Crypto 101 Fully Homomorphic Encryption CryptDB

Homomorphic EncryptionIst kunstliche Intelligenz gefahrlich?

Carine Dengler

Carine Dengler

Crypto 101 Fully Homomorphic Encryption CryptDB

Table of Contents

1 Crypto 101

2 How-to Fully Homomorphic Encryption

3 CryptDBIntroCryptDB

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Crypto 101 Fully Homomorphic Encryption CryptDB

Intro

confidentiality

authentication

integrity

non-repudiation

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Crypto 101 Fully Homomorphic Encryption CryptDB

Symmetric Encryption

(K , P, C , KeyGen, Enc , Dec)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Symmetric Encryption

(K , P, C , KeyGen, Enc , Dec)

keyspace

k ∈ K key

KeyGen outputs k ∈ K s.t. length k ≥ n

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Crypto 101 Fully Homomorphic Encryption CryptDB

Symmetric Encryption

PLAINTEXT

(K , P, C , KeyGen, Enc , Dec)

plaintext space

m ∈ P plaintext

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Crypto 101 Fully Homomorphic Encryption CryptDB

Symmetric Encryption

(K , P, C , KeyGen, Enc , Dec)

ciphertext space

c ∈ C ciphertext

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Crypto 101 Fully Homomorphic Encryption CryptDB

Symmetric Encryption

PLAINTEXT

(K , P, C , KeyGen, Enc , Dec)

encryption algorithm

Enc : K × P → C , (k ,m) 7→ c

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Crypto 101 Fully Homomorphic Encryption CryptDB

Symmetric Encryption

PLAINTEXT

(K , P, C , KeyGen, Enc , Dec)

encryption algorithm

Enc : K × P → C , (k ,m) 7→ c

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Crypto 101 Fully Homomorphic Encryption CryptDB

Symmetric Encryption

PLAINTEXT

(K , P, C , KeyGen, Enc , Dec)

encryption algorithm

Enc : K × P → C , (k ,m) 7→ c

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Crypto 101 Fully Homomorphic Encryption CryptDB

Symmetric Encryption

PLAINTEXT

(K , P, C , KeyGen, Enc , Dec)

decryption algorithm

Dec : K × C → P, (k, c) 7→ m

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Crypto 101 Fully Homomorphic Encryption CryptDB

Symmetric Encryption

(K , P, C , KeyGen, Enc , Dec)

DES (Data Encryption Standard)

AES (Advanced Encryption Standard)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Asymmetric Encryption

(K , P, C , KeyGen, Enc , Dec)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Asymmetric Encryption

(K , P, C , KeyGen, Enc , Dec)

(pk, sk) ∈ K s.t. length pk ≥ n and length sk ≥ n

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Crypto 101 Fully Homomorphic Encryption CryptDB

Asymmetric Encryption

PLAINTEXT

(K , P, C , KeyGen, Enc , Dec)

Enc(pk ,m) = c

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Crypto 101 Fully Homomorphic Encryption CryptDB

Asymmetric Encryption

PLAINTEXT

(K , P, C , KeyGen, Enc , Dec)

Dec(sk , c) = m

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Crypto 101 Fully Homomorphic Encryption CryptDB

Asymmetric Encryption

(K , P, C , KeyGen, Enc , Dec)

RSA

ElGamal

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Crypto 101 Fully Homomorphic Encryption CryptDB

Correctness

PLAINTEXTPLAINTEXT

Dec(k ,Enc(k ,m)) = m, k ∈ K ,m ∈ P

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Crypto 101 Fully Homomorphic Encryption CryptDB

Correctness

PLAINTEXTPLAINTEXT

Dec(sk ,Enc(pk,m)) = m, (pk, sk) ∈ K ,m ∈ P

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Crypto 101 Fully Homomorphic Encryption CryptDB

Kerckhoff’s Principle

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Crypto 101 Fully Homomorphic Encryption CryptDB

Table of Contents

1 Crypto 101

2 How-to Fully Homomorphic Encryption

3 CryptDBIntroCryptDB

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Crypto 101 Fully Homomorphic Encryption CryptDB

