Introduction to Practical Cryptography

1
Introduction to Practical Cryptography

• Forward Key Security
• Zero KnowledgeOblivious TransferMulti-Party
  Computation

2
Overview

• Intended as overview of specific areas
• Solutions/instances of some require background
  not covered in this class

3
Agenda

• Forward Key Security
• Zero Knowledge
• Oblivious Transfer
• Multi-party Computation

4
Forward Secure Encryption Schemes

• Attacker learns key at time t
• Should not be able to decrypt anything from prior
  to time t
• Encryption algorithm add an input, time
• Esk(m) becomes Esk(m,t)
• Current key is a function of prior key
• kt F(sk,t) G(kt-1,t)
• Given sk, can compute entire sequence of kts
• c Ekt(m)

5
Digital Signatures

• Alice has a secret key
• Everyone else has the corresponding public key
• Alice can sign message with her secret key
• Given a signature and a message, everyone can
  verify correctness using Alices public key
• Desirable property non-repudiation. If Alice
  signed a contract, she cant deny it later.

6
Digital Signature Schemes

• Three algorithms Key-Gen, Sign, Verify
• Key-Gen inputs security parameter (key size)
  k output keys (PK, SK)
• Sign inputs message M, secret key SK
  output signature S
• Verify inputs M, signature S, public key PK
  output valid/invalid
• Strong security notioneven given signatures on
  messages of its choice,adversary cannot forge
  signatures on new messages

7
Problem with Signatures

• Cant tell when a signature was really generated
• Not even if you include current time into the
  document doesnt say when it was signed
• Therefore,
• signing key disclosed Þ all signatures worthless
• (even if produced before
  disclosure)
• Inconvenience your notarized document is no
  longer valid
• Repudiation Alice gets out of past contracts by
  anonymously leaking her SK
• Key revocation doesnt help it merely informs of
  the leak

8
Fixing the Problem

• Attempt 1 re-sign all the past messages with a
  new key
• Expensive
• What if the signer doesnt cooperate?
• Attempt 2 change keys frequently, erasing past
  SK
• Disclosure of current SK doesnt affect past SK
• However, changing PK is expensive, requires
  certification
• Attempt 3 Employ time stamping authority (third
  party)

9
Forward Security

• Idea change SK but not PK And97
• Divide the lifetime of PK into T time periods
• Each SKj is for a particular time period, erased
  thereafter
• If current SKj is compromised, signatures with
  previous SKj-t remain secure
• Note cant be done without assuming secure
  erasure

10
Definitions Key-Evolving Scheme BM99

• The usual three algorithms Key-Gen, Sign, Verify
• Key-Gen inputs security parameter (key size)
  k total number T of time periods output (PK,
  SK1)
• Sign inputs message M, current secret key SKj
  output signature S (time period j included)
• Verify inputs M, time period j, signature S,
  public key PK output valid/invalid

11
Definitions Key-Evolving Scheme BM99

• The usual three algorithms Key-Gen, Sign, Verify
• Key-Gen inputs security parameter (key size)
  k total number T of time periods output (PK,
  SK1)
• Sign inputs message M, current secret key SKj
  output signature S (time period j included)
• Verify inputs M, time period j, signature S,
  public key PK output valid/invalid
• A new algorithm Update
• Update input current secret key SKj
  output new secret key SKj1

12
Definitions Forward-Security BM99

• Given
• Signatures on messages and time periods of its
  choice
• The ability to break-in in any time period b
  and get SKb
• adversary cant forge a new signature for a time
  period jltb

13
Simple Schemes Efficiency?

• Long Public and Long Private Keys
• T pairs (p1, s1), (p2,s2),. (pt, st)
• PK (p1,p2,.,pt)
• SK (s1,s2,..,st)
• Update erase si for period i
• Drawback public and private key linear in t
  number of periods

14
Anderson Long Secret Key Only

• T pairs as before and an additional pair (p,s)
• Sig(j)SIG( j pj) with key s, j 1,,t
  certificate
• Public key now p (only)
• Secret key (sj, Sig(j)) j1,..,t Still
  linear
• The public key p is like a CA key and a signature
  will include the period, the certificate, the
  message, signature on the message with periods
  secret key

