bdcalc - a calculator for large natural numbers

bdcalc is a command-line calculator and mini-programming language that works with very large unsigned integers, the natural numbers ℕ = { 0,1,2,…} used to carry out computations in cryptography. The numbers can theoretically be of unlimited size.

bdcalc has built-in functions (see below) to carry out number theoretical computations such as modular exponentiation and modular inversion, as used in the RSA and Diffie-Hellman algorithms. You can generate random prime numbers of a given bit length and test for primality.

bdcalc provides a mini-programming environment with assignments to stored variables, conditional statements (if-then-else) and control loops (for, while, repeat). See the documentation for more details.

Example session

bdcalc lets you assign numbers to variables, do arithmetic and call functions in the way you would expect.

> a=99
99
> b=a*11
1089
> ? a+b
1188
> println("a =",a,"b =",b)
a = 99 b = 1089
> p=genprime(128)
187188849449410847765185120249130623559
> !p
0x8cd342d39544246b9743c7b68d5da247
> bitlen(p)
128
> ??isprime(p)
true
> x=1;y=2;z=x+y;?z
3
> for i in (1..15) do print(prime(i), " ") done; println("")
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47

Built-in Functions

Function	Returns	Remarks
`bitlen(X)`	the bit length l of the integer `X`	2^l−1 ≤ x < 2^l
`bytelen(X)`	the length L of integer `X` in bytes
`cbrt(X)`	the truncated integer cube root of `X`	⎣³√{x} ⎦
`compl(X,n)`	the bitwise NOT of the rightmost `n` bits of `X`
`fillbytes(n,b)`†	an integer of `n` bytes each set to `b mod 256`
`gcd(X,Y,Z,...)`	the greatest common divisor of `X,Y,Z,...`
`genprime(n)`	a random prime p of length exactly `n` bits	2ⁿ⁻¹ ≤ p < 2ⁿ
`getbit(X,n)`	value 0 or 1 of bit `n` of `X` =(b_l−1…b_n…b₁b₀)
`getbyte(X,n)`	value of byte `n` of `X`
`iif(B,X,Y)`	`X` if `B` is true, otherwise `Y`	[Note 1]
`isprime(X)`	`true` if `X` is prime, otherwise `false`	[Note 2]
`jacobi(X,M)`	the Jacobi symbol `(X \| M)` in {0,1,2}, 2 ⇒ −1	[Note 3]
`max(X,Y,Z,...)`	the maximum of `X,Y,Z,...`
`min(X,Y,Z,...)`	the minimum of `X,Y,Z,...`
`modexp(X,Y,M)`	`X` raised to the power `Y` modulo `M`	x^y mod m
`modinv(X,M)`	the inverse of `X` modulo `M`	x⁻¹ mod m
`modmul(X,Y,M)`	`X` multiplied by `Y` modulo `M`	xy mod m
`modpowof2(X,n)`	`X` with all bits cleared at positions ≥ `n`	x mod 2ⁿ
`pow(X,n)`	`X` raised to the power `n`	xⁿ
`prime(n)`	the `n`-th prime number	1 ≤ n ≤ 10⁴
`randbits(n)`	a random integer r of length at most `n` bits	0 ≤ r < 2ⁿ
`random(M)`	an integer r generated at random from [0, M-1]	0 ≤ r < M
`revbytes(X,[n])`	`X` as an `n`-byte integer with its byte order reversed
`setbit(X,n,b)`	`X` =(b_l−1…b_n…b₁b₀) with bit `n` set to value `b`	b ∈ {0,1}
`setbyte(X,n,b)`	`X` with byte `n` set to value `b mod 256`
`sha1(X,n)`	the SHA-1 hash of the rightmost `n` bits of `X`	[Note 4]
`sha256(X,n)`†	the SHA-256 hash of the rightmost `n` bits of `X`	[Note 4]
`sqrt(X)`	the truncated integer square root of `X`	⎣√x ⎦
`square(X)`	the square of `X`	x²

† new in version 2.2

For notes and examples of use, see the manual.

Displaying numbers in decimal and hexadecimal

To enter a number in hexadecimal, prefix it with "0x". It will display automatically in decimal. To display a number in hexadecimal, print with the "!" command.

> 0xdeadbeefcafebabebeddeddecadedeed
295990755076957304711832595911747231469
> ! 295990755076957304711832595911747231469
0xdeadbeefcafebabebeddeddecadedeed

Example scripts

qr_find.bdscr: How to find a quadratic residue by exhaustive search (source)
discretelog.bdscr: Compute discrete logarithm (source)
primes1000.bdscr: Write out the first 1000 primes (source)
rsa_make.bdscr: Make an RSA key pair and test it (source)
rsa_make_exact.bdscr: Make an RSA key pair of exact bit length (source)
rsa_quint.bdscr: Perform RSA calculation with private key in CRT quintuple form (source)
rsacrack.bdscr: Crack RSA if used incorrectly to three recipients (source)
rDSA.bdscr: Example of the rDSA algorithm from ANSI X9.31 (source)
dsa_test.bdscr: Example of DSA from Appendix 5 FIPS PUB 186-2 (source)
rsa_quint.bdscr: Perform RSA calculation with private key in CRT quintuple form (source)
dh_gen.bdscr: Generate domain parameters for Diffie-Hellman (source)
dh_keyexch.bdscr: Perform Diffie-Hellman key exchange using parameters generated above (source)
poly1305.bdscr: Poly1305 Example and Test Vector (source)

