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CRYPTOGRAPHY AGAINST CONTINUOUS MEMORY ATTACKS Yevgeniy Dodis, Kristiyan Haralambiev, Adriana Lopez-Alt and Daniel Wichs MIT/MSR Reading Group NYU

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Motivation: Leakage-Resilient Crypto Security proofs in crypto require an adversarial attack model. e.g. adversary sees public-keys but not secret-keys. Reality: schemes broken using attacks outside of model. Side-channels: timing, power consumption, heat, acoustics, radiation. The cold-boot attack. Hackers, Malware, Viruses. Usual Crypto Response: Not our problem. Blame the Engineers… Leakage-Resilient Crypto: Let’s try to help. Primitives that provably allow some leakage of secret key.

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Modeling Leakage During an attack, adversary can query “leakage-oracle” with “leakage-functions” f : {0,1}* ! {0,1}. Learns f(sk). Need to restrict leakage so adv. does not learn sk in full. Bound number of queries during various stages of attack. Restrict type of allowed leakage-functions. sk f f(sk)

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Modeling Leakage Type: Restricted functions vs. Memory Attacks. Restricted leakage functions f. Individual bits, AC0 circuits... [Riv97, ISW03, FRR+10, …] “Only computation leaks information” [MR04, DP08] Memory Attacks [AGV09]: All efficient functions f are legal. Amount: One-Time vs. Continuous. One-time: Total number of leakage queries is bounded by L. Continuous: The secret key of the scheme gets updated periodically. Number of queries is bounded by L in each “period” but not overall.

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Prior Work OCLI Stream Cipher OCLI Signatures [DP08, Pie09, KKPR10] General Computation [ISW03, FRRTV10, JRV10]. ? All-Or-Nothing-Transform (AONT) [Riv97, Boy99, CDH+00, DSS01, KZ03] PKE [AGV09, NS09, ADNS+10, DHLW10] Signatures [ADW09, KV09, DHLW10] ID/AKA [ADW09, DHLW10] Other: [Dzi06, CLW06, ADW09, DKL09, ADNS+10, DGK+10, GKPV10] Continuous One time RestrictedMemory

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Continuous Leakage + Memory Attacks? Question: Can we get the best of both worlds? Continuous Leakage Resilience (CLR)

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Secret key can be updated using only local randomness. Public-key stays the same. Other users do not need to know about updates. Number of leakage queries bounded by L in between updates. No bound on number of queries over the lifetime of the system. No restriction on the type of leakage (memory attacks). Adversary cannot predict randomness for future updates. (No leakage during the update). Efficiency does not degrade with the number of updates. Model of CLR Security

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Initial Steps Towards CLR [ADW09]: Construct CLR signatures but updates require an “external master key” that never leaks. [BKKV10]: Construct CLR signatures, but… Requires a non-standard and non-black-box assumption. Uses PCP theorem, so not practical. (Main results to follow. Stay tuned…)

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Our Results Build the first CLR schemes under standard assumptions (K-Linear in bilinear groups). Signatures, ID schemes, Authenticated-Key-Agreement In each period, leak up to ½ of the secret key. All schemes are efficient/practical. Main technical result: CLR One-Way Relation.

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Main Technical Problem: CLR One-Way Relation pk sk R(pk, sk)=1 (pk, sk) Ã KeyGen() KeyGen() generates (pk, sk) pairs. Relation R tests if (pk, sk) are valid. Basic One-Wayness: Given pk, can’t find sk* s.t. R(pk, sk*) = 1.