15
Long Signatures Only

• Have (p,s)
• In period j get (pj,sj) and
• Let Cert(j) sig_sj-1 (j pj)
• A signature is the entire certificate chain
  signature, it is of the form (j, sig_sj(m), p1,
  Cert(1),pj, Cert(j))
• Think about it as a tree of degree one and height
  t for t periods
• (signer memory still linear in t)

16
Binary Certification tree BM

• Now (s0,p0)
• Each element certifies two children left and
  right
• We have a tree of height log t (for t periods)
• At each point only a log branch of certificate is
  used in the signature
• Only leaves are used to sign
• Keys whose children are not going to be used in
  future periods are erased.
• We get O(log t) key sizes, signature size

17
Concrete Scheme

• Use a scheme based on Fiat-Shamir variant
• Have the public key be a point x raised to 2t
• Have the initial private key at period zero be x
• Sign based on FS paradigm
• Update by squaring the current key
• The verifier is aware of the period so it knows
  that currently the private key is raised to the
  power 2t-1 (and the identification proofs are
  adjusted accordingly).

18
Replace keys Pseudorandomness

• Have future keys derived from a forward secure
  pseudorandom generator
• Pseudorandom generators which are forward secure
  are easy replace the seed with an iteration of
  the function.
• Possibilities in practice
• Block cipher use previous output as next input
• Stream cipher previous output as next state

19
Agenda

• Forward Key Security
• Zero Knowledge
• Oblivious Transfer
• Multi-party Computation

20
Zero Knowledge
Bob
Alice
I know X
Prove it
X
Some exchange, but which does not provide X

• Interactive method for Alice to prove to Bob that
  she has/knows x without revealing x to Bob.
• Motivation authentication
• Alice wants to prove her identity to Bob via some
  secret but doesn't want Bob to learn anything
  about this secret
• Login methods where password is not stored on
  server (maybe hash of password is stored)
• Alice (user/client) proves to Bob (server) that
  she knows the password without giving Bob the
  password

21
Zero Knowledge

• Interactive method for Alice to prove to Bob that
  she has/knows x without revealing x to Bob.
• Zero-knowledge proof must satisfy three
  properties
• Statement is alice knows x
• Completeness if the statement is true, the
  honest verifier (that is, one following the
  protocol properly) will be convinced of this fact
  by an honest prover.
• Soundness if the statement is false, no cheating
  prover can convince the honest verifier that it
  is true, except with some small probability.
• Zero-knowledge if the statement is true, no
  cheating verifier learns anything other than this
  fact.
• Completeness and soundness needed for any
  interactive proof

22
Zero Knowledge

• Example use enforce honest behavior while
  maintaining privacy
• Force a user to prove that its behavior is
  correct according to the protocol.
• Used in secure multiparty computation

23
Zero Knowledge

• Jean-Jacques Quisquater, et. al "How to Explain
  Zero-Knowledge Protocols to Your Children
• Peggy (prover) has uncovered the secret password
  to open a magic door in a cave. The cave is
  shaped like a circle, with the entrance in one
  side and the magic door blocking the opposite
  side.
• Victor (verifier) will pay Peggy for the secret,
  but not until he's sure that she really knows it.
  Peggy says she'll tell him the secret, but not
  until she receives the money.
• Zero knowledge need a method by which Peggy
  proves to Victor that she knows the word without
  telling it to him
• Solution
• Victor waits outside the cave, Peggy goes in.
  Label the left and right paths from the entrance
  A and B. She takes either A or B at random
  (Victor does not know which)
• Victor enters the cave and shouts the path (A or
  B at random) on which she must return
• Peggy returns along the path chosen by Victor,
  opening the door if it was not the path on which
  she had entered the cave
• If Peggy did not know the word, there is a 50
  chance she can return on the correct path repeat
  the above many times, Peggys chance of
  successfully returning becomes negligible
• (assume chance of Peggy guessing the word is
  negligible)
• If Peggy returns correctly each time, this proves
  she knows the word

door
A
B
enter
24
Agenda

• Forward Key Security
• Zero Knowledge
• Oblivious Transfer
• Multi-party Computation

