# QUANTUM COMPUTERS AND CRYPTOGRAPHY. Mark Zhandry Stanford University

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1 QUANTUM COMPUTERS AND CRYPTOGRAPHY Mark Zhandry Stanford University

2 Classical Encryption pk m c = E(pk,m) sk m = D(sk,c) m???

3 Quantum Computing Attack pk m aka Post-quantum Crypto c = E(pk,m) sk m = D(sk,c) m??? Adversary uses quantum computer to speed up attack

4 This Talk Shor s algorithm for factoring integers Other quantum attacks on classical crypto Quantum Crypto

5 Number Theory x mod N : remainder of x when divided by N x = y (mod N) : (x-y) is divisible by N Suppose p,q prime, N=pq. Let φ(n)=(p-1)(q-1) Fermat s Little Theorem: For all x relatively prime to N, x φ(n) = 1 (mod N) Efficient (classical) modular operations: x+y mod N x y mod N x -1 mod N (as long as x and N are relatively prime) x a mod N

6 RSA Cryptosystem KeyGen: Pick large (>500 bits) random primes p, q Let N = pq Let e be relatively prime to φ(n) (ex: e=3) Let d = e -1 mod φ(n) public key private key Encrypt m < N: Output c = m e mod N Decrypt c: Output m = c d mod N c d = m ed = m 1+kφ(N) = m (mod N)

7 RSA Cryptosystem If adversary can factor N, can decrypt Factoring also best known attack Typical RSA key sizes: Size of N (in bits) Classical Security Level Quantum Security Level (broken!) Exponential growth Polynomial growth

8 Shor s Algorithm

9 Factoring on a Quantum Computer Goal: Given N=pq, find p Alternative Goal: Given N, find non-trivial root of unity x x < N is a non-trivial root of unity if: x 2 = 1 (mod N) x ±1 (mod N) Observation: (x-1)(x+1) = 0 (mod N) Therefore, p and q are factors of (x-1)(x+1) Both cannot be factors of x-1 (same for x+1) GCD(N,x+1) gives p or q

10 Factoring on a Quantum Computer Goal: Given N, find non-trivial root of unity Fix a < N, a relatively prime to N. Let f a (z) = a z mod N The period of a function f is an integer r where: f(z+r) = f(z) for all integers z There is no r <r with f(z+r )=f(z) for all x Alternate Goal: Find period of f a for random a

11 Factoring on a Quantum Computer f a (z) = a z mod N f a (z + φ(n)) = a z+φ(n) = a z a φ(n) = a z (mod N) = f a (z) Therefore, the period of f a divides φ(n) Let r be the period of f a. Suppose r is even. Let x = f a (r/2) = a r/2 mod N Then x 2 = 1 (mod N) and x 1 (mod N) Fact: with probability at least 3/8 over choice of a, r is even and x -1 (mod N)

12 Factoring Strategy Pick a random integer a < N Check that a is relatively prime to N If not, GCD(a,N) is a factor of N Find period r of f a (z) = a z mod N If r is odd, abort Compute x = f a (r/2) = a r/2 mod N If x = -1 mod N, abort Otherwise, x is a non-trivial root of 1 à GCD(x-1,N) is a factor of N Only remaining step: period finding

13 Quantum Fourier Transform Given modulus n, the Quantum Fourier Transform (QFT) maps the state xi (0 x < n) to the state (log n qubits corresponding to binary expansion of x) QFT xi = p 1 nx 1 n y=0 e 2 ixy/n yi

14 Quantum Fourier Transform n=2: QFT 0i = 1 p 2 ( 0i + 1i) QFT 1i = 1 p 2 ( 0i 1i) à Exactly the Hadamard H gate n=2 k : Decompose into k QFTs for n =2 Basically implementing FFT as a quantum circuit Similar idea for smooth numbers (all factors small)

15 Period Finding on a Quantum Computer Let f be a periodic function with (unknown) period r Suppose we know multiple R of r Assume f(0) f(1) f(2)... f(r-1) Step 0: Initialize two registers to the state: 0 i = p 1 RX 1 R z=0 z,0i

16 Period Finding on a Quantum Computer Step 1: Apply f to the registers: 1 i = p 1 RX 1 R z=0 z,f(z)i Since f(z+r) = f(z): 1 i = p 1 Xr 1 R z=0 (R/r) 1 X w=0 z + rw, f(z)i

17 Period Finding on a Quantum Computer Step 2: Measure second register Output of measurement is f(z) for a random z State collapses to be consistent with measurement 2i z = p 1 (R/r) X 1 R/r Ignore second register: w=0 2i z = p 1 (R/r) X 1 R/r w=0 z + rw, f(z)i z + rwi

18 Period Finding on a Quantum Computer Step 3: Apply QFT with modulus R (ignore efficiency for now) 2i z = p 1 (R/r) X 1 R/r w=0 z + rwi z 3i = p r R (R/r) 1 X w=0 R 1 X y=0 e 2 iy(z+rw)/r yi

19 Period Finding on a Quantum Computer z 3i = p r R (R/r) 1 X w=0 = p 1 RX 1 r y=0 RX 1 y=0 e 2 izy/r e 2 iy(z+rw)/r yi r R (R/r) 1 X w=0 e 2 iryw/r 1 A yi r R (R/r) 1 X w=0 e 2 iry/r w = ( 1 if y is a multiple of R/r 0 otherwise

