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$powmod performs integer exponentiation over a modulus.

Details

For integers (B,E,M) the remainder of B^E modulo M is a result that is a non-negative integer less than M, and can obtain the result even when the E exponent is large. Should be compatible with the Powermod function in Wolfram Alpha

Synopsis

$powmod(B,E,M)

• Note: Unlike the % percent operator in $calc the result cannot be negative. If 'B' is negative, the result is calculated after casting 'B' to positive by adding 1-or-more multiple of the M modulus, so (-1,2,11) is the same as (10,2,11).
• If the exponent E is negative, the result of (B,-E,M) is obtained from ($modinv(B,M),+E,M), so $powmod(B,-1,M) is the same as $modinv(B,M)
• Inputs should be integer, so fractions in any parameter are ignored.
• While most math identifiers support numbers as large as 2^53, care should be used for this function if using a modulus greater than $sqrt($calc(2^53)), because this algorithm uses an intermediate result which can potentially be as large as (M-1)^2. When in doubt, either use this function where a %var.bf name is involved or if /bigfloat ON mode is enabled.
• In spite of using a shortcut which allows obtaining results involving large E exponents without calculating the full result of B^E, the result can be very slow for very large numbers. The main factors affecting the time for non-trivial (B,E,M) are:

1. Bit length of M
2. Bit length of E
3. Number of '1' bits returned from $base(E,10,2)

Parameters

• B - Integer Base of the exponentiation
• E - Integer Exponent
• M - Integer Modulus over which the exponentiation is found

Properties

None

Example

//var -s %x $rand(3,2026) , %A $powmod(2,%x,2027) , %y $rand(3,2026) , %B $powmod(2,%y,2027) | echo -a $powmod(%A,%y,2027) = $powmod(%B,%x,2027)

Compatibility

Added: mIRC v7.72
Added on: 27 Nov 2022
Note: Unless otherwise stated, this was the date of original functionality.
Further enhancements may have been made in later versions.

See also

• $modinv
• $gcd
• $lcm