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Main Technical Problem: CLR One-Way Relation pk sk KeyGen() generates (pk, sk) pairs. Relation R tests if (pk, sk) are valid. Rerand(sk) refreshes the secret key. Can refresh arbitrarily many times. Rerand(sk) R(pk, sk)=1 Rerand …

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Main Technical Problem: CLR One-Way Relation pk sk Adversary gets pk. Can ask for up to L bits of information about sk. What’s the 2 nd bit of sk ? What’s the 3 rd bit of SHA-1(sk) ? Security

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Main Technical Problem: CLR One-Way Relation pk sk Adversary gets pk. Can ask for up to L bits of information about sk. Key is refreshed. Security

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Main Technical Problem: CLR One-Way Relation pk Adversary gets pk. Can ask for up to L bits of information about sk. Key is refreshed. Wins if R(pk, sk*) = 1. Security sk* =

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Difficulty of Constructing CLR-OWR In a reduction, need to know many valid secret-keys in full. Nevertheless, need to solve some hard problem given a valid forgery by the adversary. Hope: forgery of different type then secret-keys we know. Adversary gets unbounded amount of information in total. Why can’t it “learn” the type?

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Outline of Construction 1. General strategy to handle continuous leakage. Reduce “continuous leakage” to “one-time leakage”. 2. Construction from (special) PKE/NIZK. 3. Instantiate components based on DDH/K-Linear.

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Strategy for CLR-OWR For each public-key pk: Valid: { sk : R(pk, sk) = 1}. Good ½ Valid Bad = Valid \ Good Can sample pk along with: samG : Samples sk 2 Good. samB : Samples sk 2 Bad. dk : isGood(sk, dk) = 0/1. Re-randomize inside Good. Given pk, samG, sk 2 Good : Rerand(sk) ¼ Sample(samG) good bad All possible sk Valid for pk

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Strategy for CLR-OWR Security Hardness of Good Keys: Given pk, samB, find sk* s.t. isGood(sk*, dk) = 1. Hardness of Bad Keys: Given pk, samG, find sk* s.t. isGood(sk*, dk) = 0. L-Leakage-Indistinguishability: Adv. gets pk, samG, samB. Challenger randomly chooses {good, bad} key sk. Adv. gets L bits of leakage on sk. Wins if produces sk* in the same category as sk. good bad Leakage Indistinguishable Re-randomizable Relation (LIRR) Only “one-time memory attack”.

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1. LIRR ) CLR-OWR Claim: An L-LIRR is a L-CLR-OWR Proof: Adversary only sees random good keys. Forgery must be a good key by “hardness of bad keys”. One-by-one, switch good keys for bad keys. Pr[forged key is good] does not decrease by “L-Leakage-Ind.”. Adversary only sees bad keys. Yet forgery is good. Contradicts “hardness of good keys”.

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Outline of Construction 1. General strategy to handle continuous leakage. Reduce “continuous leakage” to “one-time leakage”. 2. Construction from (special) PKE/NIZK. 3. Instantiate components based on DDH/K-Linear.

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Constructing LIRR Let E 1, E 2 be two PKE and ¦ be a NIZK argument-system. The LIRR scheme samples: pk = (CRS, pk 1, pk 2, c 1 ) where c 1 = Enc pk1 (m;r) for rand m. Secret-keys have form sk = (c 2, ¼ ). Valid sk: ¼ proves that “(c 1, c 2 ) encrypt same message”. Good sk: (c 1, c 2 ) actually encrypt the same message. Sample good keys with samG = (m, r). Sample bad keys with samB = TD (simulation trap-door). Distinguish good/bad with dk = (sk 1, sk 2 )

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Constructing LIRR Re-randomization. Hardness of Good Keys. Hardness of Bad Keys. L-Leakage-Indistinguishability.

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Constructing LIRR (Hardness) Claim: The “hardness of good/bad keys” holds if E 1 is “one- way secure” (weaker than semantic-sec) and ¦ is sound. Proof: Given samG = (m, r), producing a bad key, requires proving an invalid statement (break soundness). Given samB = TD, producing a good key inverts c 1 (break one-wayness of E 1 ). pk = (CRS, pk 1, pk 2, c 1 ) sk = (c 2, ¼ ). samG = (m, r), samB = TD, dk = (sk 1, sk 2 ) Valid sk: ¼ proves that “(c 1, c 2 ) encrypt same message”. Good sk: (c 1, c 2 ) actually encrypt same message.