20 Period Finding on a Quantum Computer Putting it together: 3i z = p 1 Xr 1 r y=0 e 2 izy/r (R/r)yi Measure final register à Obtain (R/r)y for a uniform y=0,...,r-1 Repeat multiple times, take GCD of all results to obtain R/r à Obtain r

21 Caveats for Integer Factorization We don t know any R φ(n) is a multiple of the period But knowing φ(n) is equivalent to knowing factors of N QFT only efficient for smooth moduli, φ(n) is probably not smooth Careful analysis shows: If R>>r, can round R to a smooth number (ex: powers of 2) and everything works out

22 Impact Of Quantum Computers Much public-key crypto is broken: RSA-based Discrete Log-based Diffie-Hellman key exchange Elliptic Curve crypto Shor s algorithm can be adapted for Dlog Must use other crypto: Ex: Lattice crypto Private-key crypto seems relatively safe Must increase key sizes to cope with Grover s algorithm

23 Quantum Channel Attacks

24 Classical Chosen Ciphertext Attack (CCA) pk m c = E(pk,m) sk m???

25 Post-Quantum Chosen Ciphertext Attack pk m c = E(pk,m) sk m???

26 Quantum Chosen Ciphertext Attack (qcca) pk m c = E(pk,m) sk m???

27 Our Results on Quantum CCA Security No known attacks on post-quantum schemes Contrived post-quantum scheme susceptible to quantum CCA attack Example of scheme resistant to quantum CCA attack Basically, use scheme that is secure, even if adversary learns decryptions on all ciphertexts other than c

28 Discussion Are these quantum channel attacks reasonable? Objection: Bob can always measure before decrypting sk pk m c = E(pk,m) c m

29 Discussion Are these quantum channel attacks reasonable? Answer: measuring entangles with environment Adversary may control environment attack restored! sk c m m c

30 Digital Signatures sk pk m,σ=s(sk,m) m,σ =??? V(pk,m,σ ) Adversary should not be able to sign m

31 Classical Chosen Message Attack (CMA) sk m σ=s(sk,m) m,σ =??? pk V(pk,m,σ ) Adversary should not be able to sign any m other than the ms he queried

32 Quantum Chosen Message Attack sk σ m m,σ =??? pk V(pk,m,σ )

33 Our Results on Quantum CMA Security Again, no attacks on existing post-quantum schemes Contrived post-quantum scheme susceptible to quantum CMA attack Two simple conversions from post-quantum signatures Hash before signing Likely to be used in practice anyway for efficiency Heuristic security guarantees against qcma attacks Use special hash function Less efficient Provable security guarantees

34 Other Results Pseudorandom Functions (funcs that look random): Simple insecure example Common post-quantum constructions are secure Message Authentication Codes (sigs with private verification): One existing post-quantum construction is insecure Other common constructions secure

35 Quantum Key Distribution (QKD)

36 Quantum Key Distribution In classical crypto, key exchange requires computational assumptions At a minimum, P NP With quantum mechanics, no longer true

37 Quantum Key Distrubution Two possible ways to encode bits in photon polarization: Rectilinear ( ): + 0! i 1! i Diagonal ( ): 0! ñi 1! ói

38 Quantum Key Distribution Measurement: Measure in + basis: i!0 i!1 ñi!0 or 1 with probability 1/2 ói!0 or 1 with probability 1/2 Measure in basis: i!0 or 1 with probability 1/2 i!0 or 1 with probability 1/2 ñi!0 ói!1

39 Quantum Key Distribution Needed properties: If measure in wrong basis, random output Cannot tell which basis used to build state Once measured, original state destroyed

40 Quantum Key Distribution Phase 1: b c i Random bit: Random basis: Encode bit in basis: i Random basis: c 0 Measure in basis: b 0 c = c 0 If, then Otherwise, b, b 0 b = b 0 uncorrelated Repeat many times

41 Quantum Key Distribution Phase 2: Random bit: b Random basis: c c c 0 Random basis: Bit: b 0 c 0 If c 6= c 0, throw away

42 Quantum Key Distribution What happens if someone evesdrops? b c i Random bit: Random basis: Encode bit in basis: i Random basis: c 0 Measure in basis: b 0 0 i Evesdropper cannot tell basis. What if guesses incorrectly? à Introduce errors, even if c = c 0

43 Example Inconsistent! 1 + i Random bit: Random basis: Encode bit in basis: i Random basis: Measure in basis: ñi Random basis: Measure in basis: Resulting state: ñi

44 Quantum Key Distribution Phase 3: Random fraction of bits If too many errors, start over on new channel

45 More Details Errors may occur even if no eavesdropping Impossible to tell if errors result from bad channel or an eavesdropper Information reconciliation: Make sure Alice and Bob have same key Eavesdropper will have some partial information about key Privacy Amplification: Reduce information to negligible Basically apply universal hash to key

46 Conclusion Quantum computer will have a large impact on crypto Additional computational power à many systems broken Quantum channel attacks Quantum protocols à security without assumptions A lot of work remains: Cryptanalysis of existing post-quantum schemes Reduce key sizes Increase distance for QKD

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