Filename: 288-privcount-with-shamir.txt
Title: Privacy-Preserving Statistics with Privcount in Tor (Shamir version)
Author: Nick Mathewson, Tim Wilson-Brown, Aaron Johnson
Created: 1-Dec-2017
Supercedes: 280
Status: Reserve

0. Acknowledgments

  Tariq Elahi, George Danezis, and Ian Goldberg designed and implemented
  the PrivEx blinding scheme. Rob Jansen and Aaron Johnson extended
  PrivEx's differential privacy guarantees to multiple counters in

  Rob Jansen and Tim Wilson-Brown wrote the majority of the experimental
  PrivCount code, based on the PrivEx secret-sharing variant. This
  implementation includes contributions from the PrivEx authors, and

  This research was supported in part by NSF grants CNS-1111539,
  CNS-1314637, CNS-1526306, CNS-1619454, and CNS-1640548.

  The use of a Shamir secret-sharing-based approach is due to a
  suggestion by Aaron Johnson (iirc); Carolin Zöbelein did some helpful
  analysis here.

  Aaron Johnson and Tim Wilson-Brown made improvements to the draft proposal.

1. Introduction and scope

  PrivCount is a privacy-preserving way to collect aggregate statistics
  about the Tor network without exposing the statistics from any single
  Tor relay.

  This document describes the behavior of the in-Tor portion of the
  PrivCount system.  It DOES NOT describe the counter configurations,
  or any other parts of the system. (These will be covered in separate

2. PrivCount overview

  Here follows an oversimplified summary of PrivCount, with enough
  information to explain the Tor side of things.  The actual operation
  of the non-Tor components is trickier than described below.

  In PrivCount, a Data Collector (DC, in this case a Tor relay) shares
  numeric data with N different Tally Reporters (TRs). (A Tally Reporter
  performs the summing and unblinding roles of the Tally Server and Share
  Keeper from experimental PrivCount.)

  All N Tally Reporters together can reconstruct the original data, but
  no (N-1)-sized subset of the Tally Reporters can learn anything about
  the data.

  (In reality, the Tally Reporters don't reconstruct the original data
  at all! Instead, they will reconstruct a _sum_ of the original data
  across all participating relays.)

  In brief, the system works as follow:

  To share data, for each counter value V to be shared, the Data Collector
  first adds Gaussian noise to V in order to produce V', uses (K,N) Shamir
  secret-sharing to generate N shares of V' (K<=N, K being the
  reconstruction threshold), encrypts each share to a different Tally
  Reporter, and sends each encrypted share to the Tally Reporter it
  is encrypted for.

  The Tally Reporters then agree on the set S of Data Collectors that sent
  data to all of them, and each Tally Reporter forms a share of the aggregate
  value by decrypting the shares it received from the Data Collectors in S
  and adding them together. The Tally Reporters then, collectively, perform
  secret reconstruction, thereby learning the sum of all the different
  values V'.

  The use of Shamir secret sharing lets us survive up to N-K crashing TRs.
  Waiting until the end to agree on a set S of surviving relays lets us
  survive an arbitrary number of crashing DCs. In order to prevent bogus
  data from corrupting the tally, the Tally Reporters can perform the
  aggregation step multiple times, each time proceeding with a different
  subset of S and taking the median of the resulting values.

  Relay subsets should be chosen at random to avoid relays manipulating their
  subset membership(s). If an shared random value is required, all relays must
  submit their results, and then the next revealed shared random value can
  be used to select relay subsets. (Tor's shared random value can be
  calculated as soon as all commits have been revealed. So all relay results
  must be received *before* any votes are cast in the reveal phase for that
  shared random value.)

  Below we describe the algorithm in more detail, and describe the data
  format to use.

3. The algorithm

  All values below are B-bit integers modulo some prime P; we suggest
  B=62 and P = 2**62 - 2**30 - 1 (hex 0x3fffffffbfffffff).  The size of
  this field is an upper limit on the largest sum we can calculate; it
  is not a security parameter.

  There are N Tally Reporters: every participating relay must agree on
  which N exist, and on their current public keys.  We suggest listing
  them in the consensus networkstatus document.  All parties must also
  agree on some ordering the Tally Reporters.  Similarly, all parties
  must also agree on some value K<=N.