25
Oblivious Transfer
Bob
Alice
0 or 1
m 1

• Alice transfers a secret bit m to Bob with
  probability ½ such that
• Bob knows whether or not he receives m
• Alice doesnt know if m transferred to Bob

26
Oblivious Transfer Example Method

• Alice oblivious transfer to Bob
• Alice
• picks 2 random primes p,q, set n pq
• encrypt message m using n (such as with RSA)
• c resulting ciphertext
• sends n and c to Bob
• Bob
• picks a ? Zn at random
• sends w a2 mod n to Alice
• Alice
• computes square roots of w S x,-x,y,-y
• picks one of the four at random, s ? S and sends
  to Bob (probability s a or a is ½)
• Bob
• if s ? a, -a Bob can factor n and obtain m (and
  will know he has m)
• Wont walk through why see Rabin cryptosystem
• Bob obtains m with probability ½ and Alice
  doesnt know result

27
Oblivious Transfer Example Method

• Works if Bob selects a at random (doesnt cheat)
• Not known if Bob gains any advantage (can cheat)
  if intentionally selects a specific a

28
Oblivious Transfer Application

• Alice and Bob will each sign a contract only if
  the other also signs it
• Idea
• If names are of equal length, could sign a letter
  at a time, alternating
• But someone must go last and could abort not
  complete last letter
• Sign small fragment at a time bit, pixel
• Dont know who will be last
• If one stops, both are approximately at same
  point
• But one could send garbage in a fragment
• Oblivious transfer solves the problem

29
Oblivious Transfer Application

• Alice and Bob will each sign a contract only if
  the other also signs it
• Idea
• If names are of equal length, could sign a letter
  at a time, alternating
• But someone must go last and could abort not
  complete last letter
• Sign small fragment at a time bit, pixel
• Dont know who will be last
• If one stops, both are approximately at same
  point
• But one could send garbage in a fragment
• Oblivious transfer solves the problem

30
Oblivious Transfer Application

• Alice and Bob each
• create 2 signatures, pick 2 random keys and
  encrypt each signature (such as with a block
  cipher)
• Alice
• LA Alice, this is my signature of the left
  half of the contract
• RA Alice, this is my signature of the right
  half of the contract
• Keys KLA, KRA
• CLA EKLA(LA), CRA EKRA(RA)
• Bob
• LB Bob, this is my signature of the left half
  of the contract
• RB Bob, this is my signature of the right half
  of the contract
• Keys KLB, KRB
• CLB EKLB(LB), CRB EKRB(RB)
• Contract is considered signed only if Alice and
  Bob each have both halves of others signature

31
Oblivious Transfer Application

• Alice sends one of KLA, KRA to Bob using
  oblivious transfer
• Bob sends one of KLB, KRB to Alice using
  oblivious transfer
• Alice and Bob exchange bits of both keys, one bit
  at a time from each key, in order, until all bits
  are sent.
• If Alice sees a mistake in key bits received,
  Alice aborts
• Likewise if Bob sees a mistake
• Bob does not know if Alice has KBL or KBR, so he
  cannot risk sending an incorrect value for either
  key
• Likewise for Alice
• Must exchange bits simultaneously otherwise,
  last one could flip last bit

32
Agenda

• Forward Key Security
• Zero Knowledge
• Oblivious Transfer
• Multi-party Computation

Some slides are modified from a presentation by
Juan Garay, Bell Labs
33
Secure MPC

• A set of parties with private inputs wish to
  compute some joint function of their inputs.
• Parties wish to preserve some security
  properties. E.g., privacy and correctness.
• Example secure election protocol
• Security must be preserved in the face of
  adversarial behavior by some of the participants

34
Secure MPC

• Multi-party computation (MPC) Goldreich-Micali-Wi
  gderson 87
• n parties P1, P2, , Pn each Pi holds a
  private input xi
• One public function f (x1,x2,,xn)
• All want to learn y f (x1,x2,,xn)
  (Correctness)
• Nobody wants to disclose his private input
  (Privacy)
• 2-party computation (2PC) Yao 82, Yao 86 n2

