SLIDE 1 Research on Quantum Computational Complexity and Quantum Cryptography at ERATOQCI Project, JST
Hiroshi IMAI ERATO QCI Project, JST
- Dept. Computer Science, Univ. Tokyo
21st Century COE Security Program and RDI, Chuo University 2003 International Symposium on Next Generation Cryptography and Related Mathematics
SLIDE 2 Complexity Theory vs. Cryptography
- Most of current cryptosystems
– Computationally secure based on Computational Complexity Theory
– Quantum Computing/Cryptography
- Quantum states ⇔ information
– Let’s start with various complexity classes
SLIDE 3
Classical Computational Complexity Classes
NEXP NEXP P Polynomial Time P NP PSPACE EXP PSPACE EXP Exponential Time Polynomial Space Nondeterministic Polynomial Time NP Intractable Tractable
SLIDE 4
Probabilistic Complexity
P ZPP RP co-RP NP co-NP BPP PP PSPACE Probabilistic Polynomial Bounded-error Probabilistic Polynomial Randomized Polynomial Zero-error Probabilistic Polynomial NEXP NEXP EXP EXP PSPACE NP P
SLIDE 5 Probabilistic Complexity
NEXP P ZPP RP co-RP NP co-NP BPP PP PSPACE
Before 2002 PRIMES 2002 PRIMES (next talk by
however, Integer Factoring ???
NEXP EXP EXP PSPACE NP P
SLIDE 6
P ZPP RP co-RP NP co-NP BPP PP PSPACE NEXP EXP BQP
Integer factoring in Quantum Polynomial Time (Shor 1994)
Bounded-error Quantum Polynomial time
Quantum Computing
NEXP EXP PSPACE NP P
SLIDE 7 Interactive Proof System
P ZPP RP co-RP NP co-NP BPP MA= AM1 PP AM= AMc≧2= AM2= IP2
Arthur-Merline Game
IP= IPpoly= AMpoly Interactive Proof EXP Mathematical Model of Cryptographic Protocol NEXP NEXP= MIP Multi-prover Interctive Proof EXP PSPACE PSPACE= IP NP P
SLIDE 8
Quantum Complexity
P ZPP RP co-RP NP(= EMA) co-NP BPP BQP MA= AM1 QMA AQMA PrQP = PP EQMA RQMA NQP= co-C=P QIP EXP NEXP= MIP= QMIP Quantum IP Quantum MIP NEXP EXP BQPSPACE= PrQPSPACE= (N)PSPACE= IP PSPACE NP P
SLIDE 9
Quantum Computing/Cryptography
Quantum Computing & I nformation
– exponential speed-up by quantum superposition – information transmission by quantum entanglements
Impacts:
– Quantum Computer destroys I T security (cryptosystem) – P. Shor: Integer Factoring, easy for Quantum Computer collapse of public key cryptosysytem (RSA crypto, etc.) – Quantum Cryptography (possible next-generation crypto.) – secure by quantum principle (physical law) – BB84, B92, etc.
SLIDE 10 Quantum Cryptography
- Aims at Unconditionally Secure cryptoprotocols
– Attempts to overcome the limit of computational secure protocols – Unconditional security by quantum power
- Measurment ⇒ state reduction
- This enales us to detect the existence of eavesdropper
- From Computational Complexity Assumptions
to Physical Principles
SLIDE 11 Existing Research on Quantum Cryptography
- Quantum Key Distribution: BB84, B92, etc.
– unconditionally secure key distribution by quantum law
- Quantum law allows detection of the eavesdropper
– unconditionally secure crypt. (one-time pad)
– Impossibility theorem? (Mayers; Lo and Chau 1997)
– Impossibility theorem??
- Almost no other cryptoprotocols by quantum information
SLIDE 12 eavesdroppter (Eve)
Q communication 1 1
Quantum effect
Correct comm.：
0,1 of ＋basis received ＋rec.
0,1 of ☓ bases Received by ☓ rec No information：
＋,☓ different rec
(with ½ probability) both can’t be used （ uncertainty p.）
s e n d e r ( Alice） r e c e i v e r ( Bob) ＋basis：
Horizontal: 0 Vertical:1 ☓basis： 45
○:: 0
135
○: 1
Verification through classical communication Detection of the eavesdroppter
＋receiver
☓receiver Select one of bases Send 0 or 1 on it
Quantum Crypto Quantum Crypto
Single photon
SLIDE 13 Need for quantum research to develop other protocols
- Digital Signature
- Secrete Sharing
- Authentification
- E-voting, E-money, E-…
- …
- Multi-party Protocol
SLIDE 14 Computationally Secure Multi-party Protocol
Computational secure multi-party protocol
Oblivious Transfer
Coin Flipping
Zero-Knowledge Proof for any Ｎ ＰProblem
Bit Commitment One-way function with trap door
SLIDE 15
Non-Interactive Quantum Statistical and Perfect Zero-Knowledge Proofs
Hirotada Kobayashi
Quantum Computation and Information (QCI) Project ERATO (Exploratory Research for Advanced Technology) JST (Japan Science and Technology Corporation) Concerning this part, cf. quant-ph/0207158 Title: Non-Interactive Quantum Statistical and Perfect Zero- Knowledge Author: Hirotada Kobayashi
SLIDE 16 Interactive Proof Systems
[Babai 1985, Goldwasser, Micali, and Rackoff 1985]
- Two players: prover, verifier
– Prover tries to convince verifier of her assertion. – Verifier must check validity of prover’s assertion. (probabilistically and efficiently) probabilistically ⇒ with bounded error efficiently ⇒ in time polynomial to input length
Peggy (Prover) Victor (Verifier) Interactive Communication
SLIDE 17 Example: Graph Non-Isomorphism
Graph Non-Isomorphism Problem (GNI) INPUT: Two graphs G1, G2 of n vertices QUESTION: For all permutation π ∈ Sn on vertices, π (G1) ≠ G2? ◎ Protocol of verifier V:
- 1. Choose an index i ∈ {1,2} of graphs
and a permutation π ∈ Sn at random. Send a graph π (Gi) to prover P to ask which of the two is isomorphic to π (Gi).
- 2. Receive an index j from P.
Accept iff i = j.
SLIDE 18
1 同型 1 2 4 4 3 3 2 1 4 非同型 2 3 4 3 1 2
SLIDE 19 Summary
- Non-interactive quantum zero-knowledge proofs
– NIQSZK, NIQPZK – Necessity of shared randomness or shared entanglement – NIQPZK of perfect completeness with shared EPR pairs
- Complete problem for NIQPZK(1, b)
– NIQPZK proofs for graph non-automorphism problem – Complete problem for BQP