  There are a number of well-known "counters", identified known by ASCII
  identifiers.  Each counter is a value that the participating relays
  will know how to count.  Let C be the number of counters.

3.1. Data Collector (DC) side

  At the start of each period, every Data Collector ("client" below)
  initializes their state as follows

      1. For every Tally Reporter with index i, the client constructs a
         random 32-byte random value SEED_i.  The client then generates
         a pseudorandom bitstream of using the SHAKE-256
         XOF with SEED_i as its input, and divides this stream into
         C values, with the c'th value denoted by MASK(i, c).

         [To divide the stream into values, consider the stream 8 bytes at a
         time as unsigned integers in network (big-endian) order. For each
         such integer, clear the top (64-B) bits.  If the result is less than
         P, then include the integer as one of the MASK(i, .) values.
         Otherwise, discard this 8-byte segment and proceed to the next

      2. The client encrypts SEED_i using the public key of Tally
         Reporter i, and remembers this encrypted value.  It discards

      3. For every counter c, the client generates a noise value Z_c
         from an appropriate Gaussian distribution. If the noise value is
         negative, the client adds P to bring Z_c into the range 0...(P-1).
         (The noise MUST be sampled using the procedure in Appendix C.)

         The client then uses Shamir secret sharing to generate
         N shares (x,y) of Z_c, 1 <= x <= N, with the x'th share to be used by
         the x'th Tally Reporter.  See Appendix A for more on Shamir secret
         sharing.  See Appendix B for another idea about X coordinates.

         The client picks a random value CTR_c and stores it in the counter,
         which serves to locally blind the counter.

         The client then subtracts (MASK(x, c)+CTR_c) from y, giving
         "encrypted shares" of (x, y0) where y0 = y-CTR_c.

         The client then discards all MASK values, all CTR values, and all
         original shares (x,y), all CTR and the noise value Z_c. For each
         counter c, it remembers CTR_c, and N shares of the form (x, y).

  To increment a counter by some value "inc":

      1. The client adds "inc" to counter value, modulo P.

         (This step is chosen to be optimal, since it will happen more
         frequently than any other step in the computation.)

         Aggregate counter values that are close to P/2 MUST be scaled to
         avoid overflow. See Appendix D for more information. (We do not think
         that any counters on the current Tor network will require scaling.)

  To publish the counter values:

      1. The client publishes, in the format described below:

         The list of counters it knows about
         The list of TRs it knows about
         For each TR:
            For each counter c:
                A list of (i, y-CTR_c-MASK(x,c)), which corresponds
                to the share for the i'th TR of counter c.
            SEED_i as encrypted earlier to the i'th TR's public key.

3.2. Tally Reporter (TR) side

  This section is less completely specified than the Data Collector's
  behavior: I expect that the TRs will be easier to update as we proceed.

  (Each TR has a long-term identity key (ed25519).  It also has a
  sequence of short-term curve25519 keys, each associated with a single
  round of data collection.)

   1. When a group of TRs receives information from the Data Collectors,
      they collectively chose a set S of DCs and a set of counters such
      that every TR in the group has a valid entry for every counter,
      from every DC in the set.

      To be valid, an entry must not only be well-formed, but must also
      have the x coordinate in its shares corresponding to the
      TR's position in the list of TRs.

   2. For each Data Collector's report, the i'th TR decrypts its part of
      the client's report using its curve25519 key.  It uses SEED_i and
      SHAKE-256 to regenerate MASK(0) through MASK(C-1).  Then for each
      share (x, y-CTR_c-MASK(x,c)) (note that x=i), the TR reconstructs the
      true share of the value for that DC and counter c by adding
      V+MASK(x,c) to the y coordinate to yield the share (x, y_final).

   3. For every counter in the set, each TR computes the sum of the
      y_final values from all clients.

   4. For every counter in the set, each TR publishes its a share of
      the sum as (x, SUM(y_final)).

   5. If at least K TRs publish correctly, then the sum can be
      reconstructed using Lagrange polynomial interpolation. (See
      Appendix A).

