On the (Im)Possibility of Key Dependent Encryption Iftach Haitner Microsoft Research TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A August 04, 2009 Thomas Holenstein Princeton University

outline  Define Key Dependent Message (KDM) secure encryption scheme  Two (impossibility) results – On fully-black-box reductions from KDM security to TDP – On strongly-black-box reductions from KDM security to “any” hardness assumption

Weak Key Dependant Message Security An encryption scheme (Enc,Dec) is KDM secure, if for any efficient A A h 1 :{0,1} n  {0,1} m Enc k (h 1 (k)) h 2 Enc k (h 2 (k)) … ¼C¼C k à {0,1} n Challenger … A h 1 :{0,1} n  {0,1} m Enc k (U m ) h 2 Enc k (U m ) k à {0,1} n Challenger A cannot find k What class of query functions (e.g., h) should be considered? In most settings, we should consider any (efficient) function

Feasibility Results  Limited output length functions: – [Hofheinz-Unruh ‘08] based on any PKE  Family of affine functions: – [Bonhe-Halevi-Hamburg-Ostrovsky ‘08] based on DDH – [Applabaum-Cash-Peikert-Sahai ‘09] based on LPN/LWE  Efficient functions ???  Any function – [Black-Rogway-Shrimpton ‘02] based on Random Oracle

Our Impossibility Results (informal) It is impossible to construct (via black-box techniques) KDM encryption scheme that is secure against  the family of poly-wise independent hash functions, based on OWF – extends to TDP  any function, based on “any assumption” We focus on the private key setting Hold also for the “many PK keys” setting

outline  Define Key Dependent Message (KDM) secure encryption scheme  Our (impossibility) results – On fully black-box reductions from KDM security to TDP – On strongly black-box reduction from KDM security to “any” hardness assumption

Black-box construction Black-box proof of security Adversary for breaking KDM ) Inverter for breaking OWF Fully-Black-Box Reduction from KDM security to OWF Adversary for KDM Inverter for OWF OWF (Enc,Dec) OWF

Black-box proof of security A R OWF ¼ Y Ã {0,1} n x 2 ¼ - 1 (y) Breaks the KDM security of (Enc ¼,Dec ¼ )

Impossibility Result for OWF Based Schemes There exists no fully-black-box reduction from KDM- secure encryption scheme to OWF, which is secure against the family of poly(n)-wise independent hash functions More formally: Let (Enc (),Dec () ) be a OWF based encryption scheme, and let v(n) = |Enc () (M)|, for M 2 {0,1} 2n. Then (Enc (),Dec () ) cannot be proved (in a black-box way) to be KDM-secure against H v(n)+n – a family of (v(n)+n)-independent hash functions from {0,1} n to {0,1} 2n

Our adversary A R OWF ¼ Y à {0,1} n x 2 ¼ - 1 (y) 1.A breaks the (weak) KDM security of (Enc ¼,Dec ¼ ) 2. ¼ is hard to invert in the presence of A. Proof: a la ’ [Simon ‘98] / [Gennaro-Trevisan ‘ 01, H-Hoch-Reingold- Segev ‘07 ] 1n1n h c k … 1) Select h à H v(n)+n 2) On input C, output (the first) k s.t. Dec k (C) = h(k)

outline  Define Key Dependent Message (KDM) secure encryption scheme  Our (impossibility) results – On fully black-box reductions from KDM security to TDP – On strongly black-box reductions from KDM security to “any” hardness assumption

Let ¡ be a cryptographic assumption (e.g., factoring is hard)  Arbitrary construction  Black-box proof of security.  The query function h is treated as a black box Strongly Black-Box Reduction from KDM security to ¡ Adversary for KDM Adversary for ¡

Strongly Black-box proof of security A R for breaking ¡ ¡ A break the KDM security of (Enc,Dec) Factoring is hard n = pq p,q 1n1n h c k … 1.h is only accessed via its input/output interface 2.Access to h is not given to a “third party”

Impossibility Result for Strongly Black-Box Reductions Assume that there exists a strongly-black-box reduction from KDM encryption scheme to ¡, which is secure against O n – the family of random functions from {0,1} n to {0,1} 2n. Then ¡ can be broken unconditionally

Our Adversary A R ¡ Breaks the KDM security of (Enc,Dec) 1) Select h à O n 2) On query C, output (the first) k s.t. Dek k (C) = h(k) 1.A breaks the (weak) KDM security of (Enc,Dec) 2. R A, ¡ can be efficiently emulated

The Emulation R ¡ hÃOnhÃOn h(x 1 ) x1x1 h(x 2 ) x2x2 … 1.Answer to h(x i ) with a random y i 2 { 0,1} 2n (while keeping consistency) 2. On query C, return (the first) x i s.t Dec x i (C) = y i Proof Idea: the probability that h(k)= Dec k (C ) for non-queried k, is 2 -2n c k A 1n1n h

Further Issues  Both bounds hold for 1-1 PRF Open questions  Prove feasibility result against larger class of functions  Extend the first impossibility result to other assumptions (e.g., “Generic Groups”)