Hardening Functions for LargeScale Distributed Computations

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Hardening Functions for Large-Scale Distributed
Computations

• Doug Szajda
• Barry Lawson
• Jason Owen

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Large-Scale Distributed Computations

• Easily parallelizable, compute intensive
• Divide into independent tasks to be executed on
  participant PCs
• Significant results collected by supervisor
• Participants may receive credits
• Money, e-cash, ISP fees, fame and glory

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Examples

• seti_at_home
• Finding Martians
• folding_at_home
• Protein folding
• GIMPS (Entropia)
• Mersenne Prime search
• United Devices, IBM, DOD Smallpox study

• DNA sequencing
• Graphics
• Exhaustive Regression
• Genetic Algorithms
• Data Mining
• Monte Carlo simulation

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The Problem

• Code is executing in untrusted environments
• Results may be corrupted either intentionally or
  unintentionally
• Significant results may be withheld
• Cheating credit for work not performed

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An Obvious Solution

• Assign Tasks Redundantly
• Collusion may seem unlikely but
• Firms solicit participants from groups such as
  alumni associations and large corporations
• Processor cycles are primary resource
• Some problems can tolerate some bad results

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Related Work

• Historical roots in result checking and
  self-correcting programs
• Golle and Mironov (2001)
• Golle and Stubblebine (2001)
• Monrose, Wyckoff, Rubin (1999)

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Related Work

• Body of literature on protecting mobile agents
  from malicious hosts
• Sander and Tschudin, Vigna, Hohl, and others
• Syverson (1998)

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Adversary

• Assumed to be intelligent
• Can decompile, analyze, modify code
• Understands task algorithms and measures used to
  prevent corruption
• Motivation may not be obvious...
• I.e. gaining credits may not be important
• E.g. business competitor
• But does not wish to be caught

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Our Approach

• Hardening functions
• Verb, not adjective
• Does not guarantee resulting computation returns
  correct results
• Does not prevent an adversary from disrupting a
  computation
• Significantly increases likelihood that abnormal
  activity will be detected

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The Model

• Computation is evaluation of algorithm f D -gt R
  for every input value x in D
• Tasks created by partitioning D into subsets Di
• Each task assigned filter function Gi

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Two General Classes

• Non-sequential
• Computed values of f in task are independent
• Sequential
• Participant given single value x0 and asked to
  compute first m elements of sequence xn f (xn-1)

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Hardening Non-sequentials

• Plant each tasks data set with values ri such
  that the following hold
• Supervisor knows f(ri) for each i
• Participant cannot distinguish ri from other data
  values regardless of number of tasks a
  participant completes

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Hardening Non-sequentials

• Participants do not know number of ri in data
  space
• For some known proportion of ri f(ri) is a
  significant result
• Nice but not necessary Same set of ri can be
  used for several tasks

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Difficulties

• ri are indistinguishable only if they generate
  truly significant results
• What is indistinguishable in theory may not be in
  practice
• E.g. DES key search Tasks given ciphertext C and
  subset Ki of key space, told to decrypt C with
  each ki and return any key that generates
  plausible plaintext

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Even Filter Function Can Be Revealing...

• E.g. Traveling Salesperson with five precomputed
  circuits of length 100, 105, 102, 113, 104
• Return any circuit whose length is any of the
  above or less than 100
• Return the ten best circuits found
• Return any circuit with length less than 120

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Optimization Problems

• Designate small proportion of tasks as initial
  distribution
• Distribute each of these tasks redundantly
• Check returned values handle non-matches
  appropriately
• Retain k best results and use them as ringers for
  remaining tasks

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Collusion

• If task in initial distribution is assigned to
  colluding adversaries, supervisor will initially
  miss this
• Honest participants not in initial distribution
  will eventually return results that do not match
• Supervisor can then determine which participants
  have been dishonest

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Size of Initial Distribution

• Probability that at least k of n best results are
  in proportion p of space is

For 109 inputs, best 105 results are in top 0.01
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Caveat

• Previous figures assume
• n, k much less than size of data space
• proportion of incorrect results is small
• Probability should be adjusted to reflect
  expected number of incorrect results returned in
  initial distribution

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The Good

• No precomputing required
• Hardening is achieved at fraction of cost of
  simple redundancy
• Ringers can be used for multiple tasks
• Additional good results can be used as ringers
• Collusion resistant since ringers can be combined
  in many ways

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The Bad

• Assuming tasks require equal time, cost of
  compute job is at least doubled...
• But, by running multiple projects concurrently,
  overall throughput rates can be reduced to factor
  of 1p times rate of unmodified job
• In some cases, implementation details can give
  away identities of ringers (or require
  significant changes to app)

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Sequential Computations

• Seeding the data is impractical
• Often the validity of returned results can only
  be checked by performing the entire task
• Ex Mersenne Primes
• nth Mersenne Number, Mn, is 2n-1

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The Strategy

• Share the work of computing N tasks among K
  participants
• K gt N is very small proportion of total number of
  participants in computation
• Assume
• Each task requires roughly m iterations
• K/N lt 2, else simple redundancy is cheaper

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The Algorithm

• Divide tasks into S segments, each containing
  roughly J m/S iterations
• Each participant in group is given an initial
  value and computes first J iterations using this
  value
• When J iterations complete, results returned to
  supervisor

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The Algorithm

• Supervisor checks correctness of redundantly
  assigned subtasks
• Supervisor permutes N values and assigns these
  values to K participants as initial value for
  next segment
• Repeat until all S segments completed

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The Numbers

• If K/N lt 2, each task assigned to no more than
  two participants, and adversary cheats in L (of
  S) segments, then in absence of collusion

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Probabilities
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Redundancy vs. P values
L 1
L 2
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Advantages

• Far fewer task compute cycles than simple
  redundancy
• Values need not be precomputed
• Method is relatively collusion resistant (unless
  supervisor picks an entire group of colluding
  participants)
• Method is tunable
• Can also be applied to non-sequential case

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Disadvantages

• Increased coordination and communication costs
  for supervisor
• Need for synchronization increases time cost of
  job
• Dial-up connectivity
• Sporadic task execution (owners using PCs)

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Disadvantages

• Strategy does not protect well against adversary
  who cheats once
• Cheating damage can be magnified
• Propagation of undetected incorrect results

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Conclusions

• Presented two strategies for hardening
  distributed metacomputations
• Non-sequential Seed data with ringers
• Sequential Share N tasks among K gt N
  participants
• Small increase in average execution time of
  modified task
• Overall computing costs significantly less than
  redundantly assigning every task