   6. If the reconstructed sum is greater than P/2, it is probably a negative
      value. The value can be obtained by subtracting P from the sum.
      (Negative values are generated when negative noise is added to small

   7. If scaling has been applied, the sum is scaled by the scaling factor.
      (See Appendix D.)

4. The document format

4.1. The counters document.

  This document format builds on the line-based directory format used
  for other tor documents, described in Tor's dir-spec.txt.

  Using this format, we describe a "counters" document that publishes
  the shares collected by a given DC, for a single TR.

  The "counters" document has these elements:

    "privctr-dump-format" SP VERSION SP SigningKey

       [At start, exactly once]

       Describes the version of the dump format, and provides an ed25519
       signing key to identify the relay. The signing key is encoded in
       base64 with padding stripped. VERSION is "alpha" now, but should
       be "1" once this document is finalized.

    "starting-at" SP IsoTime

       [Exactly once]

       The start of the time period when the statistics here were

    "ending-at" SP IsoTime

       [Exactly once]

       The end of the time period when the statistics here were

    "share-parameters" SP Number SP Number

       [Exactly once]

       The number of shares needed to reconstruct the client's
       measurements (K), and the number of shares produced (N),

    "tally-reporter" SP Identifier SP Integer SP Key

       [At least twice]

       The curve25519 public key of each Tally Reporter that the relay
       believes in.  (If the list does not match the list of
       participating Tally Reporters, they won't be able to find the
       relay's values correctly.)  The identifiers are non-space,
       non-nul character sequences.  The Key values are encoded in
       base64 with padding stripped; they must be unique within each
       counters document.  The Integer values are the X coordinate of
       the shares associated with each Tally Reporter.

    "encrypted-to-key" SP Key

       [Exactly once]

       The curve25519 public key to which the report below is encrypted.
       Note that it must match one of the Tally Reporter options above.

    "report" NL
      "----- END ENCRYPTED MESSAGE-----" NL

      [Exactly once]

      An encrypted document, encoded in base64. The plaintext format is
      described in section 4.2. below. The encryption is as specified in
      section 5 below, with STRING_CONSTANT set to "privctr-shares-v1".

    "signature" SP Signature

       [At end, exactly once]

       The Ed25519 signature of all the fields in the document, from the
       first byte, up to but not including the "signature" keyword here.
       The signature is encoded in base64 with padding stripped.

4.2. The encrypted "shares" document.

  The shares document is sent, encrypted, in the "report" element above.
  Its plaintext contents include these fields:

   "encrypted-seed" NL
      "----- END ENCRYPTED MESSAGE-----" NL

      [At start, exactly once.]

      An encrypted document, encoded in base64. The plaintext value is
      the 32-byte value SEED_i for this TR. The encryption is as
      specified in section 5 below, with STRING_CONSTANT set to

   "d" SP Keyword SP Integer

      [Any number of times]

      For each counter, the name of the counter, and the obfuscated Y
      coordinate of this TR's share for that counter.  (The Y coordinate
      is calculated as y-CTR_c as in 3.1 above.)  The order of counters
      must correspond to the order used when generating the MASK() values;
      different clients do not need to choose the same order.

5. Hybrid encryption

   This scheme is taken from rend-spec-v3.txt, section 2.5.3, replacing
   "secret_input" and "STRING_CONSTANT".  It is a hybrid encryption
   method for encrypting a message to a curve25519 public key PK.

     We generate a new curve25519 keypair (sk,pk).

     We run the algorithm of rend-spec-v3.txt 2.5.3, replacing
     "secret_input" with Curve25519(sk,PK) | SigningKey, where
     SigningKey is the DC's signing key.  (Including the DC's SigningKey
     here prevents one DC from replaying another one's data.)

     We transmit the encrypted data as in rend-spec-v3.txt 2.5.3,
     prepending pk.

Appendix A. Shamir secret sharing for the impatient

   In Shamir secret sharing, you want to split a value in a finite
   field into N shares, such that any K of the N shares can
   reconstruct the original value, but K-1 shares give you no
   information at all.

   The key insight here is that you can reconstruct a K-degree
   polynomial given K+1 distinct points on its curve, but not given
   K points.

