key cryptosystems based on composite degree residuosity classes pdf advances in cryptology eurocrypt 99 eurocrypt springer doi 10 1007 3 540 48910 x_16 paillier pascal pointcheval david 1999 efficient public key cryptosystems provably secure against active adversaries asiacrypt springer pp 165 179 doi 10 1007 978 3 540 48000 6_14 paillier pascal 1999 cryptosystems based on composite residuosity ph d thesis école nationale supérieure des télécommunications paillier pascal 2002 composite residuosity based cryptography an overview pdf cryptobytes 5 1 archived from the original pdf on october 20 2006 notes edit a b jonathan katz yehuda lindell introduction to modern cryptography principles and protocols chapman hall crc 2007 paillier pascal 1999 public key cryptosystems based on composite degree residuosity classes advances in cryptology eurocrypt 99 lecture notes in computer science vol 1592 springer pp 223 238 doi 10 1007 3 540 48910 x_16 isbn 978 3 540 65889 4 pan m sun j fang y 2011 purging the back room dealing secure spectrum auction leveraging paillier cryptosystem ieee journal on selected areas in communications 29 4 866 876 https doi org 10 1109 jsac 2011 110417 canetti ran gennaro rosario goldfeder steven makriyannis nikolaos peled udi 30 october 2020 uc non interactive proactive threshold ecdsa with identifiable aborts proceedings of the 2020 acm sigsac conference on computer and communications security association for computing machinery pp 1769 1787 doi 10 1145 3372297 3423367 isbn 9781450370899 s2cid 226228099 external links edit the homomorphic encryption project implements the paillier cryptosystem along with its homomorphic operations encounter an open source library providing an implementation of paillier cryptosystem and a cryptographic counters construction based on the same python paillier a library for partially homomorphic encryption in python including full support for floating point numbers the paillier cryptosystem interactive simulator archived 2012 02 18 at the wayback machine demonstrates a voting application an interactive demo of the paillier cryptosystem a proof of concept javascript implementation of the paillier cryptosystem with an interactive demo a googletechtalk video on voting using cryptographic methods a ruby implementation of paillier homomorphic addition and a zero knowledge proof protocol documentation v t e public key cryptography algorithms integer factorization benaloh blum goldwasser cayley purser damgård jurik gmr goldwasser micali naccache stern paillier rabin rsa okamoto uchiyama schmidt samoa discrete logarithm bls cramer shoup dh dsa ecdh x25519 x448 ecdsa eddsa ed25519 ed448 ecmqv eke elgamal signature scheme mqv schnorr speke srp sts lattice svp cvp lwe sis bliss kyber newhope ntruencrypt ntrusign rlwe kex rlwe sig falcon others ae ceilidh epoc hfe ies lamport mceliece merkle hellman naccache stern knapsack cryptosystem three pass protocol xtr sqisign sphincs theory discrete logarithm cryptography elliptic curve cryptography hash based cryptography non commutative cryptography rsa problem trapdoor function tropical cryptography standardization cryptrec ieee p1363 nessie nsa suite b cnsa post quantum cryptography topics digital signature oaep fingerprint pki web of trust key size identity based cryptography post quantum cryptography openpgp card v t e cryptography general history of cryptography outline of cryptography classical cipher cryptographic protocol authentication protocol cryptographic primitive cryptanalysis cryptocurrency cryptosystem cryptographic nonce cryptovirology hash function cryptographic hash function key derivation function secure hash algorithms digital signature kleptography key cryptography key exchange key generator key schedule key stretching keygen machines ransomware random number generation cryptographically secure pseudorandom number generator csprng pseudorandom noise prn secure channel insecure channel subliminal channel encryption decryption end to end encryption harvest now decrypt later information theoretic security plaintext codetext ciphertext shared secret trapdoor function trusted timestamping key based routing onion routing garlic routing kademlia mix network mathematics cryptographic hash function block cipher stream cipher symmetric key algorithm authenticated encryption public key cryptography quantum key distribution quantum cryptography post quantum cryptography message authentication code random numbers steganography category retrieved from https en wikipedia org w index php title paillier_cryptosystem oldid 1188810864 category public key encryption schemes hidden categories articles with short description short description matches wikidata webarchive template wayback links this page was last edited on 7 december 2023 at 21 01 utc text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers 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