Documentation

bdcalc manual in PDF

bdcalc manual in HTML

Download

Windows: Download the latest bdcalc installation program now. Use either

bdcalc-install.exe (466 kB) [md5=c7411b94c28c4897f3f764e0204237cc] or
bdcalc-install.zip (447 kB) [md5=e41d6cdc5ef99925127014e8ff07069a].

Unzip the zip file and run the setup.exe program inside it, or download the exe program directly and run it. The latest version is 2.2.1.0 released 26 July 2016.

Windows binary: bdcalc is a simple stand-alone Windows EXE file just 135 kB in size. If you just want the plain bdcalc.exe file to install yourself, plus the help and sample scripts, download bdcalc-files.zip (381 kB) [md5=1d594263ec22d75f6a72d6ea1591f764].

Linux x86 binary: Compiled with static library included. bdcalc-linux.x86.zip (529 kB) [md5=29dd079e2c5dc1b60c12893de65a3c40] (install notes)

Mac OSX binary: bdcalc-osx.zip (340 kB) [md5=c03926a7936c7970bef85dedf5c40e67] (install notes)

Please note it is a breach of copyright to put a copy of these installation files on another server or to distribute them in any manner except by providing a link to this page.

bdcalc is free software. Please read the licence conditions.

What's new in v2.2?

New version 2.2 released 17 June 2016. This fixes up a few annoyances we had with print and help and adds a couple of new functions.

Changed behaviour of print and println to insert spaces between arguments.

Added setsep command to change print separation character.

> println(10,20,30)   # New behaviour in v2.2
10 20 30
> setsep ''
> println(10,20,30)   # Old behaviour
102030
> setsep ', '
> println(10,20,30)   # Fancy stuff
10, 20, 30

Removed requirement to put quotes around arguments for help.

> help modexp   # Old behaviour - should be ``help 'modexp'``
ERROR: syntax error (incomplete line)

> help modexp   # New behaviour - no quotes required
`modexp(X,Y,M)` returns X raised to the power Y modulo M

Changed error warning from ERROR: to BDCALC ERROR:. We found ourselves opening a bdcalc interactive console, leaving it for a while, and then wondering why we couldn't type a normal command. Having the word "BDCALC" there gives you a clue.

C:\> bdcalc
BDCALC: v2.2.0 <www.di-mgt.com.au/bdcalc.html>
Enter commands: type `quit` to finish or `help` for help.
> a = 123456789
123456789
> # leave this console open and forget...
> dir file
BDCALC ERROR: syntax error at symbol 'file'
> dir file
BDCALC ERROR: syntax error at symbol 'file'
> # WTF why isn't the dir command working?

Added new functions sha256 and fillbytes. Here's a cute example to compute the SHA-256 digest of one million repetitions of the character "a" (ASCII 0x61). Ref: Test vectors for SHA-1, SHA-2 and SHA-3. But don't try and print out b - it's a huge number with 2.4 million digits! Make sure you use the verbose off statement to avoid the default behaviour of displaying the last result.
```
> verbose off
> b = fillbytes(1000000, 0x61)
> ? bytelen(b)
1000000
> !sha256(b, bytelen(b)*8)
0xcdc76e5c9914fb9281a1c7e284d73e67f1809a48a497200e046d39ccc7112cd0

> # wow, what does b look like?
> ? b     # ... OH NO! ...
```
Made second argument optional for revbytes(X,n). If the n argument is omitted it defaults to bytelen(X). Be careful, there are corner cases where this may not be what you want.
```
> showhex on   # New feature in v2.2 to display last result in hex not decimal
> a = 0xdeadbeef42
0xdeadbeef42
> revbytes(a)
0x42efbeadde
```
Added showhex statement to switch between hexadecimal and decimal displays of the result in interactive mode. See example above.
26 July 2016: Update v2.2.1 released. This fixes a stack overflow problem with large nested loops (thanks, Michael).

Acknowledgements

bdcalc uses code derived from:
- The source code in flex & bison: Text Processing Tools by John Levine. Copyright (c) 2009, Taughannock Networks.
- C99-snprintf by Patrick Powell (1995) and Holger Weiss (2008)
Thanks to lexxmark and the team behind Win flex-bison.
The installation program was created using NSIS (Nullsoft Scriptable Install System) available from http://nsis.sourceforge.net/.
The executable was compressed using UPX, the Ultimate Packer for eXecutables available from http://upx.sourceforge.net.

Contact

Any comments, feedback, questions: please send us a message.

Version 1 of bdcalc was first released on 12 May 2013 and version 2.2 on 17 June 2016 (v2.2.1 on 26 July 2016). This page last updated 21 December 2019