   So, to split a secret, we going to generate a (K-1)-degree
   polynomial.  We'll make the Y intercept of the polynomial be our
   secret, and choose all the other coefficients at random from our

   Then we compute the (x,y) coordinates for x in [1, N].  Now we
   have N points, any K of which can be used to find the original

   Moreover, we can do what PrivCount wants here, because adding the
   y coordinates of N shares gives us shares of the sum:  If P1 is
   the polynomial made to share secret A and P2 is the polynomial
   made to share secret B, and if (x,y1) is on P1 and (x,y2) is on
   P2, then (x,y1+y2) will be on P1+P2 ... and moreover, the y
   intercept of P1+P2 will be A+B.

   To reconstruct a secret from a set of shares, you have to either
   go learn about Lagrange polynomials, or just blindly copy a
   formula from your favorite source.

   Here is such a formula, as pseudocode^Wpython, assuming that
   each share is an object with a _x field and a _y field.

     def interpolate(shares):
        for sh in shares:
           product_num = FE(1)
           product_denom = FE(1)
           for sh2 in shares:
               if sh2 is sh:
               product_num *= sh2._x
               product_denom *= (sh2._x - sh._x)

           accumulator += (sh._y * product_num) / product_denom

       return accumulator

Appendix B. An alternative way to pick X coordinates

   Above we describe a system where everybody knows the same TRs and
   puts them in the same order, and then does Shamir secret sharing
   using "x" as the x coordinate for the x'th TR.

   But what if we remove that requirement by having x be based on a hash
   of the public key of the TR?  Everything would still work, so long as
   all users chose the same K value.  It would also let us migrate TR
   sets a little more gracefully.

Appendix C. Sampling floating-point Gaussian noise for differential privacy


   When we add noise to a counter value (signal), we want the added noise to
   protect all of the bits in the signal, to ensure differential privacy.

   But because noise values are generated from random double(s) using
   floating-point calculations, the resulting low bits are not distributed
   evenly enough to ensure differential privacy.

   As implemented in the C "double" type, IEEE 754 double-precision
   floating-point numbers contain 53 significant bits in their mantissa. This
   means that noise calculated using doubles can not ensure differential
   privacy for client activity larger than 2**53:
     * if the noise is scaled to the magnitude of the signal using
       multiplication, then the low bits are unprotected,
     * if the noise is not scaled, then the high bits are unprotected.

   But the operations in the noise transform also suffer from floating-point
   inaccuracy, further affecting the low bits in the mantissa. So we can only
   protect client activity up to 2**46 with Laplacian noise. (We assume that
   the limit for Gaussian noise is similar.)

   Our noise generation procedure further reduces this limit to 2**42. For
   byte counters, 2**42 is 4 Terabytes, or the observed bandwidth of a 1 Gbps
   relay running at full speed for 9 hours. It may be several years before we
   want to protect this much client activity. However, since the mitigation is
   relatively simple, we specify that it MUST be implemented.


   Data collectors MUST sample noise as follows:
     1. Generate random double(s) in [0, 1] that are integer multiples of
        TODO: the Gaussian transform in step 2 may require open intervals
     2. Generate a Gaussian floating-point noise value at random with sigma 1,
        using the random double(s) generated in step 1.
     3. Multiply the floating-point noise by the floating-point sigma value.
     4. Truncate the scaled noise to an integer to remove the fractional bits.
        (These bits can never correspond to signal bits, because PrivCount only
        collects integer counters.)
     5. If the floating-point sigma value from step 3 is large enough that any
        noise value could be greater than or equal to 2**46, we need to
        randomise the low bits of the integer scaled noise value. (This ensures
        that the low bits of the signal are always hidden by the noise.)

        If we use the sample_unit_gaussian() transform in nickm/privcount_nm:
        A. The maximum r value is sqrt(-2.0*ln(2**-53)) ~=  8.57, and the
           maximal sin(theta) values are +/- 1.0. Therefore, the generated
           noise values can be greater than or equal to 2**46 when the sigma
           value is greater than 2**42.
        B. Therefore, the number of low bits that need to be randomised is:
               N = floor(sigma / 2**42)
        C. We randomise the lowest N bits of the integer noise by replacing them
           with a uniformly distributed N-bit integer value in 0...(2**N)-1.
     6. Add the integer noise to the integer counter, before the counter is
        incremented in response to events. (This ensures that the signal value
        is always protected.)