Studied for a long time. Focus has been security.
35
Secure MPC
s3
Alice
s4
s2
s5
sn
s1
s
Alice has a secret key, s, (such as a key to a
system) Afraid she may lose s Want to give is to
someone, but dont trust anyone with entire
secret Share secret among n people s s1? s2 ?
s3 ? s4 ? s5 ? sn
36
Secure MPC
s3
X
Alice
s4
s2
s5
sn
s1
s

• Suppose need more flexibility
• ith person loses si or is malicious
• n people
• Any subset of size t can recover s
• Any subset of size lt t cannot recover any
  information about s

37
Secure MPC

• Use polynomials
• Finite field F
• (such as some Zp for prime p)
• F(x) a0x0 a1x1 a2x2 atxt
• Coefficients a0, a1, a2 at ? F
• t1 terms (degree is t)

38
Secure MPC

• Polynomials properties
• Interpolation given t1 distinct points,
  (x1,y1), (x2,y2) (xt1,yt1) , can find a0, a1,
  a2 , at
• Secrecy if have only t (or fewer) distinct
  points, cant determine anything about a0

39
Secure MPC

• Coefficients
• Pick a1, a2 .. at at random
• set a0 s
• Each person is associated with a distinct point
• Person i assigned distinct xi
• Set si f(xi) (xi,si) is point for person i

40
Secure MPC

• Polynomial construction requires honesty
• Entity choosing ais, xis is dishonest
• Collusion person 2 may give (x2,s2) to person
  10
• Verifiable secret sharing
• Each person can verify piece received is a proper
  piece
• Allows for O(log n) colluders
• Based on factoring Chor, Goldwasser, Micali,
  Awerbuch

41
Instances of 2PC

• Authentication
• Parties 1 server, 1 client.
• Function if (server.passwd client.passwd),
  then return succeed, else return fail.
• On-line Bidding
• Parties 1 seller, 1 buyer.
• Function if (seller.price lt buyer.price), then
  return (seller.price buyer.price)/2, else
  return no transaction.
• Intuition In NYSE, the trading price is between
  the ask (selling) price and bid (buying) price.

42
Instances of MPC

• Auctions
• Parties 1 auctioneer, (n-1) bidders.
• Function Many possibilities (e.g., Vickrey).
• Consider a secure auction (with secret bids)
• An adversary may wish to learn the bids of all
  parties to prevent this, requires privacy
• An adversary may wish to win with a lower bid
  than the highest to prevent this, requires
  correctness

sealed-bid, bidders submit written bids without
knowing the bid of the others. Highest bidder
wins, but pays second-highest bid. Intent
bidders bid true value.
43
MPC Protocols

• Consider 2PC (MPC is similar). Two parties P0
  and P1.
• Encode function as a Boolean circuit of ANDs and
  XORs
• Bits on each wire are shared using XOR
• Per-gate evaluation
• (Shared) Inputs x x0 ? x1 , y y0
  ? y1
• XOR No interaction needed
• ( x ? y ) ( x0 ? y0 )
  ? ( x1 ? y1 )
• AND More complex (needs interaction)
• Each party reveals his shares

44
MPC Elections

• m voters v1, v2, vm
• ith voter inputs xi
• Result function, f, of xis
• Required properties
• Only authorized voters can vote
• Each can only vote once
• Each vote is secret
• No vote can be duplicated by another voter
• Vote tally correctly computed
• Anyone can check the tally is correct
• Fault tolerate protocol works if some number of
  bad parties
• Cannot coerce a voter into revealing how he/she
  voted (no vote-buying)
• Meeting all requirements tricky, especially last
• In real life system, logistical problems main
  issues as opposed to protocols

45
MPC Digital Cash

• Required properties
• Prevent forgery
• Prevent or detect duplication
• Preserve customers anonymity
• Practical
• operationally feasible (ex. no large single
  database of all issued digital cash)

