QUANTUM COMPUTERS AND THEIR APPLICATION TO … ATTACK Private cipher attack DES use Grover’s Algorithm - like brute force attack • Grover’s Algorithm based on amplitude amplification

QUANTUM COMPUTERS AND THEIR APPLICATION TO BREAKING CIPHERS

-QUANTUM ALGORITHM RSA ATTACK

Present by Huajun Zhang

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QUANTUM COMPUTING

Quantum computing >= Classical computing most of time

Quantum computing >>>>>>>>>>> Classical computing sometime

Qubit > Bit States |0> and |1> vs 0 or 1

And vs Or

Qubits can be in a superposition of both!

Example: a register of three classical bits: 000 represent one number from 0-7 a register of three qubit bits: |000> represent 0-7 simultaneously

Quantum algorithm - develop use the quantum’s feature

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QUANTUM ATTACK

Private cipher attack DES use Grover’s Algorithm - like brute force attack

• Grover’s Algorithm based on amplitude amplification

• easy to defend by adding the length of key

public cipher attack **** RSA use Shor’s Algorithm - best factoring

• Shor’s Algorithm based on quantum Fourier transform

• Hard to defend , WHY ? �3

RSA REVIEW

RSA is the most well known public key cryptosystem It’s security base on:

1. One way trap door function 2. The hardness of factoring numbers

RSA : public key(e,N), private key(d,N)

Breaking RSA - factor a large Prime N to guess Private key

Factoring Problem!!!!!!

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 SHOR’S ALGORITHM FIVE STEPS OF SHOR

Question: find a nontrivial factor of a number N?

Step 1: random pick m from (0,N)

if gcd(N,m) = 1 continue , else we done

Step 2: Quantum period finding P of

m^1 mod N, m^2 mod N, m^3 mod N…

Step 3: if P is even continue, else go step 1

Step 4: if continue , else go step 1

Step 5: a non-trivial prime factor is

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PERIOD IN SHOR

What is period ?

In Shor’s Alogrithm

m^x mod N, x is an integer

Example: we have m=2 which has a sequence powers of two 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, … when we mod 15 on this sequence 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, … we can see the sequence of number has period of P = 4

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QUANTUM PERIOD FINDING HARDWARE IMPLEMENTATION

1. Apply the Hadamard gate to create a quantum superposition

2. Implement the function into a quantum transform

3. Perform the “quantum Fourier transform” QFT which allows a quantum computer to

perform phase estimation.

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QUANTUM PERIOD FINDING 1

What the Quantum period finding actually did?

f(x)=m^x mod N, N=21 and m=2 Let input x output f(x)

we group x with same result together

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x 0 1 2 3 4 5 6 7 8 9 10 11 12 …m^x 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 …f(x) 1 2 4 8 16 11 1 2 4 8 16 11 1 …

f(x)=1 x= 0 6 12 18 24…f(x)=2 x= 1 7 13 19 25…f(x)=4 x= 2 8 14 20 26…f(x)=8 x= 3 9 15 21 27…f(x)=16 x= 4 10 16 22 28…f(x)=11 x= 5 11 17 23 29…

QUANTUM PERIOD FINDING 2

Declare 2 quantum registers with 9 quits

1st register |x⟩ 2nd register |f(x)⟩

measure f(x)= 11, x= 5, 11, 17, 23, 29…

quantum function: = (|5⟩+|11⟩+|17⟩+|23⟩+|29⟩… ) superposition

we roughly has 6 this kind of function Perform QFT probability distribution of all possible measurements on x

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1

m|ψ⟩

PROBABILITY DISTRIBUTION OF ALL POSSIBLE MEASUREMENTS ON X

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Figure 1. Understanding Quantum Computers, Koei University, https://www.futurelearn.com/courses/intro-to-quantum-computing/0/steps/31577

TIME COMPLEXITY-ANALYSIS

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GNFS Best in Classical

SHOR Quantum challenger

time complexity

simplify e^n n^3

Exponential vs Polynomial time

CONCLUSION

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Quantum Computing will be a huge threat to current RSA cryptosystems. The core idea of shor algorithm is to reduce Factoring to Period-finding. As a result we can not against Shor by adding the key size.

PQRSA-POST QUANTUM RSA(EXTRA)

IDEA1 from pqRSA NIST 1 round submission Instead of using one extremely large RSA key It use a list of small secret primes and small encryption/verification exponent as key. Shor’s algorithm is good for factor large prime. the elliptic-curve method (ECM) methods are even faster than Shor’s method when factors are sufficiently small. Pros:

Using the disadvantage of Shor’s algorithm in factor small number. Cons:

very large key size Quantum computing can do any task that classical computer can do.

IDEA1+ from me build crypto-system recursively and make each iteration of tasks simply. We have to decryptions message dependent last decryption result. Pros:

slow down the quantum computer to breaking cipher. Cons:

encryptions and decryption will take relative long time

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WALK THROUGH EXAMPLE OF SHOR’S ALGORITHM

we have N=91=(7*13) we are trying factor it step 1

choose a random integer m=3 since gcd(N, m)=gcd(91,3)=1, move on

step 2 we generate of a sequence of numbers powers of m=3 3, 9, 27, 81, 243, 729, 2187, 6561,19683…… mod 91 3, 9, 27, 81, 61, 1, 3, 9, 27……. find period p=6

step 3 p=6 is even move on step 4

move on step 5

With the Euclidean algorithm, compute

d is a non-trivial factor of N=91

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