   This procedure is security-sensitive: changing the order of
   multiplications, truncations, or bit replacements can expose the low or
   high bits of the signal or noise.

   As long as the noise is sampled using this procedure, the low bits of the
   signal are protected. So we do not need to "bin" any signals.

   The impact of randomising more bits than necessary is minor, but if we fail
   to randomise an unevenly distributed bit, client activity can be exposed.
   Therefore, we choose to randomise all bits that could potentially be affected
   by floating-point inaccuracy.


   Although this analysis applies to Laplacian noise, we assume a similar
   analysis applies to Gaussian noise. (If we add Laplacian noise on DCs,
   the total ends up with a Gaussian distribution anyway.)

   TODO: check that the 2**46 limit applies to Gaussian noise.

   This procedure results in a Gaussian distribution for the higher ~42 bits
   of the noise. We can safely ignore the value of the lower bits of the noise,
   because they are insignificant for our reporting.

   This procedure is based on section 5.2 of:
   "On Significance of the Least Significant Bits For Differential Privacy"
   Ilya Mironov, ACM CCS 2012

   We believe that this procedure is safe, because we neither round nor smooth
   the noise values. The truncation in step 4 has the same effect as Mironov's
   "safe snapping" procedure. Randomising the low bits removes the 2**46 limit
   on the sigma value, at the cost of departing slightly from the ideal
   infinite-precision Gaussian distribution. (But we already know that these
   bits are distributed poorly, due to floating-point inaccuracy.)

   Mironov's analysis assumes that a clamp() function is available to clamp
   large signal and noise values to an infinite floating-point value.
   Instead of clamping, PrivCount's arithmetic wraps modulo P. We believe that
   this is safe, because any reported values this large will be meaningless
   modulo P. And they will not expose any client activity, because "modulo P"
   is an arithmetic transform of the summed noised signal value.


   We could round the encrypted value to the nearest multiple of the
   unprotected bits. But this relies on the MASK() value being a uniformly
   distributed random value, and it is less generic.

   We could also simply fail when we reach the 2**42 limit on the sigma value,
   but we do not want to design a system with a limit that low.

   We could use a pure-integer transform to create Gaussian noise, and avoid
   floating-point issues entirely. But we have not been able to find an
   efficient pure-integer Gaussian or Laplacian noise transform. Nor do we
   know if such a transform can be used to ensure differential privacy.

Appendix D. Scaling large counters

   We do not believe that scaling will be necessary to collect PrivCount
   statistics in Tor. As of November 2017, the Tor network advertises a
   capacity of 200 Gbps, or 2**51 bytes per day. We can measure counters as
   large as ~2**61 before reaching the P/2 counter limit.

   If scaling becomes necessary, we can scale event values (and noise sigmas)
   by a scaling factor before adding them to the counter. Scaling may introduce
   a bias in the final result, but this should be insignificant for reporting.

Appendix Z. Remaining client-side uncertainties

   [These are the uncertainties at the client side. I'm not considering
    TR-only operations here unless they affect clients.]

   Should we do a multi-level thing for the signing keys?  That is, have
   an identity key for each TR and each DC, and use those to sign
   short-term keys?

   How to tell the DCs the parameters of the system, including:
      - who the TRs are, and what their keys are?
      - what the counters are, and how much noise to add to each?
      - how do we impose a delay when the noise parameters change?
        (this delay ensures differential privacy even when the old and new
        counters are compared)
        - or should we try to monotonically increase counter noise?
      - when the collection intervals start and end?
      - what happens in networks where some relays report some counters, and
        other relays report other counters?
        - do we just pick the latest counter version, as long as enough relays
          support it?
          (it's not safe to report multiple copies of counters)

   How the TRs agree on which DCs' counters to collect?

   How data is uploaded to DCs?

   What to say about persistence on the DC side?