46
Digital Cash

• 3 protocols
• Withdrawal user can obtain a digital coin
• Payment user buys goods from vendor using
  digital coin
• Deposit vendor gives digital coin to bank to be
  credited to the vendors account
• Notation
• U user
• B Bank
• V vendor
• D digital coin of 100
• SKBx signature of B on x

47
Digital Cash - Withdrawal

• U notifies B wants to withdraw D
• B gives D to U
• D SKBI am a 100 bill, 4527
• U checks signature
• accepts D if it is valid
• else rejects D
• Bank deducts D from Us account if U accepts D

48
Digital Cash - Payment

• U pays V with D
• V checks signature
• accepts D if it is valid
• else rejects D

49
Digital Cash - Deposit

• V gives D to B
• B checks signature
• accepts D if it is valid and credits Vs account
• else rejects D

50
Digital Cash Do Properties Hold?

• Prevent forgery
• ok, infeasible under basic assumptions of
  signature schemes
• Prevent or detect duplication
• Very easy to duplicate coins, double spend
• Preserve customers anonymity
• No anonymity know U and where D was spent

51
Digital Cash Fix Properties

• Blind signature
• U presents D to B
• B signs D without seeing its contents (i.e. cant
  associate with U)
• Analogy U covers check with carbon paper and
  seals both inside an envelope. Bank signs outside
  of envelop
• But how does B know D is not fake?

52
Digital Cash Fix Properties

• RSA Blind signature
• Have key pair (e,n) (d,n)
• U picks random r mod n
• U computes D D re mod n
• presents D to B
• B signs D s (D)d mod n ( Md (re)d Mdr
  mod n )
• B deducts D from Us account
• U computes signature on M s Md mod n by
  dividing s by r
• Now anonymity no link between U and D
• But
• B can be tricked into signing fake D
• D can be duplicated and double spent

53
Digital Cash Fix Properties

• One denomination or one public key per
  denomination, feasible not many denominations
• Probability method
• U makes up 100 Ds
• Blind signature on all 100 and gives all to B
• B signs one, requires U to unblind the rest
  (reveal rs)
• U has 1/100 chance of successfully cheating
• U spends 1 remaining D, anonymous

54
Digital Cash Fix Properties

• Now have anonymity and no (or small) chance of
  cheating
• How to prevent double spending?
• B has database of all D, records Ds as they are
  returned spent from V
• not too practical large database, V must wait
  for B to ok each D at time of purchase

55
Digital Cash Fix Properties

• What if just detect double spending?
• Random identity string (RIS)
• Different for every payment of D
• Only U can create a valid RIS
• Two different RISs on same D allows B to
  retrieve Us name
• If B receives two identical Ds with different
  RIS values, U cheated
• If B receives two different Ds with same RIS
  values, V cheated

56
Digital Cash Fix Properties

• H hash
• U creates 100 Ds
• Di (Im a 100 bill, 4527i,yi,1yi,1,yi,2,
  yi,2, yi,k,yi,k)
• where
• yi,j H(xi,j) yi,j H(xi,j)
• xi,j and xi,j are randomly chosen but such that
  xi,j ? xi,j Us name ? i,j

57
Digital Cash Fix Properties

• Withdrawal
• U blinds each Di, get Di
• B has U unblind all but one Di and reveals
  appropriate xi,j and xi,j
• B checks for each
• yi,j H(xi,j) yi,j H(xi,j)
• xi,j ? xi,j Us name
• B signs remaining blind Di and gives to U

58
Digital Cash Fix Properties

• Payment
• U gives Di to V
• V checks Bs signature on Di
• V creates a challenge random bit string b1,b2,
  bk
• If bj 0, U reveals xi,j , else reveals xi,j
• V checks yi,j H(xi,j) or yi,j H(xi,j) and
  accepts if yes, else rejects
• Probability in a different payment that same RIS
  is produced is 2-k because V creates challenge at
  random
• Only U can produce valid RIS H is
  computationally infeasible to invert
• Two different RIS values on same DI leaks Us
  name have xi,j and xi,j for some j (due two
  different challenges)

