Date: 2026-06-09

# Fun with bignum: how RSA encryption works

#### 2026-06-09

Primitive types such as `int` or `double` store numbers in exactly 4 or 8 bytes, with finite precision. This suffices for most applications, but cryptography requires arithmetic on very large numbers, without loss of precision. Therefore OpenSSL uses a bignum data type which holds arbitrary sized integers and implements all basic arithmetic and comparison operators such as `+`, `-`, `*`, `^`, `%%`, `%/%`, `==`, `!=`, `<`, `<=`, `>` and `>=`.

One special case, the modular exponent `a^b %% m` can be calculated using `bignum_mod_exp`, even when `b` is too large for calculating `a^b`.

    # create a bignum
    y <- bignum("123456789123456789")
    z <- bignum("D41D8CD98F00B204E9800998ECF8427E", hex = TRUE)

    # size grows
    print(y * z)

    ## [b] 34808613111804879213872650915812112647840354642904626774

    # Basic arithmetic
    div <- z %/% y
    mod <- z %% y
    z2 <- div * y + mod
    stopifnot(z2 == z)
    stopifnot(div < z)

RSA involves a public key and a private key. The public key should be known by everyone and is used for encrypting messages. Messages encrypted with the public key can only be decrypted in a reasonable amount of time using the private key. In RSA, this asymmetry is based on the practical difficulty of factoring the product of two large prime numbers.

### RSA key generation

An RSA key-pair is generated as follows (adapted from wikipedia):

- Choose two distinct prime numbers \(p\) and \(q\). Keep these secret.
- Compute the product \(n = p*q\). This \(n\) value is public and used as the modulus.
- Compute \(\phi(n) = (p − 1)(q − 1)\).
- Choose an integer \(e\) smaller than \(\phi(n)\) such that \(e\) and \(\phi(n)\) are coprime. OpenSSL always uses \(65537\).
- Compute a value for \(d\) such that \((d * e)\pmod{\phi(n)} = 1\).

OpenSSL has a key generator that does these things for us.

    (key <- rsa_keygen(512))

    ## [512-bit rsa private key]
    ## md5: 56d13176b9433a7b0e94e636082f34a0
    ## sha256: 22b7b77681e212a986fc1c88024f32451611943c4c1b3ffca0f0eeb4be5501c2

    (pubkey <- key$pubkey)

    ## [512-bit rsa public key]
    ## md5: 56d13176b9433a7b0e94e636082f34a0
    ## sha256: 22b7b77681e212a986fc1c88024f32451611943c4c1b3ffca0f0eeb4be5501c2

Usually we would use `rsa_encrypt` and `rsa_decrypt` to perform the encryption:

    msg <- charToRaw("hello world")
    ciphertext <- rsa_encrypt(msg, pubkey)
    rawToChar(rsa_decrypt(ciphertext, key))

    ## [1] "hello world"

Let’s look at how this works under the hood.

### How RSA encryption works

The `data` field of the private key extracts the underlying bignum integers:

    key$data

    ## $e
    ## [b] 65537
    ## $n
    ## [b] 8220572180691174690365109563195809776550640669371416576017667391532305163559220843082048532113435322703309296414856413945787645610101787407346792025345043
    ## $p
    ## [b] 92095806754546869965720467060907746779897417188405310267187228594059863944141
    ## $q
    ## [b] 89261090926762701027349093148357242388331163081135867176179715056015572137823
    ## $d
    ## [b] 1537570743105610866449418084832296813112558605446615261437425681453270621098108030908463515478664832675101395932314353360857544961403892733020955367978193
    ## $dp
    ## [b] 59068238416781713110748037176529231245650671133824081233058393986949544852953
    ## $dq
    ## [b] 17208811569947460632628222102468739148520137411388249107695358343115442482283
    ## $qi
    ## [b] 58348724747271803536277990085143146849068686963419541980147359471192234845414

You can verify that the equations above hold for this key. The public key is simply a subset of the key which only contains \(n\) and \(e\):

    pubkey$data

    ## $e
    ## [b] 65537
    ## $n
    ## [b] 8220572180691174690365109563195809776550640669371416576017667391532305163559220843082048532113435322703309296414856413945787645610101787407346792025345043

In order to encrypt a message into ciphertext we have to treat the message data as an integer. The message cannot be larger than the key size. For example convert the text `hello world` into an integer:

    m <- bignum(charToRaw("hello world"))
    print(m)

    ## [b] 126207244316550804821666916

To encrypt this message \(m\) into ciphertext \(c\) we calculate \(c = m^e\pmod n\). Using the public key from above:

    e <- pubkey$data$e
    n <- pubkey$data$n
    c <- (m ^ e) %% n
    print(c)

    ## [b] 1789923328114374468131923484514737257906600123140552011073770205710384092766696359324307746750343488431953727980598638871541410681572225522363914155763641

This number represents our encrypted message! It is usually exchanged using base64 notation for human readability:

    base64_encode(c)

    ## [1] "Iiz2hecVImQTqGc9R92iLvCqEndj8VHvyYe4bVeujI3yKjzSV8Y61CBDrEW+SHT1k3pOy0o+fKNJlV7WACCnuQ=="

The ciphertext can be decrypted using \(d\) from the corresponding private key via \(m = c^d \pmod{n}\). Note that `c^d` is too large to calculate directly so we need to use `bignum_mod_exp` instead.

    d <- key$data$d
    out <- bignum_mod_exp(c, d, n)
    rawToChar(out)

    ## [1] "hello world"

The only difference with the actual `rsa_encrypt` and `rsa_decrypt` functions is that these add some additional padding to the data.