Intro

PLAINTEXT PLAINTEXT

∀f ∈ F and ci with Enc(pk,mi ) = ci ,mi ∈ P, i = 1, . . . , t :Eval(pk, f , c1, ..., ct) = cEval s.t.Dec(sk , cEval) = f (m1, ...,mt)

ElGamal

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Crypto 101 Fully Homomorphic Encryption CryptDB

Intro

(K ,P,C ,KeyGen,Enc ,Dec ,Eval , F)

∀f ∈ F and ci with Enc(pk,mi ) = ci ,mi ∈ P, i = 1, . . . , t :Eval(pk, f , c1, ..., ct) = cEval s.t.Dec(sk , cEval) = f (m1, ...,mt)

ElGamal

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Crypto 101 Fully Homomorphic Encryption CryptDB

Intro

PLAINTEXT PLAINTEXT

∀f ∈ F and ci with Enc(pk,mi ) = ci ,mi ∈ P, i = 1, . . . , t :Eval(pk, f , c1, ..., ct) = cEval s.t.Dec(sk , cEval) = f (m1, ...,mt)

ElGamal

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Crypto 101 Fully Homomorphic Encryption CryptDB

Intro

PLAINTEXT PLAINTEXT

∀f ∈ F and ci with Enc(pk,mi ) = ci ,mi ∈ P, i = 1, . . . , t :Eval(pk, f , c1, ..., ct) = cEval s.t.Dec(sk , cEval) = f (m1, ...,mt)

ElGamal

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Crypto 101 Fully Homomorphic Encryption CryptDB

Requirements

manipulations

leaked information

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Crypto 101 Fully Homomorphic Encryption CryptDB

Starting Point

f can be expressed as a boolean circuit

evaluate gates for x , y ∈ {0, 1}AND(x , y) = xyOR(x , y) = 1− (1− x)(1− y)NOT(x , y) = 1− x

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Crypto 101 Fully Homomorphic Encryption CryptDB

Starting Point

f can be expressed as a boolean circuit

evaluate gates for x , y ∈ {0, 1}AND(x , y) = xyOR(x , y) = 1− (1− x)(1− y)NOT(x , y) = 1− x

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Crypto 101 Fully Homomorphic Encryption CryptDB

Somewhat Homomorphic Encryption Scheme (Symmetric)

KeyGen(n) = p s.t. p ∈ Z, p odd

Enc(p,m) = m′ + pq, m ∈ {0, 1} with m′ n-bit lengthnumber s.t. m′ mod 2 = m mod 2, q random number

Dec(p, c) = (c mod p) mod 2

c mod p = m′ noise associated to c

m′ ∈ (−p2 ,

p2 )

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Crypto 101 Fully Homomorphic Encryption CryptDB

Somewhat Homomorphic Encryption Scheme (Symmetric)

KeyGen(n) = p s.t. p ∈ Z, p odd

Enc(p,m) = m′ + pq, m ∈ {0, 1} with m′ n-bit lengthnumber s.t. m′ mod 2 = m mod 2, q random number

Dec(p, c) = (c mod p) mod 2

c mod p = m′ noise associated to c

m′ ∈ (−p2 ,

p2 )

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Crypto 101 Fully Homomorphic Encryption CryptDB

Somewhat Homomorphic Encryption Scheme (Symmetric)

KeyGen(n) = p s.t. p ∈ Z, p odd

Enc(p,m) = m′ + pq, m ∈ {0, 1} with m′ n-bit lengthnumber s.t. m′ mod 2 = m mod 2, q random number

Dec(p, c) = (c mod p) mod 2

c mod p = m′ noise associated to c

m′ ∈ (−p2 ,

p2 )

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Crypto 101 Fully Homomorphic Encryption CryptDB

Somewhat Homomorphic Encryption Scheme (Symmetric)

KeyGen(n) = p s.t. p ∈ Z, p odd

Enc(p,m) = m′ + pq, m ∈ {0, 1} with m′ n-bit lengthnumber s.t. m′ mod 2 = m mod 2, q random number

Dec(p, c) = (c mod p) mod 2

c mod p = m′ noise associated to c

m′ ∈ (−p2 ,

p2 )

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Crypto 101 Fully Homomorphic Encryption CryptDB

Somewhat Homomorphic Encryption Scheme (Symmetric)

