Chap 12

1

Key Establishment Protocols

September 4, 2001

Yongdae Kim

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Contents

v Classification and framework

v Key transport based on symmetric encryption

v Key agreement based on symmetric techniques

v Key transport based on public-key encryption

v Key agreement based on asymmetric techniques

v Secret sharing

v Conference keying

v Analysis of key establishment protocols

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Classification and concepts

v Key establishment: a shared secret becomes available to two or more parties, for subsequent cryptographic use.n key transport protocolw one party creates, and securely transfers it to the other(s).

n key agreement protocol: key establishment technique in whichw a shared secret is derived by two (or more) partiesw as a function of information contributed by each of these

w (ideally) such that no party can predetermine the resulting value

n Key pre-distributionw resulting established keys are completely determined a priori by initial

keying material

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Classification and concepts (cnt.)

v Use of trusted serversn key establishment protocols involve a centralized or trusted party,

for either or both initial system setup and on-line actionsn trusted third party, trusted server, authentication server, key

distribution center (KDC), key translation center (KTC), and certification authority (CA).

v secure key establishment n each party in a key establishment protocol be able to determine the

true identity of the other(s) which could possibly gain access to the resulting key, implying preclusion of any unauthorized additional parties from deducing the same key

n secrecy of key, and identification of those parties with access to it

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Classification and concepts (cnt.)

evidence an identified party possesses a given keyexplicit key

authentication

evidence that a key is possessed by some partykey confirmation

identity of party which may possibly share a key(implicit) key

authentication

identity of the source of datadata origin

authentication

identity of a party, and aliveness at a given instantentity

authentication

depends on context of usageauthentication

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Classification and concepts (still)

v (Implicit) Key authenticationn one party is assured that no other party aside from a specifically

identified second party may gain access to a particular secret keyn independent of the actual possession of such key by the second

party, or knowledge of such actual possession by the first party

v Key confirmationn one party is assured that a second (possibly unidentified) party

actually has possession of a particular secret key

v Explicit key authenticationn both (implicit) key authentication and key confirmation holdn Possession of key: (keyed) one-way hash, encryption, ZK

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Classification and concepts (god…)

v authenticated key establishmentn key establishment + key authentication

v identity-basedn identity information of the party involved is used as public key

v message-independentn messages sent by each party are independent of any per-session

time-variant data (dynamic data) received from other parties

n Message-independent protocols include non-interactive protocols (zero-pass and one-pass protocols)

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Motivation for use of session key

v Defn ephemeral secret, i.e., one whose use is restricted to short time

period after which all trace of it is eliminated

v Motivationn to limit available ciphertext for cryptanalytic attackn to limit exposure, with respect to both time period and quantity of

data, in the event of (session) key compromise

n to avoid long-term storage of a large number of distinct secret keys (in the case where one terminal communicates with a large numberof others), by creating keys only when actually required;

n to create independence across communications sessions or applications

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Key Establishment characteristics

v nature of the authentication: Any combination of entity authentication, key authentication, and key confirmation.

v reciprocity of authentication: unilateral or mutual authenticationv key freshnessv key control: key distribution vs. key agreementv efficiency

n number of message exchanges (passes) required between partiesn bandwidth required by messages (total number of bits transmitted)n complexity of computations by each party (as it affects execution time)n possibility of precomputation to reduce on-line computational complexity.

v third party requirementsn requirement of an on-line (real-time), off-line, or no third partyn degree of trust required in a third party

v type of certificate usedv non-repudiation: type of receipt keying material has been exchanged

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Assumptions and Adversaries

v Attacksn passive attack: adversary simply records data and analyzen active attack: adversary modifies or injects messages

v What are the attacker’s roles?n deduce a session key using information gained by eavesdropping;n participate covertly in protocol initiated by one party, and influence

it by altering messages so as to be able to deduce the keyn initiate one or more protocol executions, and combine messages

from one with another, so as to carry out one of the above attacksn without being able to deduce the session key, deceive a legitimate

party regarding the identity of the party with which it shares a keyn In entity authentication, adversary’s objective is to arrange that one

party receives messages which satisfy that party that the protocol has been run successfully with a party other than the adversary.

