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Cryptographic primitives

Cryptographic primitives. Outline. Preliminary Oblivious Transfer (OT) Random share based methods Homomorphic Encryption ElGamal. Assumptions. Semi-honest party assumption Parties honestly follow the security protocol Parties might be curious about the transferred data

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Cryptographic primitives

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  1. Cryptographic primitives

  2. Outline • Preliminary • Oblivious Transfer (OT) • Random share based methods • Homomorphic Encryption • ElGamal

  3. Assumptions • Semi-honest party assumption • Parties honestly follow the security protocol • Parties might be curious about the transferred data • Malicious party assumption • The malicious party can do anything • Transfer false data • Turn down the protocol • Collusion • Often, we can handle • semi-honest + integrity verification

  4. Public-key encryption • Let (G,E,D) be a public-key encryption scheme • G is a key-generation algorithm (pk,sk)  G • Pk: public key • Sk: secret key • Terms • Plaintext: the original text, denoted as m • Ciphertext: the encrypted text, denoted as c • Encryption: c = Epk(m) • Decryption: m = Dsk(c) • Concept of one-way function: knowing c, pk, and the function Epk, it is still computationally intractable to find m. *Check literature for different implementations

  5. 1-out-of-2 Oblivious Transfer (OT) • Setting • Sender has two messages m0 and m1 • Receiver has a single bit {0,1} and wants to learn m , but does not want the sender know which bit is selected. • Outputs • Sender knows nothing about  • Receiver obtain m and learns nothing ofm1-

  6. Assume that a public-key can be sampled without knowledge of its secret key (knowing pk only): • The protocol is simplified with this assumption • Knowing pk but not knowing sk – tricky to do that • Both parties are honest

  7. Protocol for Oblivious Transfer • Receiver (with input ): • Receiver chooses one key-pair (pk,sk) and one public-key pk’ (oblivious key generation), but does not know sk’. • Receiver sets pk = pk, pk1- = pk’ • Receiver sends pk0,pk1 to sender • Sender (with input m0,m1): • Sends c0=Epk0(m0), c1=Epk1(m1) • Receiver: • Decrypts c using sk and obtains m. • Note: receiver can decrypt for pk but not for pk1-

  8. Generalization • Similarly, we can define 1-out-of-k oblivious transfer • Protocol remains the same: • Choose k-1 public keys for which the secret key is unknown • Choose 1 public-key and secret-key pair

  9. Random share based method • Settings • have >2 parties • No collusion – semi-honest • Let f be the function that the parties wish to compute • Represent f as an arithmetic circuit with addition and multiplication gates • Theoretically, any function can be implemented with addition and multiplication • Goal: securely compute additions and multiplications

  10. Random Shares Paradigm • Let a be some value: • Party 1 holds a, distributes random values ai to other parties (and party 1 knows a-ai) • Party i receives ai • Note that without knowing a-ai, and all random shares ai , nothing of a is revealed. • We say that the parties hold random shares of a.

  11. Securely computing addition (xor for one-bit data) • Party 1,2,3 own a,b,c respectively • Generate random shares: • Party 1 has shares a1 , b1 andc1 • Party 2 has shares a2 , b2 andc2 • Party 3 has shares a3 , b3 andc3 • Note: a1+a2 +a3 =a, b1+b2 +b3 =b, and c1+c2 +c3 =c • To compute random shares of output d=a+b+c • Party 1 locally computes d1=a1+b1+c1 • Party 2 locally computes d2=a2+b2+c2 • Party 3 locally computes d3=a3+b3+c3 • Note: d1+d2 +d3 =d • The result shares do not reveal the original value of a,b,c

  12. Multiplication (2 parties) • Simplified 2party case • Assume a, b are binary bit • We work on “and” operation instead –multiplication for one bit numbers • Input wires to gate have values a and b: • Party 1 has shares a1 and b1 • Party 2 has shares a2 and b2 • Wish to compute c = ab = (a1+a2)(b1+b2) • Party 2’s shares are unknown to Party 1,but there are only 4 possibilities (depending on correspondence to 00,01,10,11)

  13. Multiplication (cont) • Party 1 prepares a table as follows: • Row 1 corresponds to Party 2’s input 00 • Row 2 corresponds to Party 2’s input 01 • Row 3 corresponds to Party 2’s input 10 • Row 4 corresponds to Party 2’s input 11 • Let rbe a random bit chosen by Party 1: • Row 1 contains the value ab+r when a2=0,b2=0 • Row 2 contains the value ab+r when a2=0,b2=1 • Row 3 contains the value ab+r when a2=1,b2=0 • Row 4 contains the value ab+r when a2=1,b2=1

  14. Concrete Example • Assume: a1=0, b1=1 • Assume: r=1

  15. The Protocol • The parties run a 1-out-of-4 oblivious transfer protocol • Party 1 plays the sender: message i is row i of the table. • Party 2 plays the receiver: it inputs 1 if a2=0 and b2=0, 2 if a2=0 and b2=1, and so on… • Output: • Party 2 receives c2=ab+r – this is its output • Party 1 outputs c1=r

  16. Problems with OT and RS • Theoretically, any function can be computed with addition and multiplication gates • However, as we have seen, it is not efficient at all • Huge communication/computational cost for the multiplication protocol

  17. Homomorphic encryption • They are “probabilistic encryptions” • using randomly selected numbers in generating keys and encryption • properties • Homomorphic multiplication • Epk(m0)  Epk(m1) = Epk(m0*m1) • Without knowing the secret key, we can still calculate m0*m1 • Implementations: ElGamal method • Homomorphic addition • Epk(m0)Epk(m1) = Epk(m0+m1) • Implementation: Paillier method

  18. ElGamal method • System parameters (P,g) • Input 1n • P is a uniformly chosen prime |P|>n • g: a random number called “generator” • keys • Private key (P,g,x), x is randomly chosen • Public key pk=(P, g, y): y = gx mod P (one way function, computationally intractable to guess x given (P,g,y) ) • Encryption: • E(pk, m, k) = (gk mod P, mgk mod P), k is a random number, m is plaintext

  19. ElGamal’s Homomorphic property • For two ciphertext • E(pk, m0, k0)= (gk0 mod P, m0gk0 mod P) = (a0,b0) • E(pk, m1,k1) = (gk1 mod P, m1gk1 mod P) = (a1,b1) • E(pk, m0*m1, k0+k1) = (gk0+k1 mod P, m0*m1*gk0+k1 mod P) = (a0*a1, b0*b1)

  20. Paillier Encryption • System parameters (p,q) • Input 1N • n=pq, and g = n+1 • Encryption • c=gmrn mod n2. • Decryption

  21. Paillier homomorphic property • E(m1, r1) * E(m2, r2) mod n2= E(m1+m2 mod n) • Multiplication is implemented in partially encrypted form

  22. Summary • Three basic methods • Oblivious Transfer • Random share • Homomorphic encryption • We will see how to use them to construct privacy preserving algorithms

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