KeyGen(n) = p s.t. p ∈ Z, p odd

Enc(p,m) = m′ + pq, m ∈ {0, 1} with m′ n-bit lengthnumber s.t. m′ mod 2 = m mod 2, q random number

Dec(p, c) = (c mod p) mod 2

c mod p = m′ noise associated to c

m′ ∈ (−p2 ,

p2 )

Carine Dengler

Crypto 101 Fully Homomorphic Encryption CryptDB

Somewhat Homomorphic Encryption Scheme (Symmetric)

KeyGen(n) = p s.t. p ∈ Z, p odd

Enc(p,m) = m′ + pq, m ∈ {0, 1} with m′ n-bit lengthnumber s.t. m′ mod 2 = m mod 2, q random number

Dec(p, c) = (c mod p) mod 2

c mod p = m′ noise associated to c

m′ ∈ (−p2 ,

p2 )

Carine Dengler

Crypto 101 Fully Homomorphic Encryption CryptDB

Somewhat Homomorphic Encryption Scheme (Symmetric)

KeyGen(n) = p s.t. p ∈ Z, p odd

Enc(p,m) = m′ + pq, m ∈ {0, 1} with m′ n-bit lengthnumber s.t. m′ mod 2 = m mod 2, q random number

Dec(p, c) = (c mod p) mod 2

c mod p = m′ noise associated to c

m′ ∈ (−p2 ,

p2 )

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Crypto 101 Fully Homomorphic Encryption CryptDB

Operations

Add(c1, c2) = c1 + c2, Sub(c1, c2) = c1 − c2,Mult(c1, c2) = c1c2

Mult(c1, c2) = m′1m

′2 + pq′ with m′

i noise associated to ci ,i = 1, 2

f +(c1, c2, ..., ct) = f +(m′1,m

′2, ...,m

′t) + pq with m′

i noiseassociated to ci , i = 1, 2, ..., t

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Crypto 101 Fully Homomorphic Encryption CryptDB

Operations

Add(c1, c2) = c1 + c2, Sub(c1, c2) = c1 − c2,Mult(c1, c2) = c1c2

Mult(c1, c2) = m′1m

′2 + pq′ with m′

i noise associated to ci ,i = 1, 2

f +(c1, c2, ..., ct) = f +(m′1,m

′2, ...,m

′t) + pq with m′

i noiseassociated to ci , i = 1, 2, ..., t

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Crypto 101 Fully Homomorphic Encryption CryptDB

Operations

Add(c1, c2) = c1 + c2, Sub(c1, c2) = c1 − c2,Mult(c1, c2) = c1c2

Mult(c1, c2) = m′1m

′2 + pq′ with m′

i noise associated to ci ,i = 1, 2

f +(c1, c2, ..., ct) = f +(m′1,m

′2, ...,m

′t) + pq with m′

i noiseassociated to ci , i = 1, 2, ..., t

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Crypto 101 Fully Homomorphic Encryption CryptDB

Operations

Add(c1, c2) = c1 + c2, Sub(c1, c2) = c1 − c2,Mult(c1, c2) = c1c2

Mult(c1, c2) = m′1m

′2 + pq′ with m′

i noise associated to ci ,i = 1, 2

f +(c1, c2, ..., ct) = f +(m′1,m

′2, ...,m

′t) + pq with m′

i noiseassociated to ci , i = 1, 2, ..., t

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Crypto 101 Fully Homomorphic Encryption CryptDB

Operations

Add(c1, c2) = c1 + c2, Sub(c1, c2) = c1 − c2,Mult(c1, c2) = c1c2

Mult(c1, c2) = m′1m

′2 + pq′ with m′

i noise associated to ci ,i = 1, 2

f +(c1, c2, ..., ct) = f +(m′1,m

′2, ...,m

′t) + pq with m′

i noiseassociated to ci , i = 1, 2, ..., t

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Crypto 101 Fully Homomorphic Encryption CryptDB

Operations

Add(c1, c2) = c1 + c2, Sub(c1, c2) = c1 − c2,Mult(c1, c2) = c1c2

Mult(c1, c2) = m′1m

′2 + pq′ with m′

i noise associated to ci ,i = 1, 2

f +(c1, c2, ..., ct) = f +(m′1,m

′2, ...,m

′t) + pq with m′

i noiseassociated to ci , i = 1, 2, ..., t

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Crypto 101 Fully Homomorphic Encryption CryptDB