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PFS and Known Key Attacks

v perfect forward secrecyn compromising long-term key do not compromise past session keysn Idea of PFS is that previous traffic is locked securely in the pastn May be provided by generating session keys by DH key

agreement, wherein DH exponentials are based on short-term keys

n If long-term secrets are compromised, future session can be impersonated

v known-key attackn compromise of past session keys allows either a passive adversary

to compromise future session keys, or impersonation by an activeadversary in the future.

n in some environments, the probability of compromise of session keys may be greater than that of long-term keys.

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Contents






v Secret sharing

v Conference keying


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Key Transport(Symmetric Key Encryption)

messagestimestampServer

3noKTCProtocol 13.12

4noKDCOtway-Rees

5noKDCNeedham-Schroeder shared-key

4yesKDCKerberos

3nononeShamir’s no-key protocol

1-3optionalnonepoint-to-point key update

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Point-to-Point Key Update

v Key Transport with one passn A → B: EK(rA)n Implicit key authenticationn Additional fieldw timestamp, sequence number: freshnessw redundancy: explicit key authentication, message modification attackw target identifier: prevent undetectable message replay

n Hence A → B: EK(rA, tA, B)n Mutual authentication: A → B: EK(rB, tB, A): K = f(rA, rB)

v Key Transport with challenge-responsen B → A: nB : for freshnessn A → B: EK(rA, nA, nB, B)n B → A: EK(rB, nB, nA, A)n Cannot provide PFS

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Point-to-Point Key Update

v Authenticated Key Exchange Protocol 2 (AKEP2)n A → B: rA

n B → A: (B, A, rA, rB), hK(B, A, rA, rB)n A → B: (A, rB), hK(A, rB)

n W = h’K’(rB)

v AKEP1n B → A: (B, A, rA, rB, (r, W ⊕ h’K’(r)), hK(B, A, rA, rB, (r, W ⊕ h’K’(r))n Optimization: r = rB

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Shamir’s no key algorithm

v Protocoln A → B: KA mod pn B → A: (KA)B mod pn A → B: (KAB) A-1

mod p

v Propertyn Provide key transportn No a priori information is requiredn Not necessarily modular exponentiation, but not one-time pad

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Kerberos

v Basicn A, B, a trusted server share long-term pairwise secret keys a priorin Server either plays the role of KDC and itself supplies the session

key, or serves as a key translation center (KTC)n A and B share no secret, while T shares a secret with each

n Goal: for B to verify A’s identity, establishment of a shared key

v Descriptionn A requests from T credentials to allow it to authenticate itself to Bn T plays the role of a KDC, returning to A a session key encrypted

for A and a ticket encrypted for B

n The ticket contains the session key and A’s identityw authentication of A to B when accompanied by appropriate message

created by A containing a timestamp encrypted under that session key

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Kerberos (cnt.)

v Protocoln A → T: A, B, NA NA: freshnessn T → A: EKBT(k, A, L), EKBT(k, NA, L, B): L: lifetime

n A → B: EKBT(k, A, L), Ek(A, TA, Asubkey)

n B → A: Ek(TA, Bsubkey) Optional mutual authentication: (4)

v Propertiesn Since timestamps are used, the hosts on which this protocol runs must

provide both secure and synchronized clocks

n If initial shared keys are password-derived, protocol is no more secure than secrecy of such password or their resistance to password-guessing attack

n Asubkey and Bsubkey allow transfer of a key from A to Bn Lifetime is intended to allow A to re-use the ticket

w A creates new authenticator with new timestamp and same session key k

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Needham-Schroeder

v important primarily for historical reasonsv Protocol

n A → T: A, B, NA

n T → A: EKAT(NA, B, k, EKBT(k, A))n A → B: EKBT(k, A)

n B → A: Ek(NB)

n A → B: Ek(NB-1)

v Propertiesn The protocol provides A and B with a shared key k with key authentication n (4) and (5) provide entity authentication of A to B. B to A can be obtained

using redundancy check on NB upon decrypting message (4).