59
Digital Cash Fix Properties

• Deposit
• V gives Di, s, RIS to B
• B verifies signature, checks if already returned
• If already in database, B compares RIS values
• If different, U double spent
• If equal, V trying to deposit twice

60
Other MPC Applications

• Database Query
• Alice has a string q Bob has a database of
  strings
• Alice wants to know whether there exists a string
  ti in Bob's database that matches q (exact or
  close)
• Privacy
• Bob cannot know Alice's q or the response
• Alice cannot know the database contents except
  for what can be known from the query result

61
Other MPC Applications

• Profile Matching
• Alice has a database of known hacker's behaviors
• Bob has a hacker's behavior from a recent
  break-in
• How can Bob check if his hacker is in Alices
  database while
• Not disclosing the hacker's actual behavior to
  Alice doing so that might disclose the
  vulnerability in his system.
• Not obtaining contents of Alices database it
  contains confidential information.

62
Other MPC Applications

• Companies want to cooperate in preventing
  intrusions into their networks.
• Need to share data patterns, but this is
  sensitive information
• Real data for IDS/security research
• None exists
• Outdated/now irrelevant MIT Lincoln Laboratory
  IDS Evaluation Data Set 1998-2000

63
Other MPC Applications

• In general, many database applications
• Simple queries
• Determining intersection
• Sharing of patterns without revealing actual
  content

64
Backup
65
On-line Bidding Definition of Security

• Correctness
• seller.output buyer.output f (seller.price,
  buyer.price)
• Privacy The transcript carries no additional
  information about seller.price and buyer.price.

66
Privacy is a little tricky

• On-line Bidding Function
• if (seller.price lt buyer.price),
• then return (seller.price buyer.price)/2,
• else return no transaction.
• If seller.price buyer.price, then both parties
  can learn each others private input.
• If seller.price gt buyer.price, then both parties
  should learn nothing more than this fact.
• Privacy Each party should only learn whatever
  can be inferred from the output (which can be a
  lot sometimes).

67
Fair Secure Multi-Party Computation (FMPC)
Parties P1, P2, , Pn (some corrupted), each
holding private input xi, wish to compute y
f(x1, x2,, xn) privately and correctly.
Security is about absolute information gain. At
the end of the protocol, each party learns y (and
anything inferable from y).
Fairness is about relative information gain. At
the end of the protocol, either all parties learn
y, or no party learns anything.
Important in MPC crucial in some
applications (e.g., two-party contract signing).

68
Security vs. Fairness

• The problem of secure MPC/2PC is well-studied and
  well-understood.
• The problem of fair MPC/2PC less developed
• Security and fairness different concepts
• Fair without being secure

69
Security ? Fairness
On-line Bidding Function if (seller.price lt
buyer.price), then return (seller.price
buyer.price)/2 else return no transaction.
E.g., in an unfair on-line bidding protocol, the
seller may learn the output (and thus
buyer.price) before the buyer learns
anything.
70
Cheating with Unfair Protocols

• A cheating seller
• Initiate protocol with price x (originally
  999,999).
• Run until getting the output (buyer hasnt got
  the output yet).
• if (output no transaction), then abort
  (e.g., announce network failure), set x ? x-1,
  and repeat.

• A cheating seller can
• find out the buyers price (destroys privacy) and
• achieve maximum profit (destroys correctness)
• (the actual function computed is return
  buyer.price)

? The lack of fairness completely voids the
security!
71
Fairness Positive Results

• n parties, t corrupted
• t ? n/3 possible with p2p channels
• computational GMW87
• information-theoretic BGW88, CCD88
• n/3 ? t ? n/2 possible with broadcast channel
• computational GMW87
• information-theoretic RB89

72
Unfortunately

• Fairness is impossible with corrupted majority (t
  ? n/2)

Intuition (2 parties) Party sending the last
message may abort early.

• Consequently, many security definitions do
  not consider fairness,
• or only consider partial fairness BG90,
  BL91, FGHHS02, GL02.