Somewhat Homomorphic Encryption Scheme(Asymmetric)

KeyGen(n) = (pk, sk)

sk = p

pk list of encryptions of 0

Enc(pk ,m) = m +∑

pki∈I pki with I random subset

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Crypto 101 Fully Homomorphic Encryption CryptDB

Somewhat Homomorphic Encryption Scheme(Asymmetric)

KeyGen(n) = (pk, sk)

sk = p

pk list of encryptions of 0

Enc(pk ,m) = m +∑

pki∈I pki with I random subset

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Crypto 101 Fully Homomorphic Encryption CryptDB

Somewhat Homomorphic Encryption Scheme(Asymmetric)

KeyGen(n) = (pk, sk)

sk = p

pk list of encryptions of 0

Enc(pk ,m) = m +∑

pki∈I pki with I random subset

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Crypto 101 Fully Homomorphic Encryption CryptDB

Somewhat Homomorphic Encryption Scheme(Asymmetric)

KeyGen(n) = (pk, sk)

sk = p

pk list of encryptions of 0

Enc(pk ,m) = m +∑

pki∈I pki with I random subset

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Crypto 101 Fully Homomorphic Encryption CryptDB

Noise

the noise becomes too large (|f +(m′1, ...,m

′t)| >

p2 )

decryption removes noise

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Crypto 101 Fully Homomorphic Encryption CryptDB

Noise

the noise becomes too large (|f +(m′1, ...,m

′t)| >

p2 )

decryption removes noise

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Crypto 101 Fully Homomorphic Encryption CryptDB

Solution

bootstrappable

circular-secure

F = {Dec ,DecAdd,DecSub,DecMult}c1 = Enc(pk,m)

sk = Enc(pk, ski )

Recrypt(pk,Dec , sk , c1)

c1 = Enc(pk , c1i )c = Eval(pk ,Dec , sk , c1)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Solution

bootstrappable

circular-secure

F = {Dec ,DecAdd,DecSub,DecMult}c1 = Enc(pk,m)

sk = Enc(pk, ski )

Recrypt(pk,Dec , sk , c1)

c1 = Enc(pk , c1i )c = Eval(pk ,Dec , sk , c1)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Solution

F = {Dec ,DecAdd,DecSub,DecMult}

c1 = Enc(pk,m)

sk = Enc(pk, ski )

Recrypt(pk,Dec , sk , c1)

c1 = Enc(pk , c1i )c = Eval(pk ,Dec , sk , c1)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Solution

F = {Dec ,DecAdd,DecSub,DecMult}c1 = Enc(pk,m)

sk = Enc(pk, ski )

Recrypt(pk,Dec , sk , c1)

c1 = Enc(pk , c1i )c = Eval(pk ,Dec , sk , c1)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Solution

F = {Dec ,DecAdd,DecSub,DecMult}c1 = Enc(pk,m)

sk = Enc(pk, ski )

Recrypt(pk,Dec , sk , c1)

c1 = Enc(pk , c1i )c = Eval(pk ,Dec , sk , c1)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Solution

F = {Dec ,DecAdd,DecSub,DecMult}c1 = Enc(pk,m)

sk = Enc(pk, ski )

Recrypt(pk,Dec , sk , c1)

c1 = Enc(pk , c1i )c = Eval(pk ,Dec , sk , c1)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Operations

c1 = Enc(pk,m1), c2 = Enc(pk,m2)

c1 = Enc(pk, c1i ), c2 = Enc(pk , c2i )

c = Eval(pk,DecAdd, sk , c1, c2)

c = Enc(pk, (m1 + m2) mod 2)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Operations

c1 = Enc(pk,m1), c2 = Enc(pk,m2)

c1 = Enc(pk, c1i ), c2 = Enc(pk , c2i )

c = Eval(pk,DecAdd, sk , c1, c2)

c = Enc(pk, (m1 + m2) mod 2)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Operations

c1 = Enc(pk,m1), c2 = Enc(pk,m2)

c1 = Enc(pk, c1i ), c2 = Enc(pk , c2i )

c = Eval(pk,DecAdd, sk , c1, c2)

c = Enc(pk, (m1 + m2) mod 2)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Operations

c1 = Enc(pk,m1), c2 = Enc(pk,m2)

c1 = Enc(pk, c1i ), c2 = Enc(pk , c2i )

c = Eval(pk,DecAdd, sk , c1, c2)

c = Enc(pk, (m1 + m2) mod 2)