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Constructing LIRR Re-randomization. Hardness of Good Keys. Hardness of Bad Keys. L-Leakage-Indistinguishability.

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Constructing LIRR (Re-randomization) To re-randomize sk = (c 2, ¼ ) need to: Re-randomize the ciphertext c 2. (same message, fresh rand.) Update the NIZK proof ¼ (new statement, fresh rand). pk = (CRS, pk 1, pk 2, c 1 ) sk = (c 2, ¼ ).

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Constructing LIRR (Re-randomization) Assume E 1, E 2, ¦ are homomorphic over some groups. Hom. Encryption: Enc(m; r) + Enc(m’; r’) = Enc(m + m’; r + r’). Hom. Language: x 2 L, x’ 2 L ) x + x’ 2 L. Hom. NIZK for Hom. Language. If ¼, ¼ ’ are proofs of x, x’ then ¼ + ¼ ’ is a proof of x+x’. Notion of re-randomizable/malleable NIZKS defined in [BCC+09]. Language “(c 1, c 2 ) encrypt the same message” is homomorphic. Notice (c 1, c 2 + Enc(0;r)) = (c 1, c 2 ) + (Enc(0;0), Enc(0;r) ). pk = (CRS, pk 1, pk 2, c 1 ) sk = (c 2, ¼ ).

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Constructing LIRR Re-randomization. (If E 1, E 2, ¦ are “homomorphic”) Hardness of Good Keys. Hardness of Bad Keys. L-Leakage-Indistinguishability.

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To get leakage indistinguishability, need “something” from E 2. Given L bits of leakage on ctext c, can’t produce related ctext c’. Adversary gets pk. Chooses m 0, m 1. Challenger chooses b Ã {0,1}. Sets c = Enc(m b ). Adversary gets L bits of leakage on ciphertext c. Adversary makes 1 decryption query on any c*. Wins if guesses b. Call this L-“Leakage-of-Ciphertext Non-Malleability” (L-LoC-NM). Notice: Non-malleable Enc ) |m|-LoC-NM ) CPA. But we need L-LoC-NM + Homomorphic encryption! Is it possible? Constructing LIRR (Leakage Ind.) pk = (CRS, pk 1, pk 2, c 1 ) sk = (c 2, ¼ ).

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Constructing LIRR Re-randomization. (If E 1, E 2, ¦ are “homomorphic”) Hardness of Good Keys. Hardness of Bad Keys. L-Leakage-Indistinguishability. (If E 2 is L-LoC-NM)

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Outline of Construction 1. General strategy to handle continuous leakage. Reduce “continuous leakage” to “one-time leakage”. 2. Construction from (special) PKE/NIZK. 3. Instantiate components based on DDH/K-Linear.

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Homomorphic NIZKs Groth-Sahai NIZKs are homomorphic for “systems of equations” over prime-order groups (with a pairing). Secure under K-linear assumption for any K ¸ 1. 1-linear = DDH. Can only hold for “asymmetric pairings” (two base groups are different). 2-linear is a standard assumption for symmetric and asymmetric pairings. K-linear assumption gets weaker as K grows.

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Homomorphic Encryption For E 1, use ElGamal (generalized to K-linear). For E 2, use “Cramer-Shoup-Lite” Encryption (generalized). Full Cramer-Shoup: CCA-2 secure. Not homomorphic. Enc(m;r) = (g r, h r, (f 0 ) r m, (f 1 ) r (f 2 ) r a ) where a = H(g r, h r ) Cramer-Shoup-Lite: Only CCA-1 secure. Is homomorphic. Enc(m;r) = (g r, h r, (f 0 ) r m, (f 1 ) r ) In general, CCA-1 security does not imply L-LoC-NM. But proof technique extends to show L-LoC-NM for L ¼ ¼|c|. Can generalize scheme to get L-LoC-NM for L ¼ (1- ² )|c|. Enc(m;r) = (g r, h r, (f 0 ) r m, (f 1 ) r, (f 2 ) r, (f 3 ) r, …).