73
Fairness After the Impossibility Result
We still need (some form of) fairness, so tweak
model/definition

• Optimistic approach (tweak the model) M97,
  ASW98, CC00,
• Adds a trusted party as an arbiter in case of
  dispute.
• Needs to be (constantly) available.

• Gradual Release approach (tweak the definition)
  Blum83, D95,
• BN00,
• No trusted party needed.
• Parties take turns releasing info
  little-by-little.
• Still somewhat unfair, but we can quantify and
  control the
• amount of unfairness.

74
The Gradual Release Approach

• Reasonably studied
• Initial idea by Blum 83
• Subsequent work ,Damgard 95, Boneh-Naor 00,
  Garay-Pomerance 03, Pinkas 03,

• Not quite well-understood
• Ad hoc security notions
• Limited general constructions (only 2PC)
• Few practical constructions

75
Security and Fairness

• A typical gradual release protocol (e.g., BN00,
  GP03, P03) consists of two phases
• Computation phase Normal computation.
• Revealing phase Each Pi gradually reveals a
  secret si then each Pi computes
  the result y from s1, s2,, sn.

76
Observation on Existing MPC Protocols
Many (unfair) MPC protocols (e.g., GMW87,
CDN01, CLOS02) share the same structure Sharing
phase Parties share data among themselves
(simple sharing, or (n, t) threshold
sharing) Evaluation phase Gate-by-gate
evaluation (all intermediate data are
shared or blinded) Revealing phase Each
party reveal its secret share (all
parties learn the result from the shares)

77
FCPFO Commit-Prove-Fair-Open

• Commit phase Every party Pi commits to a value
  xi.
• Prove phase Every party Pi proves a relation
  about xi.
• Open phase Open x1, x2,, xn simultaneously.

Simultaneous opening guarantees fairness
either all parties learn all the committed
values, or nobody learns anything.
Using FCPFO, the revealing phase becomes fair,
and so does the MPC protocol.


78
Time-lines Towards realizing FCPFO

accelerator 1
accelerator 2
accelerator k

• A time-line An array of numbers (head, ,
  tail).
• Time-line commitments
• TL-Commit(x) (head, tail x)
• Perfect binding.
• Hiding (2k steps to compute tail from head).
• Gradual opening Each accelerator cuts the
  number of steps by half.

79
A time-line, mathematically BN00,GJ02,GP03

accelerator 1
accelerator 2

g
g22k
g22k-1
g2(2k-12k-2)

• N safe Blum modulus,
• N pq, where p, q, (p-1)/2, (q-1)/2 are
  all primes.
• g a random element in ZN.
• head g, tail g22k

80
A time-line, mathematically (contd)

accelerator 1
accelerator 2

g
g22k
g22k-1
g2(2k-12k-2)
g22i
Gi

• Can move forward m positions by doing m
  squarings
• Knowing ?(N), one can compute Gi efficiently,
  for any i.
• Hard to move backward, not knowing factorization
  of N
• inefficient to move forward (step-by-step) ?
  point far away is unknown

81
Fair exchange using time-lines

• START Alice has a, Bob has b.
• COMMIT
• Alice sends TL-Commit(a) to Bob,
• Bob sends TL-Commit(b) to Alice.
• OPEN Take turns to gradually open the
  commitments.

Alice
Bob
82
Fair exchange using time-lines (contd)
Alice
Bob

• ABORT If Bob aborts and force-opens in t steps,
  Alice can do it as well in 2t steps.

83
Realizing FCPFO using time-lines

• Setup A master time-line T ?N g Gj,
  j1,,k? in CRS.
• Commit Each party Pi
• Derives a time-line Ti ?N gi Gij?
• TL-commits to xi (gi Gik xi),
• Prove Standard ZK proof.
• Open In round m, each party Pi reveals Gim
  with ZK proof
• if any party aborts, enter panic mode.
  
• Panic mode Depends on current round
  m
• If (k-m) is large, then abort.
• (A does not have enough time to force-open.)
• If (k-m) is small, then force-open.
• (A has enough time to force-open as well.)

84
Putting things together
Plug FCPFO into existing MPC protocols ? Fair
MPC protocols