Rejection Sampling Schemes for Extracting Uniform Distribution from - PowerPoint PPT Presentation

Rejection Sampling Schemes for Extracting Uniform Distribution from Biased PUFs Rei Ueno , Kohei Kazumori, and Naofumi Homma Tohoku University

Background PUF circuit One PUF cell response (Latch PUF) Example of PUF signal Set wafer Silicon input outputting one bit PUF circuit constructing secure and trustable systems Chip #1 Chip #2 • Physically unclonable functions (PUFs) play essential role for • Generate hardware-intrinsic random number like fingerprint • Exploit process variations for physical unclonability and tamper evidence • Major applications of PUF • Entity authentication (Strong PUF) • Cryptographic key generation (Weak PUF) R 1 n -bit R 2 Even for same input and same circuit construction, PUF responses vary due to process variation (i.e., R 1 ≠ R 2 ≠ …) 2

PUF-based key generation enrolled key from noisy PUF response Reconstruction Enrollment ́ ́ • Fuzzy extractor (FE) is commonly used for reconstructing x x PUF PUF ECC s c w (Noisy) RNG Helper data encode w c s k ECC Helper data KDF key decode k KDF key (key derivation function) • Helper data is stored in common nonvolatile memory (NVM) • NVM is usually non-tamper resistant, and helper data is considered public • We should consider conditional entropy for key generation • A s -bit key generation is realized only if 3

Problem of PUF bias: Entropy leakage • If PUF response is unbiased, (i.e., seed length) • But significantly decreases with PUF bias increase • Entropy leakage • If PUF is biased, random seed should be set longer than s such that • But required PUF size rapidly grows with PUF bias, especially when p 1 > 0.58 PUF size required for reliable C W 128-bit key generation 1 – p 1 0 0 (Values are from [DGV+16]) p 1 p 1 PUF 0.54 0.58 0.62 0.66 1 1 1 – p 1 bias p 1 p 1 : probability Bit-error 0.100 0.098 0.096 0.092 of 1 in X rate Channel diagram of FE [HO17] PUF size 1,530 2,550 5,100 13,005 4

Debiasing = 1 0 1 ≠ ≠ = ≠ 1 PUF size FE w/o debiasing FEs w/ debiasing Figure is based on graph in presentation slide of [DGV+16] 1 0 0 0 biased PUF response 1 0 1 1 1 1 1 1 0 1 • Extract unbiased bit string from • Realize secure key generation even from PUFs with nonnegligible biases • Efficiency has been evaluated through PUF size required for reliable 128-bit key gen. PUF bias p 1 • Example of debiasing: von Neumann corrector (VNC) • Values of 1 and 0 are extracted with an identical probability of p 1 p 0 • Debiasing data d is used for reproducing z at reconstruction PUF response ! : Debiased data $ : Debiasing data ": 5

Conventional debiasing-based FEs Trivial debiasing biased PUFs 2019 2016 [DGV+16] Tight bounds of min-entropy loss, motivation for debiasing [SUHA17] Ternary VNC- based FEs [S17] [AWSO17] secure key Maskless debiasing (MD) [HO17] Coset coding (CC)-based FE, FE is modeled as wire-tap channel [USH19] Biased masking (BM)-based FE [KW19] Selection and balancing schemes generation from solution for for SRAM PUF 2010 developed for improving efficiency [YD10] Index-based syndrome (IBS) [LSHT10] First von Neumann corrector (VNC)- based debiasing [HMSS12] Generalized first explicit IBS 2012 2014 2015 2017 [KLRW14] Report on entropy loss in PUF- based key generation [MLSW15] VNC-based FEs, • Various debiasing-based FEs have been • Efficient FE reduces PUF and NVM sizes • How far can we go? 6

This work scheme based on rejection sampling and FE construction of 128-bit key generation in comparison with conventional FEs • Acceptance-or-Rejection (AR)-based FE: New debiasing • Extract uniform distribution with highest efficiency among conventional FEs • Implemented with solely an RNG at enrollment, and no critical additional operation is required at reconstruction performed on client device • First FE which can tolerate local biases depending on cell addresses (for example, found in some SRAM PUFs) • Extended to ternary PUF response for improved efficiency (see our paper) • Performance of proposed FE is evaluated through simulation • AR-based FE achieves smallest PUF and/or NVM sizes (i.e., hardware cost) for various PUFs • At most 55% and 72% smaller PUF and/or NVM sizes than counterparts 7

