# Introduction to openfhe.R: Fully Homomorphic Encryption in R

#### Balasubramanian Narasimhan

#### 2026-08-19

A statistician fitting a Cox model on hospital records, or a data scientist pooling case counts across institutions, often hits the same wall: the data cannot leave the institution that collected it. Privacy regulations, consent agreements, and plain institutional caution all conspire to keep the rows where they are. The usual workaround — summary statistics shipped between sites — works for some questions and falls short for others, especially anything that needs a joint likelihood, an iterative optimizer, or a model owner who is not allowed to see the raw features.

Fully Homomorphic Encryption (FHE) gives us another option. A data holder encrypts the data once, ships the encrypted copy to whoever is doing the computation, and that party can add, multiply, and otherwise manipulate the encrypted values without ever seeing the underlying numbers. Only the key holder can decrypt the final answer. The `openfhe.R` package wraps the OpenFHE C++ library so that R users can do this from the same environment they already use for analysis.

## A few terms from the cryptography side

The package’s function names mirror the underlying C++ API, which means a handful of crypto-world words show up in arguments and return values. None of them require a cryptography background to use; this is the dictionary you need to read the rest of this vignette.

- Plaintext. The encoded form of the numeric input — not raw R values but a packed object the encryption step knows how to read. You build one from a numeric vector with `make_packed_plaintext()` (for the integer schemes) or `make_ckks_packed_plaintext()` (for real-valued data). Think of it as the moral equivalent of `as.numeric()` for the encryption pipeline.
- Ciphertext. What you get after encrypting a plaintext. All the encrypted-arithmetic functions operate on ciphertexts and return ciphertexts; the only way back to numbers is `decrypt()` plus an unpacker like `get_packed_value()` or `get_real_packed_value()`.
- CryptoContext. A parameter bundle returned by `fhe_context()`. It pins down the scheme, the precision budget, and the layout of the slots. Almost every other call takes a context as the `cc =` argument.
- Public and secret key. Standard public-key roles. `key_gen()` returns a `KeyPair` with `@public` and `@secret` slots. Encryption uses the public key, decryption uses the secret key.
- Evaluation keys. Auxiliary keys that authorize specific encrypted-side operations — multiplication of two ciphertexts, summation across slots, rotation. They are generated alongside the secret key when you ask for them (`eval_mult = TRUE` to `key_gen()`, for example) and travel with the public key when you ship encrypted data.
- Slots. Internally an encrypted vector has many more positions than the data you put in — typically thousands. You use the first few and call `set_length()` on the decrypted result to trim back.

That is the whole vocabulary and the rest of this vignette describes the package facilities using only those terms.

## Five Short Demonstrations

The rest of this introduction walks five short demonstrations, in the order most R users will meet them:

1. Encrypted integer arithmetic. The simplest case — adding and multiplying integer vectors that the computing party never sees. Useful for counts, ranks, and exact tallies.
2. Encrypted real arithmetic. Most statistical work lives here: regression coefficients, likelihoods, gradients, sample variances. The CKKS scheme handles real numbers with controlled approximation error.
3. Boolean operations on encrypted bits. When the computation needs comparisons, branching, or per-bit logic, a different scheme (BinFHE) takes over.
4. Serialization. Distributed protocols need to send encrypted values and keys across processes or across the network. Every object in `openfhe.R` can be written to a file and read back.
5. Threshold (multi-party) decryption. No single party needs to hold the full secret key. Two or more parties can each hold a share and jointly decrypt the result — the foundation for federated computation across institutions.

Each step builds on the previous one. By the end you will have seen the full set of building blocks that the rest of the package’s vignettes — and the companion `homomorpheR` application package (version >= 1.0) — assemble into worked examples.

## Encrypted integer arithmetic

The BFV scheme handles exact integer arithmetic modulo a plaintext modulus. There is no rounding error: an addition of two encrypted vectors decrypts to exactly the same vector the cleartext addition would have produced.