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Summary of CLR-OWR Construct L-CLR-OWR based on the K-linear assumption. Relative leakage L/|sk| > 1/(K+1). 1/2 for DDH, 1/3 for 2-linear. “Practical Efficiency”: Constant number of group elements, group operations.

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Applications Build CLR signatures, ID schemes, Authenticated-Key- Agreement (AKA) generically from CLR-OWR, using prior techniques [ADW09, KV09, DHLW10]. Preserve leakage L and “relative leakage” L/|sk|. Can even preserve efficiency!

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Extensions Leakage-of-Randomness of Computation: So far, only assumed that sk leaks. For signatures (and ID/AKA) need to hide “local randomness”. Can be done. Standard model for ID. RO model for Sigs/AKA. Bounded-Retrieval Model (BRM): Allow “huge” amount of leakage, by increasing secret-key size. Does not effect efficiency (pk size, signature size, computation). (More extensions in the paper.)

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Follow-up Results of [BKKV10] Show how to do CLR Public-Key Encryption, IBE. Under 2-linear assumption. Leads to alternate construction of signatures (worse leakage). Show all CLR OWR/Signature/Encryption schemes also allow small (logarithmic) leakage of the update process.

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Thank you! ePrint Report 2010/196

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Homomorphic Encryption For E 1, use ElGamal (generalized to K-linear). For E 2, use (generalization) of “Cramer-Shoup-Lite”. CS-Lite is CCA-1 but not CCA-2 secure. It is homomorphic. In general, CCA-1 security does not imply L-LoC-NM. But “proof strategy” for CCA-1 generalizes to L-LoC-NM. Based on “1-universal hash proof system” (HPS). Used by [NS09] for leakage-resilient PKE. CS-Lite uses 2 copies of HPS. Use more copies for bigger L. Can leak L/|c| = 1- ² fraction of ciphertext.

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CS-Lite L-LoC-NM secure Use 3 properties of CS-Lite. “malformed ciphertexts” statistically hide the message (without sk). Can’t distinguish malformed from normal ciphertext (even with sk). To produce a malformed ciphertext, must know > L bits about sk. Make challenge-ciphertext malformed. Message is hidden, unless decryption queries give info about sk. Only decryption of malformed ciphertexts reveals info about sk. Argue that decryption query is normal. To produce malformed ctxt, need > L bits of info about sk. Only learn L bits from leakage queries on challenge ciphertext.

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Use “Cramer-Shoup-Lite” encryption: Let g, h be generators of a group G of prime-order p. sk = ((x 0, y 0 ), (x 1, y 1 ), …, (x n, y n )) pk = (f 0 = g x0 h y0, f 1 = g x1 h y1, …, f n = g xn h yn ) Enc(m) = ( g r, h r, (f 0 ) r m, (f 1 ) r, …, (f n ) r ) To decrypt: (g r ) xi (h r ) yi = (f i ) r Theorem: The “Cramer-Shoup-Lite” encryption scheme is L-LoC-NM where L ¼ n log(p). It is homomorphic. Constructing LIRR (Leakage Ind.)

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Constructing LIRR (Re-randomization) If E 1, E 2, ¦ are homomorphic, then can re-randomize sk. To Re-randomize sk = (c 2, ¼ ) : Choose c 2 ’ = Enc pk2 (0; r 2 ’). Let c 2 * = c 2 + c 2 ’. Let ¼ ’ be proof that (0, c 2 ’) encrypt same msg. 0 = Enc pk1 (0; 0). So ¼ * = ¼ + ¼ ’ proves (c 1, c 2 *) encrypt the same message. Since (c 1, c 2 *) = (c 1, c 2 ) + (0, c 2 ’) And (c 2 *, ¼ * ) is a perfectly re-randomized key. pk = (CRS, pk 1, pk 2, c 1 ) sk = (c 2, ¼ ).

