Author: Balasubramanian Narasimhan
Date: 2025-04-08

# Distributed Query Count using Non-Cooperating Parties

#### Balasubramanian Narasimhan

#### 2025-04-08

## Introduction

We demonstrate the use of non-cooperating parties to run a distributed query count computation using the `homomorpheR` package a simulated data set containing:

- `sex` (F, M) for female/male
- `age` between 40 and 70
- `bm` a biomarker

    set.seed(130)
    sample_size  <- c(60, 15, 25)
    query_data  <- local({
        tmp  <- c(0, cumsum(sample_size))
        start  <- tmp[1:3] + 1
        end  <- tmp[-1]
        id_list  <- Map(seq, from = start, to = end)
        lapply(seq_along(sample_size),
               function(i) {
                   id  <- sprintf("P%4d", id_list[[i]])
                   sex <- sample(c("F", "M"), size = sample_size[i], replace = TRUE)
                   age <- sample(40:70, size = sample_size[i], replace = TRUE)
                   bm <- rnorm(sample_size[i])
                   data.frame(id = id, sex = sex, age = age, bm = bm, stringsAsFactors = FALSE)
               })
    })

### Site 1

    str(query_data[[1]])

    ## 'data.frame':    60 obs. of  4 variables:
    ##  $ id : chr  "P   1" "P   2" "P   3" "P   4" ...
    ##  $ sex: chr  "F" "F" "F" "M" ...
    ##  $ age: int  43 67 66 59 63 52 55 46 43 60 ...
    ##  $ bm : num  1.252 -1.01 0.551 -0.379 0.284 ...

### Site 2

    str(query_data[[2]])

    ## 'data.frame':    15 obs. of  4 variables:
    ##  $ id : chr  "P  61" "P  62" "P  63" "P  64" ...
    ##  $ sex: chr  "M" "F" "M" "F" ...
    ##  $ age: int  53 64 45 61 55 65 65 51 53 47 ...
    ##  $ bm : num  0.698 -0.447 -0.224 1.086 1.188 ...

### Site 3

    str(query_data[[3]])

    ## 'data.frame':    25 obs. of  4 variables:
    ##  $ id : chr  "P  76" "P  77" "P  78" "P  79" ...
    ##  $ sex: chr  "M" "M" "M" "M" ...
    ##  $ age: int  69 70 41 63 42 57 43 55 68 46 ...
    ##  $ bm : num  -1.682 1.363 1.273 -0.68 0.625 ...

## Aggregated Query

If the data were all aggregated in one place, it would very simple to query it. Let us run a sample query on this aggregated data set for the condition `age < 50 & sex == 'F' & bm < 0.2`

    do.call(rbind, query_data) %>%
        filter(age < 50 & sex == 'F' & bm < 0.2) %>%
        nrow()

    ## [1] 11

## Distributed Computation

Assume now that the data `query_data` is distributed between three sites none of whom want to share actual data among each other or even with a master computation process. They wish to keep their data secret but are willing, together, to provide the sum of the total count. They wish to do this in a manner so that the master process is unable to associate the contribution to the likelihood from each site.

The overall query count for for the entire data is the sum of the counts at each site. How can this count be computed while preventing the master from knowing the individual contributions of each site?

We will use two non-cooperating parties, say NCP1 and NCP2, to accomplish this. These parties do not talk to each other, but do talk to the sites and the master process. Site \(i\) sends \(E(c_i + r_i)\) to NCP1 and \(E(c_i - r_i)\) to NCP2, where \(c_i\) is the actual count, \(E(c_i)\) denotes the encrypted value of \(c_i\) and \(r_i\) is a random quantity generated anew for each site. NCP1 can compute \(\sum_{i=1}^3E(c_i + r_i)\) and NCP2 can compute \(\sum_{i=1}^3E(c_i - r_i)\), but individually, neither has a handle on \(l = \sum_{i=1}^3 c_i\).

The master process can retrieve \(\sum_{i=1}^3E(c_i + r_i)\) and \(\sum_{i=1}^3E(c_i - r_i)\) from NCP1 and NCP2 respectively. Each is an encrypted value of the sum of counts from all sites, obfuscated by random terms, and hence is random to the master. However, the master using the associative and homomorphic properties of \(E(.)\), can compute:

\[ \sum_{i=1}^3E(c_i + r_i) +\sum_{i=1}^3E(c_i - r_i) = \sum_{i=1}^3E(c_i + r_i + c_i - r_i) = \sum_{i=1}^3E(2c_i) = E(2c) \]

since \(c = c_1 + c_2 + c_3\) is the grant total count. The master can now decrypt the result and obtain \(c\)!