Bias models Global and cell-wise biases Typical example of cell-wise-based PUF • Global bias model • All bits in PUF response have an identical bias of p 1 (with corresponding p 0 ) • All conventional debiasing scheme employed global bias model • Cell-wise bias model (or local bias model) • Each bit has unique bias depending on cell address i • Expected value of biases are considered equal to global bias (i.e., ) i = 0 1 2 3 4 5 6 7 8 9 i = 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 3 2 7 9 3 4 5 2 1 8 Local bias p 1, i = Grobal bias p 1 = 0.44 . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 8 8 8 8 2 2 2 2 2 Local bias p 1, i = Grobal bias p 1 = 0.50 . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 PUF response x 1 : 0 0 1 1 0 0 1 0 0 1 PUF response x 1 : 0 1 1 1 0 0 0 0 0 1 PUF response x 2 : 0 0 0 1 1 1 1 1 0 0 PUF response x 2 : 1 1 0 0 0 1 1 1 1 0 PUF response x 3 : 1 0 1 1 0 0 0 0 1 1 PUF response x 3 : 1 0 1 1 1 1 0 0 0 1 0 PUF response x 4 : 0 1 0 1 0 1 1 0 0 1 PUF response x 4 : 1 1 1 1 1 0 1 0 0 0 PUF response x 5 : 0 0 1 1 0 0 0 0 0 1 PUF response x 5 : 1 0 1 1 1 0 0 0 0 0 8

Rejection sampling (Scaled) proposal Accept Overview of rejection sampling otherwise, reject it distribution Target Reject • Method for deriving target distribution from proposal one • Target distribution: Distribution which is needed, but not directly available • Proposal distribution: Easily available distribution distribution hp prop ( x ) Step (1): Obtain sample a from p prop ( x ) hp prop ( a ) Step (2): Draw random number b from [0, p prop ( a )] p tar ( x ) p tar ( a ) Step (3): Accept the sample if b < p tar ( a ) ; Sample a • Application to PUF debiasing • Target distribution: Uniform distribution • Proposal distribution: PUF response (i.e., p 1, i -biased Bernoulli distribution) 9

reject “1” cells are rejected (i.e., discarded) with Debiased always accepted Minor value is bit string “0” cells are always accepted and probability of Rejected with (i.e., target distribution) rejection sampling Distribution after (Frequency) Occurrence probability 1 PUF 0 (i.e., proposal distribution) as Bernoulli distribution Biased PUF response (Frequency) Occurrence probability Rejection 1 0 sampling Extraction of uniform distribution from biased PUFs reject response • Key idea: Bit-wise rejection sampling • Rejection sampling is applied to i -th cell with biases p 1, i , p 0, i for all i • Expected length of debiased bit string is longer than conventional schemes 1 1 1 0 1 1 1 0 1 1 0 reject reject 1 – p 0, i / p 1, i 1 0 1 0 1 0 1 p 0, i p 0, i p 0, i p 1, i Example of debiasing ( p 1, i = 0.70 for all i ): Value of i -th cell Value of i -th cell probability of 1 – p 0, i / p 1, i = 0.57 10

Proposed scheme: AR-based FE Enrollment of AR-based FE Reconstruction of AR-based FE extraction (ACE) operations are applied to PUF response • Reproducible rejection sampling (RRS) and accepted cell x x′ PUF PUF (Noisy) d RRS ACL data d ACL data ACE u ECC s c w u′ Helper data RNG encode ECC w c′ s k Helper data KDF key decode k KDF key • RRS operation generates debiased bit string and accepted cell location (ACL) data d • Naïve rejection sampling is not reproducible • ACL data enables us to reproduce debiased bits at ACL at reconstruction • We proved there is no entropy leakage from pair of helper and ACL data 11

RRS and ACE operations̶Implementation Input PUF Step (4) : Obtain debiased bit string Step (3) : Generate ACL data as Step (1) : Take bit- Step (0) : Generate bit- • RRS operation performs rejection sampling with reproducibility • First generate ACL data d , and then extract debiased bit string • Implemented using an RNG and bit-parallel operations in enrollment server Step (2) : Generate random number r p 1, i = 0.6 0.3 0.7 0.4 0.5 0.9 0.6 0.4 0.1 0.8 0.3 ( i -th bit has a bias of min( p 1, i , p 0, i )/max( p 1, i , p 1, i ) ) 1 1 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 1 0 0 1 0 response x d = h ∨ r 1 0 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 0 0 1 1 0 string h , where i -th bit is Boolean value of p 1, i ≥ p 0, i as extraction of x i with d i = 1 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 1 parallel XOR of x and h (as y ) • ACE operation extracts bit value of cells indicated by ACL data • No additional computation is required in reconstruction 12