Kernels

Extracts one byte of an encrypted word at an encrypted position — random access inside a number, with neither the number nor the position visible.

Adds two encrypted integers modulo a third, without the intermediate sum overflowing. Modular arithmetic where every operand, including the modulus, is hidden.

Multiplies two encrypted integers modulo a third, with the full-width product never materializing in the clear or overflowing.

Shifts an encrypted integer left by an encrypted number of positions, which is multiplication by a power of two chosen by data nobody can read.

Shifts an encrypted integer right by an encrypted number of positions — integer division by a power of two, with the discarded bits genuinely discarded.

Extends a shorter signed encrypted integer to full width, preserving its sign. The step that lets narrow encrypted values participate in wide arithmetic.

Signed integer division of encrypted operands, truncating toward zero. Division is the operation encrypted arithmetic lacks outright, and signedness doubles the number of cases.

The remainder of signed encrypted division, taking its sign from the dividend. The companion of signed division, and wrong in a different way if the sign rule is missed.

Bitwise exclusive or over two encrypted integers. Trivial on a bit-level scheme, awkward on an arithmetic one, which is exactly why it is worth measuring.

Returns one when an encrypted value is zero and zero otherwise. The smallest possible predicate, and the one every encrypted branch is eventually built from.