A `CryptoContext` holds the scheme parameters; from it we generate a public/secret key pair (and the evaluation keys that authorize multiplication of encrypted values).

    library(openfhe.R)

    cc <- fhe_context("BFV",
      plaintext_modulus    = 65537,
      multiplicative_depth = 2
    )
    keys <- key_gen(cc, eval_mult = TRUE)

With keys in hand, encrypt two integer vectors. In `openfhe.R`, encryption takes a plaintext object — a packed, scheme-aware representation of the input — rather than a raw R vector, so the workflow is “pack, then encrypt”.

    x <- c(1, 2, 3, 4, 5, 6, 7, 8)
    y <- c(10, 20, 30, 40, 50, 60, 70, 80)

    ct_x <- encrypt(keys@public, make_packed_plaintext(cc, x), cc = cc)
    ct_y <- encrypt(keys@public, make_packed_plaintext(cc, y), cc = cc)

`ct_x` and `ct_y` are the encrypted versions of `x` and `y`. A computing party with no access to `keys@secret` can still add and multiply them.

    ct_sum  <- ct_x + ct_y
    ct_prod <- ct_x * ct_y

Decryption requires the secret key. The `set_length()` call trims the result to the original input length, since the underlying packed representation has many more slots than we used.

    result_sum  <- decrypt(ct_sum,  keys@secret, cc = cc)
    result_prod <- decrypt(ct_prod, keys@secret, cc = cc)
    set_length(result_sum,  8)
    set_length(result_prod, 8)

    sum_vec  <- get_packed_value(result_sum)
    prod_vec <- get_packed_value(result_prod)
    sum_vec
    #> [1] 11 22 33 44 55 66 77 88
    prod_vec
    #> [1]  10  40  90 160 250 360 490 640

The decrypted sum is exactly `x + y` and the decrypted product is exactly `x * y` — element by element, no approximation. The first entries are 11 and 10, matching `1 + 10` and `1 * 10`. BFV is the right tool whenever the analytic quantity is an integer count, an exact aggregate, or anything that should be reproduced bit-for-bit.

## Encrypted real arithmetic

Most statistical work involves real numbers. The CKKS scheme encrypts vectors of doubles and supports addition, multiplication, and scalar operations with bounded approximation error. The error is controlled by a `scaling_mod_size` parameter at context construction; for the values below it sits well under one part in a million.

    cc <- fhe_context("CKKS",
      multiplicative_depth = 1,
      scaling_mod_size     = 50,
      batch_size           = 8
    )
    keys <- key_gen(cc, eval_mult = TRUE)

Encrypt a real-valued vector and run a few operations on it.

    x  <- c(0.25, 0.5, 0.75, 1.0, 2.0, 3.0, 4.0, 5.0)
    ct <- encrypt(keys@public, make_ckks_packed_plaintext(cc, x), cc = cc)

    ct_doubled <- ct + ct
    ct_squared <- ct * ct
    ct_scaled  <- ct * 4.0

Decryption returns a packed plaintext; `get_real_packed_value()` extracts the underlying doubles.

    result <- decrypt(ct_doubled, keys@secret, cc = cc)
    set_length(result, 8)
    doubled_vec <- get_real_packed_value(result)
    doubled_vec
    #> [1]  0.5  1.0  1.5  2.0  4.0  6.0  8.0 10.0

The decrypted “double” of `x` matches `2 * x` to within the CKKS tolerance. The maximum elementwise error is

    max_err <- max(abs(doubled_vec - 2 * x))
    max_err
    #> [1] 1.527667e-13

1.53^{-13} — small enough to be invisible at the precisions reported in most statistical work. CKKS is the scheme to reach for when the encrypted computation is a likelihood, a gradient step, a regression coefficient, or anything else where a real-valued answer with bounded error is acceptable.

## Boolean operations on encrypted bits

Some computations need comparisons or branching — “is this encrypted value greater than zero?” — which do not translate cleanly into polynomial operations on real numbers. For those tasks OpenFHE provides a separate scheme (BinFHE) that operates on encrypted single bits, each a 0 or 1.

The unit of computation here is the logic gate: a basic Boolean operation — `AND`, `OR`, `NOT`, and a few relatives — that takes one or two bits in and returns one bit. Wiring gates together, the output of one feeding the input of the next, builds a Boolean circuit that computes a larger function; a comparator that tests whether one encrypted number exceeds another, or an adder, is just a fixed arrangement of gates. BinFHE evaluates one gate at a time on encrypted bits, and every gate evaluation runs an internal refresh, so a circuit can be made arbitrarily deep — at the cost of each gate being comparatively expensive.

    ctx <- bin_fhe_context(BinFHEParamSet$STD128, BinFHEMethod$GINX)
    sk  <- bin_key_gen(ctx)
    bin_bt_key_gen(ctx, sk)

Encrypt two bits and evaluate `AND` and `OR` on them.

