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HMQV: A High-Performance Secure * Diffie-Hellman Protocol

HMQV: A High-Performance Secure * Diffie-Hellman Protocol. Hugo Krawczyk IBM Research. * Proven secure (just in case you heard rumors to the contrary…). . Outline of the Talk. The authenticated Diffie-Hellman problem Implicitly authenticated KE protocols

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HMQV: A High-Performance Secure * Diffie-Hellman Protocol

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  1. HMQV: A High-Performance Secure*Diffie-Hellman Protocol Hugo Krawczyk IBM Research *Proven secure (just in case you heard rumors to thecontrary…) 

  2. Outline of the Talk • The authenticated Diffie-Hellman problem • Implicitly authenticated KE protocols • The MQV Protocol: does it deliver its wish list? • Some remarkable ideas but some design weaknesses as well • HMQV: A proven secure variant of MQV • Random oracle, computational assumptions (CDH), model-based • Rationale, validation, design (and even implementation) guide • The essential role of formal analysis • Unfortunately: no time for analysis idea (really interesting part)

  3. gx gy Diffie-Hellman Exchange [DH’76] AliceBob • both parties compute the secret key K=gxy=(gx)y=(gy)x • assumes authenticated channels (+ DDH assumption) • open to m-i-t-m in a realistic unauthenticated setting

  4. Authenticated Diffie-Hellman • Bind key to identities via PKs (or other means) • Non-trivial: innumerable proposals, many broken • NOT that a good protocol must be complex or inefficient, only that it is incredibly easy to design them wrong • No need to compromise for weak protocols anymore • What does it mean for a KE protocol to be secure? What are the attacker’s capabilities? • Many works/approaches: much beyond preventing obvious impersonation and key recovery attacks (known-key attacks, ephemeral vs static, UKS, PFS, KCI,...)

  5. Implicitly Authenticated DH [MTI’86] • Minimalist approach: Keep a plain DH exchange, but give Alice and Bob public keys (possibly with certificates) • Authentication via session key computation • No transmitted signatures, MAC values, etc • Session key must involve long-term and ephemeral keys: K=F(PKA,PKB,SKA,SKB ,gx,gy,x,y) • Ability to compute key  authentication • Possible and simple but tricky: many insecure proposals, also in standards (e.g. UM [BJM97])

  6. MQV [MQV’95,LMQSV’00] • Most attractive among implicitly authenticated DH • Performance: just ½ exponentiation (25%) more than DH (with NO added bandwidth except if public keys transmitted) • Broad array of security goals considered: m-i-t-m, known-key attacks, UKS, PFS, KCI (hard to achieve with implicit auth),… • Widely standardized: ANSI, IEEE, ISO, NIST • NSA: “next generation cryptography”(including protection of “classified or mission critical national security information“) • But is MQV secure? In what sense? Can be improved?

  7. The MQV Protocol • Basic DH + special key computation • Notation: G=<g> of prime order q; g in supergroup G’ (eg. EC, Z*p) • Alice’s PK is A=ga and Bob’s is B=gb, • Exchanged ephemeral DH values are X=gx, Y=gy • From which two values are computed: d=LSB(X), e=LSB(Y) where LSB(X)= 2L + X mod 2L for L=|q|/2 (this is the ½ exponentiation) • Both compute σ=g(x+da)(y+eb) as σ = (YBe)x+da = (XAd)y+eb • Session key is K=KDF(σ) (KDF unspecified: “prevent weak bits”) • Magic, isn’t it? Is it secure?Why?Can it be formally analyzed?

  8. The MQV Protocol • Actual computation of σ involves co-factor h=|G’|/q • σ’ = (YBe)x+da = (XAd)y+eb • σ = (σ’)h • Adds an exponentiation: small for ECC, large for Z*p (can replace with a q-order test) • Omitted in above description for simplicity: does not help against any of the weaknesses described here

  9. Design Weaknesses in MQV • MQV cannot be proved generically (for any q-order CDH G) • For each group G there are representations of its element under which the protocol is totally insecure • E.g. if 80 lsb’s have 50 bits of entropy then can break in 250 • E.g. a simple padding can break the protocol. Also note that subgroup G usually very sparse (e.g. |G|= 2160 |Zp*|=21024 ) • Proof (if at all possible) must involve details of representation • An authentication failure: UKS [DVW] • Key bound to the wrong identities

  10. UKS Attack on MQV • UKS: key-identity misbinding • Alice and Bob engage in a KE with Eve as m-i-t-m • Alice and Bob end computing the same key K (unknown to Eve) • But Alice binds the key to Bob while Bob binds it to Eve • Can UKS be prevented in MQV? • Require “proof of possession” (PoP) of private key at CA • PoP hard to do it right and not even a solution [Kaliski] • Key confirmation? Not sufficient if ephemeral state leaked • The essential identity-key binding is missing (cf 2-msg HMQV)

  11. How About KCI Attacks? • A KCI attack: Using Alice’s private key, Eve can imperso- nate other parties to Alice (the reverse is unavoidable) • Resisting KCI is a major argument for MQV: does it hold? • The answer is NO if KDF(σ) is invertible (but one-wayness not a reqt in MQV: “the requirement that KDF be preimage resistant appears unnecessary”) • KCI resistance w/non-invertible KDF?? Yes for HMQV.