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Crypto 101 Fully Homomorphic Encryption CryptDB

Fully Homomorphic Encryption Scheme

express f as circuit

arrange its gates into levels

evaluate the levels sequentially

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Crypto 101 Fully Homomorphic Encryption CryptDB

Fully Homomorphic Encryption Scheme

express f as circuit

arrange its gates into levels

evaluate the levels sequentially

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Crypto 101 Fully Homomorphic Encryption CryptDB

Fully Homomorphic Encryption Scheme

express f as circuit

arrange its gates into levels

evaluate the levels sequentially

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Crypto 101 Fully Homomorphic Encryption CryptDB

But . . .

the noise of Dec is � p2

Dec(p, c) = (c mod p) mod 2 = (c − b cp e) mod 2 ⇔LSB(c) XOR LSB(b cp e)b cp e

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Crypto 101 Fully Homomorphic Encryption CryptDB

But . . .

the noise of Dec is � p2

Dec(p, c) = (c mod p) mod 2 = (c − b cp e) mod 2 ⇔LSB(c) XOR LSB(b cp e)b cp e

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Crypto 101 Fully Homomorphic Encryption CryptDB

But . . .

the noise of Dec is � p2

Dec(p, c) = (c mod p) mod 2 = (c − b cp e) mod 2 ⇔LSB(c) XOR LSB(b cp e)b cp e

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Crypto 101 Fully Homomorphic Encryption CryptDB

But . . .

the noise of Dec is � p2

Dec(p, c) = (c mod p) mod 2 = (c − b cp e) mod 2 ⇔LSB(c) XOR LSB(b cp e)b cp e

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Crypto 101 Fully Homomorphic Encryption CryptDB

But . . .

the noise of Dec is � p2

Dec(p, c) = (c mod p) mod 2 = (c − b cp e) mod 2 ⇔LSB(c) XOR LSB(b cp e)b cp e

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Crypto 101 Fully Homomorphic Encryption CryptDB

Hint

make the decryption function simpler

include a hint about the secret integer p

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Crypto 101 Fully Homomorphic Encryption CryptDB

Hint

KeyGen with parameters α, β

(pk, sk) with sk = p

Y = {y1, ..., yβ}, yi ∈ Q s.t. ∃J ⊂ {1, ..., β} with∑j∈J yj = 1

p mod 2 and | J |= α

x ∈ {0, 1}β with Hamming weight α

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Crypto 101 Fully Homomorphic Encryption CryptDB

Hint

KeyGen with parameters α, β

(pk, sk) with sk = p

Y = {y1, ..., yβ}, yi ∈ Q s.t. ∃J ⊂ {1, ..., β} with∑j∈J yj = 1

p mod 2 and | J |= α

x ∈ {0, 1}β with Hamming weight α

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Crypto 101 Fully Homomorphic Encryption CryptDB

Hint

KeyGen with parameters α, β

(pk, sk) with sk = p

Y = {y1, ..., yβ}, yi ∈ Q s.t. ∃J ⊂ {1, ..., β} with∑j∈J yj = 1

p mod 2 and | J |= α

x ∈ {0, 1}β with Hamming weight α

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Crypto 101 Fully Homomorphic Encryption CryptDB

Hint

KeyGen with parameters α, β

(pk, sk) with sk = p

Y = {y1, ..., yβ}, yi ∈ Q s.t. ∃J ⊂ {1, ..., β} with∑j∈J yj = 1

p mod 2 and | J |= α

x ∈ {0, 1}β with Hamming weight α

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Crypto 101 Fully Homomorphic Encryption CryptDB