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Constructing CLR-OWR from Components Constructed L-CLR-OWR from an L-LIRR. Constructed L-LIRR from PKE E 1, E 2, NIZK ¦ such that: E 1 is “one-way secure” and homomorphic. E 1 is L-LoC NM and homomorphic. ¦ is a homomorphic NIZK for the homomorphic language: “(c 1, c 2 ) encrypt the same message” Left to do: instantiate PKE E 1, E 2, NIZK ¦.

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Outline of CLR-OWR Construction 1. Reduce to “one-time memory attacks” on LIRR. 2. Build LIRR from simpler generic components. 3. Instantiate the components.

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Constructing LIRR (Re-randomization) To re-randomize sk = (c 2, ¼ ) need to: Re-randomize the ciphertext c 2. (same message, fresh rand.) Update the NIZK proof ¼ (new statement, fresh rand). Possible if E 1, E 2, ¦ are homomorphic over some groups. Hom. Encryption: Enc(m; r) + Enc(m’; r’) = Enc(m + m’; r + r’). New notion of “homomorphic NIZK” Influenced by re-randomizable and malleable NIZKs [BCC+09]. Can be instantiated using Groth-Sahai NIZKS. For some very specific “homomorphic” languages over groups. Requires the use of “K-linear assumption” in bilinear groups. pk = (CRS, pk 1, pk 2, c 1 ) sk = (c 2, ¼ ).

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2. Constructing LIRR (Re-randomization) To re-randomize sk = (c 2, ¼ ) need to: Re-randomize the ciphertext c 2. (same message, fresh rand.) Update the NIZK proof ¼ (new statement, fresh rand). Need: E 1, E 2, ¦ are homomorphic over some groups. Hom. Encryption: Enc(m; r) + Enc(m’; r’) = Enc(m + m’; r + r’). Hom. Relation: (x, w) 2 R, (x’, w’) 2 R ) (x + x’, w + w’) 2 R. Hom. NIZK: ¼, ¼ ’ proofs for x, x’ ) ¼ + ¼ ’ is a proof for x+x’. pk = (CRS, pk 1, pk 2, c 1 ) sk = (c 2, ¼ ). samG = (m, r), samB = TD, dk = (sk 1, sk 2 ) Valid sk: ¼ proves that “(c 1, c 2 ) encrypt same message”. Good sk: (c 1, c 2 ) actually encrypt same message.

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2. Constructing LIRR (Rerandomization) To Re-randomize sk = (c 2, ¼ ) : Choose c 2 ’ = Enc pk2 (0; r 2 ’). Let c 2 * = c 2 + c 2 ’. Let ¼ ’ be proof that (0, c 2 ’) encrypt same msg. 0 = Enc pk1 (0; 0). So ¼ + ¼ ’ is a proof that (c 1, c 2 *) encrypt the same message. Since (c 1, c 2 *) = (c 1, c 2 ) + (0, c 2 ’) And (c 2 + c 2 ’, ¼ + ¼ ’) is a (perfectly) re-randomized key. pk = (CRS, pk 1, pk 2, c 1 ) sk = (c 2, ¼ ). samG = (m, r), samB = TD, dk = (sk 1, sk 2 ) Valid sk: ¼ proves that “(c 1, c 2 ) encrypt same message”. Good sk: (c 1, c 2 ) actually encrypt same message.

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Prior Work on Leakage-Resilience Memory Attacks [AGV09] : PKE: [AGV09, NS09, ADNS+10, DHLW10] Signatures: [ADW09, KV09, DHLW10] ID/AKA: [ADW09, DHLW10] Extensions of Memory Attacks: Bounded Retrieval Model: [Dzi06, CLW06, ADW09, ADNS+10] Aux. Input : [DKL09, DGK+10] Noisy/Graceful: [NS09, GKPV10] Continuous Leakage: Only Computation Leaks Information: [MR04, DP08] Stream Cipher [DP08, Pie09] Signatures [KKPR10], Other [MR04, JRV10]. Other Restrictions: [FRRTV10, ISW03].

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