This is pictorially shown below.

The red arrows show the master proposing a value \(\beta\) to each of the sites, which reply back to NCP1 and NCP2. The master then retrieves the values from NCP1 and NCP2 and sums them.

### A Modified Topology

The drawback of the above scheme is that channels of communication have to be established from each site to the master process and also to the two non-cooperating parties NCP1 and NCP2. If the number of participating sites in a computation changes, then both the master and NCP1 and NCP2 have to be made aware of the change.

It would be simpler if only NCP1 and NCP2 can talk to both the master and the sites. Such a situation would arise, for example, when the sites are all participating in a disease specific registry. The parties NCP1 and NCP2 would probably be set up once and any new site that has to be onboarded needs only to be known to P1 and P2. This has the added advantage of hiding the number of sites, which could even be 1!

Such a communication topology would mean that the \(\beta\) values have be funneled to the sites through NCP1 and NCP2 and that can be easily accomplished. The picture below shows this configuration and looks more complicated than it actually is.

To summarize, the modified scheme has several characteristics:

- The master only communicates with NCP1 and NCP2
- NCP1 and NCP2 are the only parties communicating with both the master and sites
- NCP1 and NCP2 are the only ones that know how many sites are participating
- New sites can be added and only NCP1 and NCP2 need to account for them while the master remains oblivous to the number of sites; so the scheme works even with one site
- It appears that there is unnecessary communication of the same information, i.e. \(\beta\) is being sent twice to each site from each of the NCP1 and NCP2. This is easily mitigated by engineering either by using a broker between NCP1 and NCP2, or the sites caching their results for a short period to avoid recomputation.

## Implementation

The above implementation assumes that the encryption and decryption can happen with real numbers which is not the actual situation. Instead, we use rational approximations using a large denominator, \(2^{256}\), say. In the future, of course, we need to build an actual library is built with rigorous algorithms guaranteeing precision and overflow/undeflow detection. For now, this is just an ad hoc implementation.

Also, since we are only using homomorphic additive properties, a partial homomorphic scheme such as the Paillier Encryption system will be sufficient for our computations.

We define classes to encapsulate our sites, non-cooperating parties and a master process.

### The Site Class

Our site class will compute the count on site data.

    Site <-
        R6::R6Class(
                "Site",
                private = list(
                    ## name of the site
                    name = NA,
                    ## local data
                    data = NA,
                    result_cache = NULL,
                    filterCondition = NA,
                    local_query_count = function() {
                        ## Check if value is cached
                        result  <- private$result_cache
                        if (is.null(result)) {
                            ## We need to run the query
                            pubkey <- self$pubkey
                            ## Generate random offset for int and frac parts
                            offset.int <- random.bigz(nBits = 256)
                            ## 2. Add to count
                            data  <- private$data
                            filter_expr  <- eval(parse(text = paste("rlang::expr(", private$filterCondition, ")")))
                            data %>%
                                dplyr::filter(!! filter_expr) %>%
                                nrow() ->
                                result.int
                            result  <- list(
                                int1 = pubkey$encrypt(result.int - offset.int),
                                int2 = pubkey$encrypt(result.int + offset.int)
                            )
                            private$result_cache  <- result
                        }
                        result
                    }
                ),
                public = list(
                    ## Common denominator for approximate real arithmetic
                    den = NA,
                    ## The master's public key; everyone has this
                    pubkey = NA,
                    initialize = function(name, data) {
                        private$name <- name
                        private$data <- data
                    },
                    setPublicKey = function(pubkey) {
                        self$pubkey <- pubkey
                    },
                    setDenominator = function(den) {
                        self$den = den
                    },
                    setFilterCondition = function (filterCondition) {
                        private$filterCondition  <- filterCondition
                    },
                    ## query count,
                    query_count = function(party) {
                        result  <- private$local_query_count()
                        if (party == 1) result$int1 else result$int2
                    }
                )
            )

### The Non-cooperating Parties Class

The non-cooperating parties can communicate with the sites. So they have methods for adding sites, passing on public keys from the master etc. The `query_count` method for this class merely calls each site to compute the result and adds them up before sending it on to the master, so that the master has no idea of the individual contributions.