n If acceptable for A to re-use key k with B, A may securely cache (3) with kw To prevent replay of (4), Ek(NA’) should be appended to message (3), and (4)

should be replaced by Ek(NA’�1, NB) allowing A to verify B’s knowledge of k

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Needham-Schroeder vs. Kerberos

v Kerberos lifetime parameter is not present

v (3) (corresponds to Kerberos ticket) is double-encrypted

v authentication here employs nonce rather than timestamp

v since B has no way of knowing if k is fresh, should k ever be compromised, any party knowing it may both resend message (3) and compute a correct message (5) to impersonate A to Bn This situation is ameliorated in Kerberos by the lifetime parameter

which limits exposure to a fixed time interval.

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Otway-Rees protocol

v Protocoln A → B: M, A, B, EKAT(M, A, B, NA) M: Another noncen B → T: M, A, B, EKAT(M, A, B, NA), EKBT(M, A, B, NB)n T → B: EKAT(k, NA), EKBT(k, NB)

n B → A: EKAT(k, NA)

v Propertiesn Only 4 roundsn NA could be eliminated in (1), (2), and replaced by M in (3), (4)

n Could provide key confirmation and entity authentication (5 round)w B → A: EKAT(k, NA), Ek(NA, NB)

w A → B: Ek(NB)

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Contents






v Secret sharing

v Conference keying


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Key Agreement(Symmetric key encryption)

v KDS is said to be j-secure if coalition of j or fewer users can do no better at computing the key shared by two than a party which guesses key without any pieces whatsoever

v Blom KDS bound: In any j-secure KDS(m-bit session key), secret data by each user must be at least m(j + 1) bits

v Blom’s schemen engineered to provide unconditional security against coalitions of a

specified maximum sizen initial keying material assigned to each user (row of S, correspond

to k keys) allows computation of larger number of derived keys (a row of K, providing n keys), one per each other user

n Storage savings results from choosing k less than n

n derived keys of different user pairs are not statistically independent

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v Blom’s schemen Summary: each user is given initial secret keying material and public datan Result: each pair of users Ui, Uj computes m-bit pairwise secret key Kij

1. k X n public generator matrix G of an (n, k) MDS code over Fq of order q

2. trusted party T creates a random secret k X k symmetric matrix D over Fq

3. T gives to Ui secret key Si, defined as row i of the n X k matrix S = (DG)T

n Si: k-tuple over Fq of k lg(q) bits, allowing Ui to compute entry in row i of (DG)TG

4. Ui and Uj compute common secret Kij = Kji of bitlength m = lg(q) as follows n Using Si and column j of G, Ui computes the (i, j) entry of the K = (DG)TG.n Using Sj and column i of G, Uj similarly computes the (j, i) entry (K: symmetric)

v Do not explain in detail

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v Blom’s schemen Summary: each user is given initial secret keying material and public datan Result: each pair of users Ui, Uj computes m-bit pairwise secret key Kij

1. k X n public generator matrix G of an (n, k) MDS code over Fq of order q

2. trusted party T creates a random secret k X k symmetric matrix D over Fq

3. T gives to Ui secret key Si, defined as row i of the n X k matrix S = (DG)T

n Si: k-tuple over Fq of k lg(q) bits, allowing Ui to compute entry in row i of (DG)TG

4. Ui and Uj compute common secret Kij = Kji of bitlength m = lg(q) as follows n Using Si and column j of G, Ui computes the (i, j) entry of the K = (DG)TG.n Using Sj and column i of G, Uj similarly computes the (j, i) entry (K: symmetric)

v Do not explain in detail

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Contents






v Secret sharing

v Conference keying


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Key Transport based on PKC

#msgentity authen.Sign required

2unilateralyesBeller-Yacobi (2-pass)