Hint

KeyGen with parameters α, β

(pk, sk) with sk = p

Y = {y1, ..., yβ}, yi ∈ Q s.t. ∃J ⊂ {1, ..., β} with∑j∈J yj = 1

p mod 2 and | J |= α

x ∈ {0, 1}β with Hamming weight α

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Crypto 101 Fully Homomorphic Encryption CryptDB

Hint

Encrypt(pk,m) = m

z s.t. zi = cyi mod 2, i ∈ {1, ..., β}

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Crypto 101 Fully Homomorphic Encryption CryptDB

Hint

Dec(x , z , c) = LSB(c)XOR LSB(b∑

xizie)∑xizi =

∑cxiyi = c

p mod 2

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Crypto 101 Fully Homomorphic Encryption CryptDB

Hint

Dec(x , z , c) = LSB(c)XOR LSB(b∑

xizie)∑xizi =

∑cxiyi = c

p mod 2

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Crypto 101 Fully Homomorphic Encryption CryptDB

Hint

Dec(x , z , c) = LSB(c)XOR LSB(b∑

xizie)∑xizi =

∑cxiyi = c

p mod 2

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Crypto 101 Fully Homomorphic Encryption CryptDB

Hint

replace multiplication by summation

if α is small enough, the noise is small enough

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Crypto 101 Fully Homomorphic Encryption CryptDB

Table of Contents

1 Crypto 101

2 How-to Fully Homomorphic Encryption

3 CryptDBIntroCryptDB

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Crypto 101 Fully Homomorphic Encryption CryptDB

Intro

Database-backed Application

AS DBMS

PLAINTEXT

QUERY

RESULT

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Crypto 101 Fully Homomorphic Encryption CryptDB

Intro

Threats

AS DBMS

PLAINTEXT

QUERY

RESULT

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Crypto 101 Fully Homomorphic Encryption CryptDB

Intro

Threats

AS DBMS

PLAINTEXT

QUERY

RESULT

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Crypto 101 Fully Homomorphic Encryption CryptDB

Intro

Key Ideas

SQL-aware encryption strategy

adjustable query-based encryption

chaining encryption keys to user passwords

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Crypto 101 Fully Homomorphic Encryption CryptDB

Intro

Architecture

AS DBMSPROXY

QUERY

QUERY

RESPONSERESULT

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

SQL-aware Encryption

DATA

ONION EQ

DATA

ONION SEARCH

DATA

ONION ORD

DATA

ONION ADD

Random (RND)

Homomorphic encryption (HOM)

Word search (SEARCH)

Deterministic (DET)

Join (JOIN and OPE-JOIN)

Order-preserving encryption (OPE)

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

Adjustable Query-based Encryption

DATA

ONION EQ

DATA

ONION SEARCH

DATA

ONION ORD

DATA

ONION ADD

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

Adjustable Query-based Encryption

DATA

ONION EQ

DATA

JOIN

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

Query Execution

AS PROXY

QUERY

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

Query Execution

PROXY

QUERY QUERY

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

Query Execution

PROXY DBMS

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

Query Execution

AS DBMSPROXY

QUERY

RESPONSE

RESULT

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

Access Policy

principal

annotations

delegation rules

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

Key Chaining

BOB

SYMSYM

SYM SYM

ALICE

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

Key Chaining

SYM

SYM

ALICE

BOB

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

Security

sensitive data

revealed information

compromise

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

Further Reading I

Buchmann, J.Einfuhrung in die Kryptographie

Katz, J., Lindell, Y.Introduction to Modern Cryptgraphy

Beutelspacher A., Schwenk, J., Wolfenstetter, K.-D.Moderne Verfahren der Kryptographie

Petrlic, R., Sorge C.Datenschutz. Einfuhrung in technischen Datenschutz,Datenschutzrecht und angewandte Kryptographie

I Illustrations in section Crypto 101 based on descriptions in theworks listed above.

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Crypto 101 Fully Homomorphic Encryption CryptDB

CryptDB

Further Reading II

Gentry, C.Computing Arbitrary Functions of Encrypted DataIllustrations in section Fully Homomorphic Encryption basedon descriptions in this work.

Popa, R. A., Redfield, C. M. S., Zeldovich, N., Balakrishnan,H.CryptDB: Protecting Confidentiality with Encrypted QueryProcessingIllustrations in section CryptDB based on descriptions andillustrations in this work.

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