    NCParty <-
        R6::R6Class(
                "NCParty",
                private = list(
                    ## name of the site
                    name = NA,
                    ## NC party number
                    number = NA,
                    ## filter condition
                    filterCondition = NA,
                    ## The master
                    master = NA,
                    ## The sites
                    sites = list()
                ),
                public = list(
                    ## The master's public key; everyone has this
                    pubkey = NA,
                    ## The denoinator for rational arithmetic
                    den = NA,
                    initialize = function(name, number) {
                        private$name <- name
                        private$number  <- number
                    },
                    setPublicKey = function(pubkey) {
                        self$pubkey <- pubkey
                        ## Propagate to sites
                        for (site in sites) {
                            site$setPublicKey(pubkey)
                        }
                    },
                    setDenominator = function(den) {
                        self$den <- den
                        ## Propagate to sites
                        for (site in sites) {
                            site$setDenominator(den)
                        }
                    },
                    setFilterCondition = function(filterCondition) {
                        private$filterCondition  <- filterCondition
                        ## Propagate to sites
                        for (site in sites) {
                            site$setFilterCondition(filterCondition)
                        }
                    },
                    addSite = function(site) {
                        private$sites  <- c(private$sites, list(site))
                    },
                    ## sum of all counts
                    query_count = function() {
                        pubkey  <- self$pubkey
                        results  <- lapply(sites, function(x) x$query_count(private$number))
                        ## Accumulate the integer and fractional parts
                        n  <- length(results)
                        enc_sum <- pubkey$encrypt(0)
                        for (result in results) {
                            enc_sum  <- pubkey$add(enc_sum, result)
                        }
                        enc_sum
                    }
                )
            )

### The Master Class

The master process

    Master  <-
        R6::R6Class(
                "Master",
                private = list(
                    ## name of the site
                    name = NA,
                    ## Private and public keys
                    keys = NA,
                    ## Non cooperating party 1
                    nc_party_1 = NA,
                    ## Non cooperating party 2
                    nc_party_2 = NA,
                    ## filter condition
                    filterCondition = NA
                ),
                public = list(
                    ## Denominator for rational arithmetic
                    den  = NA,
                    initialize = function(name, filterCondition) {
                        private$name <- name
                        private$keys <- PaillierKeyPair$new(1024) ## Generate new public and private key.
                        self$den <- gmp::as.bigq(2)^256  #Our denominator for rational approximations
                        private$filterCondition  <- filterCondition
                    },
                    setNCParty1  = function(site) {
                        private$nc_party_1 <- site
                        private$nc_party_1$setPublicKey(private$keys$pubkey)
                        private$nc_party_1$setDenominator(self$den)
                        private$nc_party_1$setFilterCondition(private$filterCondition)
                    },
                    setNCParty2  = function(site) {
                        private$nc_party_2 <- site
                        private$nc_party_2$setPublicKey(private$keys$pubkey)
                        private$nc_party_2$setDenominator(self$den)
                        private$nc_party_2$setFilterCondition(private$filterCondition)
                    },
                    ## Query count
                    query_count = function() {
                        pubkey  <- private$keys$pubkey
                        privkey  <- private$keys$getPrivateKey()
                        result1  <- private$nc_party_1$query_count()
                        result2  <- private$nc_party_2$query_count()
                        ## Sum it
                        enc_sum <- pubkey$add(result1, result2)
                        final_result <- as.integer(privkey$decrypt(enc_sum))
                        ## Since we 2c, we divide by 2.
                        final_result / 2
                    }
                )
            )

## Example

We are now ready to use our sites in the computation.

### 1. Create sites

    site1 <- Site$new(name = "Site 1", data = query_data[[1]])
    site2 <- Site$new(name = "Site 2", data = query_data[[2]])
    site3 <- Site$new(name = "Site 3", data = query_data[[3]])

    sites  <- list(site1 = site1, site2 = site2, site3 = site3)

### 2. Create Non-cooperating parties

    ncp1  <- NCParty$new("NCP1", 1)
    ncp2  <- NCParty$new("NCP1", 2)

We add sites to the non-cooperating parties.

    for (s in sites) {
        ncp1$addSite(s)
        ncp2$addSite(s)
    }

### 3. Create the master process

    master  <- Master$new(name = "Master",
                          filterCondition = "age < 50 & sex == 'F' & bm < 0.2")

We next connect the master to the non-cooperating parties.

    master$setNCParty1(ncp1)
    master$setNCParty2(ncp2)

At this point the communication graph has been defined between the master and non-cooperating parties and the non-cooperating parties and the sites.

### 4. Perform the Query

    cat(sprintf("Query Count is %d\n", master$query_count()))

    ## Query Count is 11

## References