  12. Leaked ephemeral exponents • Yet another property that requires KDF to be one-way: protection against disclosure of ephemeral exponents (revealing x or y should not compromise K or other sessions) • Property claimed by MQV but we show not to hold if KDF is not one-way

  13. PFS (perfect forward secrecy) • PFS = eventual disclosure of long-term private keys does not compromise past session keys • A fundamental property of DH (but also with mitm?) • Also claimed as a (central) property of MQV • We show: no implicity authenticated DH protocol can enjoy PFS • Hence: MQV does NOT provide PFS except for sessions where the attacker was passive (also HMQV) • This (and not UKS) is the real reason to augment the protocol with a third msg and MACs

  14. HMQV: A secure MQV variant • As in MQV: basic DH (X=gx, Y=gy), PKs: A=ga, B=gb • Both compute σ=g(x+da)(y+eb) as σ = (YBe)x+da = (XAd)y+eb • d=H(X,”Bob”) e=H(Y,”Alice”)(here H outputs |q|/2 bits) • Session key K=H(σ) • Differences with MQV • Definition of d, e: binds id’s, randomizes representation • H(σ): integral (and essential) part of the protocol (OW,RO) • “HMQV = Hashed MQV” (note: 2.5 exponentiations)

  15. HMQV Analysis • In the KE model of Canetti and Krawczyk [CK’01] • Attacker may access private keys, session keys, session-state information (“exposed session”) • Any unexposed session is secure (key is indist from random) • In addition: extensions to capture PFS, KCI • [CK’01] Prove that secure KE in this model  secure communications (“secure channels”) • Note: protocol must specify what resides in state and what in protected memory (such as private keys)

  16. Basic Security of HMQV • Thm: Under the CDH assumption and in the random oracle model, HMQV (basic 2-msg or 3-msg with KC) is a secure KE protocol in the Canetti-Krawczyk KE model • The thm applies when σ and the ephemeral x,y are specified to be in protected memory, same as the private key (cf DSA) • The Thm includes resistance to KCI and PFS (the latter only in the case of 3-msg variant with KC), and of course UKS, known-key attacks, key recovery, etc

  17. Security in the face of ephemeral disclosure • Under Gap-DH, KEA1 and in the random oracle model HMQV is secure also if ephemeral x,y disclosed • Menezes: pointed out that proof assumes X, Y in G=<g>  security and proof hold if parties are required to check peer’s DH value is of prime order • Checks needed only against ephemeral disclosure and one-pass. Other cases and all proofs remain unchanged. • When needed test adds an exponentiation, cost depends on underlying group (always performed in MQV) • Establishes a clear security/performance trade-off (possible only with analysis, also an implementation guide: protect σ)

  18. Cryptography is a Science! • Intuition, ideas, cryptanalysis, new attacks… all necessary and important but: • Formal analysis as main confidence tool • Not a Panacea: never stronger than the model it is based on • But well-defined mechanisms and properties: can be verified (not just “trust me, I have not been able to break it”) • Formal analysis as main design tool • Guides us to choose secure mechanisms, compose them right, discern between the essential, desirable and dispensable • Result is efficiency, simplicity, rationale, even impl’n guidance e.g. avoid PoP and PK before DH, selective prime tests, essential hashing, KC for PFS (not for UKS), levels of protection,etc

  19. A message to standard bodies • No reason to compromise on protocol quality • No need to “live” with UKS attacks or mandate avoidable PoPs • No need to depend on heuristics only • Cannot “afford” ignoring formal study (if we only had it for encryption and hash functions…) • Standardize on proven protocols (especially when they are equally, or even more, efficient than alternatives) • In other words: DEMAND HIGH STANDARDS…

  20. Analysis: the really interesting part • Main tool: “challenge-response signatures” in which signer and verifier generate same signature • DH values serve as challenges, key = hash (signature) • Signature scheme: “exponential Schnorr” + FiatShamir http://eprint.iacr.org/2005/176

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