Towards Total Encryption

Adding and Scalar Multiplying With Homomorphic Encryption

Towards Total Encryption

Adding and Scalar Multiplying With Homomorphic Encryption

I predict in a decade, we will process encrypted values. This will allow us to preserve the privacy of data but still use mathematical operations. For example, we could run machine learning models on encrypted data, without revealing the core data used underneath. For this we are likely to look towards full homomorphic encryption, and which supports all of the arithmetic operators. But many current applications just need to support adding and subtraction functions for homomorphic encryption. This will support partial homomorphic encryption. For this, Paillier provided a useful method [here]:

And here is a quick presentation:

Its citation count shows that although it was first published in 1999, it has been a slow-burner, and its impact has generally increased over the years:

The method

Here is the method:

One of the most up-to-date libraries is Kryptology, and which supports Paillier adding. The outline coding using the library from the Kryptology library. Normally we would generate the public and private key with:

pub, sec, _ := paillier.NewKeys()

In this case, though, we will manually create the keys, in order to save time in processing them. In the following example we will add two values and also scalar multiply one of the values [link]:

package main

import (
 "fmt"
 "math/big"
 "os"

"github.com/coinbase/kryptology/pkg/paillier"
)

func main() {

var sec paillier.SecretKey
 var pub paillier.PublicKey

// pub, sec, _ := paillier.NewKeys()

pub.N, _ = new(big.Int).SetString("22203902867524505059996239340306362808852805402888214954381553003002718752808306965243974655390219346481612755387890570991182385566928749760445875916800573782909761881261515602762049819293013811136510263722491329215251675663091154175860620927146517652389408089110716148633480085801107700968384078929774277970426932561081560231010426294975678729992804063220974701278229766883426991469078323539488917623430196595127834729964807458110080684240115196595760172158113810254192728271785178985307185853395355962836026777351498860874006114137632167254987479651229489157192247478252351962954320801263428208801271515398015887801", 10)
 pub.N2 = new(big.Int).Mul(pub.N, pub.N)

sec.N = pub.N
 sec.N2 = pub.N2
 sec.Lambda, _ = new(big.Int).SetString("11101951433762252529998119670153181404426402701444107477190776501501359376404153482621987327695109673240806377693945285495591192783464374880222937958400286891454880940630757801381024909646506905568255131861245664607625837831545577087930310463573258826194704044555358074316740042900553850484192039464887138985064438949068643503538028104882092520753872226183177268421975002892541203962036580811698912148170624870917841537346985483337432253649017635885024033744586145456966646545432316660152135614842196027111652507352356170725302821683928442675667004336523805058723372589095589316741830468728743532156406225121890756346", 10)
 sec.Totient, _ = new(big.Int).SetString("22203902867524505059996239340306362808852805402888214954381553003002718752808306965243974655390219346481612755387890570991182385566928749760445875916800573782909761881261515602762049819293013811136510263722491329215251675663091154175860620927146517652389408089110716148633480085801107700968384078929774277970128877898137287007076056209764185041507744452366354536843950005785082407924073161623397824296341249741835683074693970966674864507298035271770048067489172290913933293090864633320304271229684392054223305014704712341450605643367856885351334008673047610117446745178191178633483660937457487064312812450243781512692", 10)
 sec.U, _ = new(big.Int).SetString("11108720355657041776647490476262423100273444107890028034525926371450220865341816894255404548947647952186614924131107824022774134150177636593890919577479093776330856863330832621779073688637362125712752218152358601679371477743308049274665882845748928958872854339383876978533793119837835146167726752137945969833510158025058440018805812646143848801020667307117190186323497007267362203459968413761903710129427082211518450427328378521338965975284389455704146581178957343955351641425309976491396722227064973929033480821132777267246266490713852090985090755453080857556125803448778856380844874040092451545474091429770838805", 10)
 // fmt.Printf("Pub N: %v\n", pub.N)
 // fmt.Printf("Pub N2: %v\n", pub.N2)
 // fmt.Printf("Sec: Lambda: %v\n", sec.Lambda)
 // fmt.Printf("Sec: Totient: %v\n", sec.Totient)
 // fmt.Printf("U: %v\n", sec.U)

v1 := "111"
 v2 := "123"
 argCount := len(os.Args[1:])

if argCount > 0 {
  v1 = os.Args[1]
 }
 if argCount > 1 {
  v2 = os.Args[2]
 }

val1, _ := new(big.Int).SetString(v1, 10)
 val2, _ := new(big.Int).SetString(v2, 10)

cipher1, c1, _ := pub.Encrypt(val1)

cipher2, c2, _ := pub.Encrypt(val2)

cipher3, _ := pub.Add(cipher1, cipher2)
 cipher4, _ := pub.Mul(val1, cipher2)

decrypted1, _ := sec.Decrypt(cipher3)
 decrypted2, _ := sec.Decrypt(cipher4)

fmt.Printf("\na=%s, b=%s\n", val1, val2)

fmt.Printf("\nEnc(a)=%v, Enc(b)=%v\n", c1, c2)

fmt.Printf("%s + %s = %s\n", val1, val2, decrypted1)
 fmt.Printf("%s * %s = %s\n", val1, val2, decrypted2)

A sample run is [link]:

a=99, b=101

Enc(a)=3194871588628358748966290700084048039504351559895566068649699226686196823202719691848442538154334259086870718517154309238960535353238124681826437650609974756182111331187136550288888243913621355453394195988893907331662212619099956597214048344379155251813838360379613014234504733115070813852665958338054399570730908960729062476752496268730534679455242536966348295264398620170743424134258798655234402342376007577814664497900979829030250365742441838215291615628708192832598514895136556284052555200143916946043349922329083954234944551751946938497382693177826236713968527218819113690486875483513214352456619925368986523829, Enc(b)=13589527499232325752634778817851281121651040055415530301574438693923234689704613260352777781803577016195096649735936270615387802707921560074089079322940073435224175548073476004757540792879509712875530692314042199000397816797600039616264877447516437328780537103533071545942061490755982844224226236302168781174648600938374827556758739664948513788576346004713439291103166197919603791687898854044026637077370502518171971262488604799791153729247987748432317839412374043015668538315916284242023004655533679040068212527422566840965831377168708156964822692509744524741001528815590877128498757010895632375885634162380520158219
99 + 101 = 200
99 * 101 = 9999

Conclusions

If you want to see the first principles generation of Paillier, try here. If you want to see more on Krytopology, try here:

https://asecuritysite.com/kryptology/

References

[1] Paillier, P. (1999, May). Public-key cryptosystems based on composite degree residuosity classes. In International conference on the theory and applications of cryptographic techniques (pp. 223–238). Springer, Berlin, Heidelberg.