4mutualyesBeller-Yacobi (4-pass)

3mutualyesX.509 (3-pass) – random #’s

2mutualyesX.509 (2-pass) – timestamps

1data origin onlyyessigning encrypted keys

1data origin onlyyesseparate signing, encrypting

1data origin onlyyesencrypting signed keys

3mutualnoNeedham-Schroeder PK

1nonobasic PK encryption (1-pass)

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Key Transport using PKC without signature

v Needham-Schroedern Algorithmw A → B: PB(k1, A)

w B → A: PA(k2, B)

w A → B: PB(k2)

n Propertiesw Mutual authentication, mutual key transport

v Modified NSn Algorithmw A → B: PB(k1, A, r1)w B → A: PA(k2, r1, r2)

w A → B: r2

n Removing third encryption

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Combining PK encryption and signature

v Encrypting signed keysn A → B: PB(k, tA, SA(B, k, tA))n Problem: Data for encryption is too large

v Encrypting and signing separatelyn A → B: PB(k, tA), SA(B, k, tA)

n Acceptable only if no information regarding plaintext data can be deduced from the signature

v Encrypting signed keysn A → B: tA, PB(A, k), SA(B, tA, PB(A, k))n Prevent the above problem

n Can provide mutual authentiation

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Combining PK and signature (cnt.)

v Assurances of X.509 strong authenticationn identity of A, and that the token received by B was constructed by An the token received by B was specifically intended for B;n the token received by B has “freshness”

n the mutual secrecy of the transferred key.

v X.509 strong authenticationn DA=(tA, rA, B, data1, PB(k1)), DB=(tB, rB, A, rA, data2, PA(k2)), n A → B: certA, DA, SA(DA)

n B → A: certB, DB, SB(DB)

v Commentsn Since protocol does not specify inclusion of an identifier within the

scope of the encryption PB within DA, one cannot guarantee that the signing party actually knows (or was the source of) plaintext key

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Hybrid Key Transport using PKE

v Beller-Yacobi (4 pass)n Propertiesw mutual authentication, explicit key authenticationw for applications where there is imbalance in processing powerw identity of the weaker remains concealed from eavesdroppers

n Algorithmw B → A : certB = (IB, nB, GB) : certificate generated with RSA w A → B : PB(K) =K3 mod nB

w B → A : EK(m, {0}t) : Encryption with symmetric key encryptionw A → B : EK((v, w), certA) : DSA signature with precomputation

n Commentw To achieve mutual authentication, each party carry out at least one

private-key operation, and one or two public-key operationsw careful selection of two separate public-key schemesw RSA public operation and ElGamal private-key operation are cheap

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Hybrid Key Transport using PKE (cnt.)

v Beller-Yacobi (2 pass)n Algorithmw precompute x, v = gx mod nS select random challenge m

w verify certB via PT(GB) ← send m, certBw compute (v, w) =SA(m, IB) certB = (IB, nB, GB)

w send PB(v), Ev(certA, w) → recover v, set K = vw certA = (IA, uA, GA) verify certA, signature (v, w)

n Properties: slightly weaker authentication assurancesw B obtains entity authentication of A and obtains a key K that A alone

knows, while A has key authentication with respect to Bw For A to obtain explicit key authentication of B, a third message may be

added whereby B exhibits knowledge through use of K on a challenge or standard message (e.g., {0}t )

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Contents






v Secret sharing

v Conference keying


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Key Agreement (Asymmetric technique)

