# Large Sized Integer Values Class `"bigz"` encodes arbitrarily large integers (via GMP). A simple S3 class (internally a `raw` vector), it has been registered as formal (S4) class (via `setOldClass`), too. ```r as.bigz(a, mod = NA) NA_bigz_ ## S3 method for class 'bigz' as.character(x, b = 10, ...) is.bigz(x) ## S3 method for class 'bigz' is.na(x) ## S3 method for class 'bigz' print(x, quote=FALSE, initLine = is.null(modulus(x)), ...) c_bigz(L) ``` ## Arguments - `a`: either `integer`, `numeric` (i.e., `double`) or `character` vector. If character: the strings either start with `0x` for hexadecimal, `0b` for binary, `0` for octal, or without a `0*` prefix for decimal values. Formatting errors are signalled as with `stop`. - `b`: base: from 2 to 36 - `x`: a big integer number (vector), of class `"bigz"`. - ``...``: additional arguments passed to methods - `mod`: an integer, numeric, string or bigz of the internal modulus, see below. - `quote`: (for printing:) logical indicating if the numbers should be quoted (as characters are); the default used to be `TRUE` (implicitly) till 2011. - `initLine`: (for printing:) logical indicating if an init ial line (with the class and length or dimension) should be printed. The default prints it for those cases where the class is not easily discernable from the print output. - `L`: a `list` where each element contains `"bigz"` numbers, for `c_bigz()`, this allows something like an `sapply()` for `"bigz"` vectors, see `sapplyZ()` in the examples. ## Returns An object of (S3) class `"bigz"`, representing the argument (`x` or `a`). ## Details Bigz's are integers of arbitrary, but given length (means: only restricted by the host memory). Basic arithmetic operations can be performed on bigzs as addition, subtraction, multiplication, division, modulation (remainder of division), power, multiplicative inverse, calculating of the greatest common divisor, test whether the integer is prime and other operations needed when performing standard cryptographic operations. For a review of basic arithmetics, see `add.bigz`. Comparison are supported, i.e., `"=="`, `"!="`, `"<"`, `"<="`, `">"`, and `">="`. `NA_bigz_` is computed on package load time as `as.bigz(NA)`. Objects of class `"bigz"` may have a modulus , accessible via `modulus()`, currently as an attribute `mod`. When the object has such a modulus $m$, arithmetic is performed **modulo m**, mathematically within the ring Z/mZ . For many operations, this means ``` result <- mod.bigz(result, m) ## == result %% m ``` is called after performing the arithmetic operation and the result will have the attribute `mod` set accordingly. This however does not apply, e.g., for `/`, where $a / b = a b^(-1)$ and $b^(-1)$ is the **multiplicate inverse** of $b$ with respect to ring arithmetic, or `NA` with a warning when the inverse does not exist. The warning can be turned off via `options("gmp:warnModMismatch" = FALSE)` Powers of bigzs can only be performed, if either a modulus is going to be applied to the result bigz or if the exponent fits into an integer value. So, if you want to calculate a power in a finite group (modulo c ), for large $c$ do not use `a ^ b %% c`, but rather `as.bigz(a,c) ^ b`. The following rules for the result's modulus apply when performing arithmetic operations on `bigz`s: * If none of the operand has a modulus set, the result will not have a modulus. * If both operands have a different modulus, the result will not have a modulus, except in case of `mod.bigz`, where the second operand's value is used. * If only one of the operands has a modulus or both have a common (the same), it is set and used for the arithmetic operations, except in case of `mod.bigz`, where the second operand's value is used. ## Note ``` x <- as.bigz(1234567890123456789012345678901234567890) ``` will not work as converts the number to a double, losing precision and only then convert to a `"bigz"` object. Instead, use the syntax ``` x <- as.bigz("1234567890123456789012345678901234567890") ``` ## References The GNU MP Library, see [https://gmplib.org](https://gmplib.org) ## Author(s) Immanuel Scholz ## Examples ```r ## 1+1=2 a <- as.bigz(1) a + a ## Two non-small Mersenne primes: two <- as.bigz(2) p1 <- two^107 -1 ; isprime(p1); p1 p2 <- two^127 -1 ; isprime(p2); p2 stopifnot( is.na(NA_bigz_) ) ## Calculate c = x^e mod n ## -------------------------------------------------------------------- x <- as.bigz("0x123456789abcdef") # my secret message e <- as.bigz(3) # something smelling like a dangerous public RSA exponent (n <- p1 * p2) # a product of two primes as.character(n, b=16)# as both primes were Mersenne's.. ## recreate the three numbers above [for demo below]: n. <- n; x. <- x; e. <- e # save Rev <- function() { n <<- n.; x <<- x.; e <<- e.} # first way to do it right modulus(x) <- n c <- x ^ e ; c ; Rev() # similar second way (makes more sense if you reuse e) to do it right modulus(e) <- n c2 <- x ^ e stopifnot(identical(c2, c), is.bigz(c2)) ; Rev() # third way to do it right c3 <- x ^ as.bigz(e, n) ; stopifnot(identical(c3, c)) # fourth way to do it right c4 <- as.bigz(x, n) ^ e ; stopifnot(identical(c4, c)) # WRONG! (although very beautiful. Ok only for very small 'e' as here) cc <- x ^ e %% n cc == c # Return result in hexa as.character(c, b=16) # Depict the "S4-class" bigz, i.e., the formal (S4) methods: if(require("Rmpfr")) # mostly interesting there showMethods(class="bigz") # an sapply() version that works for big integers "bigz": sapplyZ <- function(X, FUN, ...) c_bigz(lapply(X, FUN, ...)) # dummy example showing it works (here): zz <- as.bigz(3)^(1000+ 1:999) z1 <- sapplyZ(zz, function(z) z^2) stopifnot( identical(z1, zz^2) ) ```

biginteger function