# Presentation: 505517

## Multi-Photon Quantum Cryptography

- George Khoury
- Dirk Bouwmeester
- UC Santa Barbara

## Outline

- Review of classical cryptography
- Multi-photon QC proposal:
- Protocol
- Advantages
- Challenges

## The Players

- Alice

- Bob

- Eve

## Vernam Cipher

- Alice and Bob share secret key *k*
- Unconditionally secure if *k* is secret
- Message		*m*			101010101...
- Key			*k*			001001111...
- Ciphertext	*C* = *m**k*		10**0**01**1010**...
- Decrypt		*m* = *C**k*		101010101...

## Vernam Cipher

- Inconvenient: |k| ≥ |m|
- Really inconvenient: k used only once
*C*1 = *m*1*k*
*C*2 = *m*2*k*
*C*1*C*2 = *m*1*k**m*2*k = m*1*m*2

## Public Key Cryptography

- Bob distributes public key, keeps secret key
- No shared secret necessary
- Maybe computationally secure

- Alice encrypts with public key

- Only Bob’s secret key decrypts

## Quantum Cryptography

- Bits encoded in quantum states
- Security from quantum mechanics
- Eve’s tampering disturbs quantum state
- Can’t prevent, only detect, Eve

## Multi-photon pulses

- Security risk in single-photon protocols
- Hard to avoid

## Multi-photon QC Protocol

## Singlet States

- (*n*h)Bob = (*n*v)Alice
- (*n*h+*n*v)Bob = (*n*v+*n*h)Alice

## Alice and Bob randomly measure in non-orthogonal bases

- Basis

- Result

- Basis

- Result

- L
- C
- C
- C
- L
- C
- L
- L
- C
- L

- L
- C
- L
- L
- L
- C
- L
- C
- C
- L

- 2
- 1
- 1
- 3
- 3
- 1
- 2
- 1
- 1
- 0

- 2
- 1
- 0
- 1
- 3
- 1
- 2
- 2
- 1
- 0

- PBS

- EOM

- PBS

- EOM

*n*ah

- Alice (Bob) records *n*h(*n*v), *n*h+*n*v
- Correlated results if they use the same basis

*n*av

*n*bv

*n*bh

## Key sifting

| Alice’s raw key | 2 | 1 | 1 | 3 | 3 | 1 | 2 | 1 | 1 | 0 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Alice’s Basis | L | C | L | C | L | C | L | L | C | L |
| Bob’s raw key | 2 | 0 | 0 | 1 | 3 | 1 | 2 | 2 | 1 | 0 |
| Bob’s Basis | L | L | C | L | L | L | L | C | C | L |
| Sifted Key | 2 |  |  |  | 3 |  | 2 |  | 1 | 0 |

## Error estimation

- Alice and Bob announce random subset of sifted key
- Ideally, no errors
- QBER = error rate in sifted key
- Assume all errors due to Eve

## Error correction + Privacy Amplification

- Classical error correction eliminates Bob’s errors
- Privacy amplification reduce’s Eve’s information to zero. Possible if

## What about losses?

- Linear losses (independent of intensity) modeled by beamsplitter in front of detectors

## Simple Eavesdropping Strategy

- Assume Eve controls the source
- Eve replaces 	  	with
- Constrained to mimic expected probabilities

## Secrecy Capacity

- For losses < 35%, using two photon pulses increases capacity

## Technical Challenges

- Bright source of entangled photons
- Efficient photon counters
- Low loss transmission links

## Entangled Photon Source

- a

- b

- UV pump

- BBO

- λ/2

## Visible Light Photon Counter

- Developed by Boeing for Fermilab
- High quantum efficiency photon counter

- Courtesy Don Lincoln, Fermilab

## VLPC Operation

- Si

- n+

- As-doped

- VB

- As atoms form impurity band 50 meV below conduction band
- Photon absorbed in Si creates electron-hole pair

## VLPC Operation

- Si

- n+

- As-doped

- VB

- Hole accelerates, ionizes As e- into CB

- These electrons accelerate, ionize As
- As+ charge limits avalanche

## Conclusion

- Single photon pulses are not necessary if you have photon counters
- Multi-photon pulses can increase bit rate
- Efficiency necessary currently unobtainable over long distance

## No information without disturbance

- Eve would like *U* such that:

- But *U* is unitary:

## Entropy, Mutual Information

## RSA details

- Choose two large primes *p*, *q*
- Compute *n* = *pq*, *φ*(*n*)* *= (*p*-1)(*q*-1)
- Choose *e* rel. prime to *φ*
- Compute *d*, s.t. *ed* = 1 (mod *φ*)
- Public key = (*e*, *n*)		Secret key = (*d*,*n*)
*E*(*M*)=*M**e* (mod *n*)
*M*(*E*) = *E**d* (mod *n*)

## Error Correction

- Shannon limit:
- e.g. Hamming code:
- Check bits compute parity of different subsections of message
- “1” means error in that subset
- Intersection of all subsets w/ error is the error location

## Privacy Amplification

- Replace sets of bits with their XOR
- If Eve knows only one of the bits, she has no idea about the XOR

- 1  ? = ?
- 0  ? = ?

## Message Authentication

- Alice and Bob agree on a set of functions F : M → T
- Also share secret key, which chooses an *f* from F
- Alice sends *m*, *t* = *f*(*m*)
- Bob computes *f*(*m*), compares to *t*

## Message Authentication

- Eve knows *f* is in F’ : *f*(*m*1) = *t*1
- F’ has |M| / |T| members
- |M| / |T|2 also satisfy *f*(*m*2) = *t*2
- Eve can only guess which function in F’ is being used, but only 1 / |T| will be correct
- Trick is to find a *small* set F (so that key is small)

## Electronics

- VLPC

- -*V*b

- # disc.

- x4

- counter

- RAM

- clock

- addr

- data

- write