3mutualmutual-implicitSTS

2nonemutual-implicitGunther

2nonemutual-implicitMTI/A0

1noneunilateralElGamal key agreement

2nonenoneDiffie-Hellman

#msgentity authenticationkey authentication

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Diffie-Hellman and ElGamal

v Diffie-Hellmann Setup: prime p, generator g of Zp*n A → B : gx mod pn B → A : gy mod p

n Propertiesw fixed exponent: zero-pass key agreement with special certificates

w Zp*, F2m

w Signature is required

v ElGamaln A → B : gx mod pn no entity authentication or key confirmation

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MTI/A0

v Protocoln A → B : gx mod pn B → A : gy mod pn A: k = (gy)aPKb

x = gya gbx = gya+bx

n B: k = (gx)bPKay

n source-substitution attack: C is not actually able to compute k itself,but rather causes B to have false beliefs.w C registers A’s public key as its ownw When A sends B, C replaces A’s certificate with its own

w C forwards B’s response gy to A

w B concludes that subsequently received messages encrypted by k =gbx+ay originated from C, it is only A who knows k and can originate such messages

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STS

v Algorithmn A → B : gx mod pn B → A : gy mod p, Ek(SB(gy, gx))n A → B : Ek(SA(gx, gy))

v Propertiesn Encryption under key k provides mutual key confirmation plus

allows the conclusion that the party knowing the key is that which signed the exponentials.

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Gunther’s implicitly-certified ID-based PK

v Algorithmn SUMMARY: TTP creates an implicitly-certified, publicly-recoverable DH

PK for A, and transfers to A the corresponding private key.

1. TTP selects p and g of Zp*, a random integer t, gcd(t, p �1) = 1 as its private key, and publishes its public key u = gt mod p

2. TTP assigns to each A DN IA and a random integer kA with (kA, p�1) = 1, then computes PA = gkA mod pn PA is A’s reconstruction public, allowing other parties to compute PA

a below.

n The gcd condition ensures that PA itself is a generator

3. T solves the following equation for an h(IA) = t PA + kA a (mod p � 1)

4. T securely transmits to A the pair (r, s) = (PA, a) (ElGamal signature on IA)

5. Any other party can then reconstruct A’s public key PAa(=gkA a ) by

computing PAa = gh(IA) u�PA mod p

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DH with Implicitly-certified keys

v Algorithmn A → B : IA, PA

n B → A : IB, PB, (PA)y mod pn A → B : (PB)x mod p

v Propertiesn Subject to known key attacks

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Contents






v Secret sharing

v Conference keying


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Secret Sharing

v Motivationn To safeguard cryptographic keys from loss, desirable to create backupn The greater number of copies made, the greater risk of security exposure;

the smaller the number, the greater the risk that all are lost

n address this issue by allowing enhanced reliability without increased riskn facilitate distributed trust or shared control for critical activities by gating the

critical action on cooperation by t of n users.

v Basic idean to start with a secret, and divide it into pieces called shares which are

distributed amongst users such that the pooled shares of specific subsets of users allow reconstruction of the original secret

n may be viewed as a key pre-distribution technique, facilitating one-time key establishment, wherein the recovered key is pre-determined

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Secret Sharing (cnt.)

v Trivial (n, n) schemen S = Σ Si

n Shouldn’t split r bit key into r/t pieces

v Threshold schemesn Def: A (t, n) threshold scheme (t ≤ n) is a method by whichw a trusted party computes secret shares Si, 1 ≤ i ≤ n from an initial secret

S and securely distributes Si to user Pi such that the following is true: w any t or more users who pool their shares may easily recover S

w but any group knowing only t � 1 or fewer shares may not

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v Shamir’s threshold schemen based on polynomial interpolation, and that a uni-variate polynomial

y = f(x) of degree t � 1 is uniquely defined by t points (xi, yi)n since these define t linearly independent equations in t unknownsn Algorithmw Setup: T begins with a secret integer S it wishes to distribute among n

users.n T chooses a prime p >max(S, n), and defines a0 = S, selects t�1

random coefficients a1, …, at�1 defining the polynomial over Zp, f(x) = Σt�1

j=0 ajxj

n T computes Si = f(i) mod p for all i, and securely transfers the shareSi to Pi

w Pooling of shares: Group of t or more users pool shares, which provide t distinct points allowing computation of aj’s by Lagrange interpolation