    ct_a <- bin_encrypt(ctx, sk, 1L)
    ct_b <- bin_encrypt(ctx, sk, 0L)

    ct_and <- eval_bin_gate(ctx, BinGate$AND, ct_a, ct_b)
    ct_or  <- eval_bin_gate(ctx, BinGate$OR,
                            bin_encrypt(ctx, sk, 1L),
                            bin_encrypt(ctx, sk, 0L))

    and_bit <- bin_decrypt(ctx, sk, ct_and)
    or_bit  <- bin_decrypt(ctx, sk, ct_or)
    and_bit
    #> [1] 0
    or_bit
    #> [1] 1

`AND(1, 0)` decrypts to 0 and `OR(1, 0)` decrypts to 1, as expected. The `binfhe-boolean-circuits` vignette explores this scheme in much more detail — including comparators, arbitrary-function evaluation, and multi-input gates.

## Serialization

Federated and threshold protocols need to move encrypted values and keys between processes, machines, or institutions. Every object the package produces — contexts, keys, ciphertexts, evaluation keys — has a binary serialization format. The example below writes a few objects to a temporary directory and reads them back; in a real protocol the same calls would write to network sockets or storage buckets.

    tdir <- tempdir()

    fhe_serialize(cc, file.path(tdir, "context.bin"))
    fhe_serialize(keys@public, file.path(tdir, "pubkey.bin"))
    fhe_serialize(ct, file.path(tdir, "ciphertext.bin"))
    serialize_eval_keys(file.path(tdir, "mult_keys.bin"), "mult")

    cc2 <- fhe_deserialize(file.path(tdir, "context.bin"),    "CryptoContext")
    ct2 <- fhe_deserialize(file.path(tdir, "ciphertext.bin"), "Ciphertext")

`cc2` and `ct2` are independent objects that behave exactly like the originals — a different R session, possibly on a different machine, can decrypt or further process the encrypted data once it has the appropriate keys.

## Threshold decryption

The previous examples gave one party the full secret key. In a federated setting, no single party should hold enough key material to decrypt on their own. OpenFHE supports threshold decryption: two or more parties each generate a key share, encryption uses the joint public key, and decryption is a small protocol where each party contributes a partial result.

The flow below sketches the two-party version. Party A generates initial keys, Party B daisy-chains its own keys off A’s public key, and encryption is done under B’s public key (which carries both contributions).

    cc_mp <- fhe_context("BFV",
      plaintext_modulus    = 65537,
      multiplicative_depth = 2,
      features             = c(Feature$MULTIPARTY)
    )

    kp_a <- key_gen(cc_mp)
    kp_b <- multiparty_key_gen(cc_mp, kp_a@public)

    ct_mp <- encrypt(kp_b@public,
                     make_packed_plaintext(cc_mp, 1:8),
                     cc = cc_mp)

Decryption proceeds in three steps: a lead partial decryption from the first party, a main partial decryption from the second party, and a fusion step that combines the partials into the final plaintext.

    partial_a <- multiparty_decrypt_lead(cc_mp, kp_a@secret, ct_mp)
    partial_b <- multiparty_decrypt_main(cc_mp, kp_b@secret, ct_mp)
    result_mp <- multiparty_decrypt_fusion(cc_mp, partial_a, partial_b)
    set_length(result_mp, 8)
    mp_vec <- get_packed_value(result_mp)
    mp_vec
    #> [1] 1 2 3 4 5 6 7 8

The decrypted vector matches `1:8` — neither party alone could have produced it, but together they recovered the plaintext. This is the foundation that the worked applications build on. Cox regression across hospitals, consensus ADMM for distributed convex optimization, secure inference, and privacy-preserving aggregation all reduce to “encrypt under a joint key, run the optimizer over the encrypted channel, threshold-decrypt the final answer”.

## Further Exploration

Two further vignettes in `openfhe.R` itself dig into the schemes in depth:

- `ckks-bootstrapping` — what to do when a long CKKS computation runs out of multiplicative depth, including the multi-party variant that composes with the threshold flow above.
- `binfhe-boolean-circuits` — comparators, arbitrary-function evaluation, and the full set of BinFHE operations.

For worked statistical applications — Cox regression, MLE, CVXR-based consensus ADMM, encrypted regression, secure inference, and privacy-preserving aggregation, plus differentially private variants — see the companion application package `homomorpheR` (version >= 1.0). The two packages are siblings: `openfhe.R` provides the encryption primitives, `homomorpheR` hosts the statistical demos that run on top of them, ordered from simple aggregation through to advanced threshold and differential privacy workflows.