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v Lagrange interpolation

n f(xs) = ys

v Propertiesn perfect: Given knowledge of any t � 1 or fewer shares, the shared

secret remain equally probablen ideal: The size of one share is the size of the secret

n extendable for new users: New shares (for new users) may be computed and distributed without affecting shares of existing users.

n varying levels of control possible: Providing a single user withmultiple shares bestows more control upon that individual

n no unproven assumptions

∑ ∏= ≠≤≤ −

−=

t

i ijtj ji

ji xx

xxyxf

1 ,1

)( ∏∑≠≤≤= −

==ijtj ij

ji

t

iii xx

xcycS

,11

where,

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v detection of cheaters,and verifiable secret sharing. These schemes respectively address cheating by one or more group members, and the distributor of the shares

v Proactive secret sharing: secret shares are periodically updated to provide robustness against intrusion

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Conferencing Keying

v A conference keying protocol is a generalization of two-party key establishment to provide three or more parties with a shared secret key

v Cliques, BD, TGDH, STR

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Contents






v Secret sharing

v Conference keying


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Attack strategies and classic flaws

v Intruder-in-the-middlen “man-in-the-middle” attack on unauthenticated DH

v Reflection attackn Original protocol1. A → B : rA

2. B → A : Ek(rA, rB)3. A → B : rB

n Attack1. A → E : rA

2. E → A : rA : Starting a new session3. A → E : Ek(rA, rA’) : Reply of (2)4. E → A : Ek(rA, rA’) : Reply of (1)5. A → E : rA’n Can be prevented by using different keys for different sessions

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Attack strategies and classic flaws (cnt.)

v Interleaving attacksn To provide freshness and entity authenticationn Flawed protocol

1. A → B : rA

2. B → A : rB, SB(rB, rA, A)

3. A → B : rA’, SA(rA’, rB, B)

n Attack1. E → B : rA

2. B → E : rB, SB(rB, rA, A)3. E → A : rB

4. A → E : rA’, SA(rA’, rB, B)

5. A → E : rA’, SA(rA’, rB, B)

n Due to symmetric messages (2), (3)

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Analysis methods

v ad hoc and practical analysis (Provide heuristic security)n convincing arguments that any successful attack requires resource

level greater than the resources of the perceived adversaryn May uncover protocol flaws establishing that a protocol is badn Subtle flaws in protocols typically escape ad hoc analysis

v reducibility from hard problemsn proving that any successful protocol attack leads directly to the

ability to solve a well-studied reference problemn provably secure protocoln A challenge is to establish that all possible attacks have been

taken into account, and can be equated to solving the identifiedreference problems

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Analysis methods

v complexity-theoretic analysisn Model of computation is defined, and adversaries are modeled as

having polynomial power. Security proof relative to the model isthen constructed

n The existence of underlying cryptographic primitives with specified properties is typically assumed.

n An objective is to design cryptographic protocols which require the fewest cryptographic primitives, or the weakest assumptions.

n As the analysis is asymptotic, care is required to determine when proofs have practical significance

n Polynomial attacks which are feasible under such a model may in practice be computationally infeasible

n Despite these issues, complexity-theoretic analysis is invaluable for formulating fundamental principles and confirming intuition.

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Analysis methods

v information-theoretic analysisn mathematical proofs involving entropy relationships to prove protocols are

unconditionally secure

n Adversaries are modeled to have unbounded computing resources

n not applicable to most practical schemes for several reasonsw many schemes can at best be computationally securew typically involve keys of impractically large size, or can only be used once

v 5. formal methodsn logics of authentication (BAN), term re-writing systems, expert systems,

and other methods combining algebraic and state-transition technique

n utility in finding flaws and redundancies in protocols

n the “proofs” provided are proofs within the specified formal system, and cannot be interpreted as absolute proofs of security

n Absence of discovered flaws does not imply the absence of flaws

Chap 12

Documents

key secrecy of key

possession of key

resulting key

key key confirmation

given key

key initiate

key establishment technique

conceptsv key establishment