exponentiation.exponent module

Module containing all logic for securely performing exponentiation using the MPyC framework.

exponentiation.exponent.convert_exponents_to_other_base(old_base, new_base, exponents)

Convert the exponent of a given exponentiation for a given new base so that the result is the same.

More specifically, the following invariant is preserved: \(base_{old}\^exponent == base_{new}\^return_value\)

Parameters:

• old_base (Union[int, float]) – Original base of the exponentiation.

• new_base (float) – Desired base of the exponentiation.

• exponents (List[TypeVar(SecureNumberType, bound= SecureNumber)]) – List of original exponents.

Return type:

List[TypeVar(SecureNumberType, bound= SecureNumber)]

Returns:

List of exponents that satisfy the stated invariant.

exponentiation.exponent.maximal_exponent(base, stype, bits_buffer=0)

Returns largest exponent x such that base**x fits within the precision of the field.

Parameters:

• base (float) – base of the exponentiation

• stype (Union[Type[SecureFixedPoint], Type[SecureInteger]]) – SecureNumberType class

• bits_buffer (int) – Introduce a buffer so that base**x * 2 ** bits_buffer fits in the precision of the field

Return type:

int

Returns:

largest exponent x such that base**x fits within the precision of the given types field

exponentiation.exponent.minimal_exponent(base, stype, bits_buffer=0)

Returns smallest exponent x such that base**x fits within the precision of the field.

Parameters:

• base (float) – base of the exponentiation

• stype (Union[Type[SecureFixedPoint], Type[SecureInteger]]) – SecureNumberType class

• bits_buffer (int) – Introduce a buffer so that base**x * 2 ** -bits_buffer fits in the precision of the field (only for SecureFixedPoint)

Return type:

int

Returns:

smallest exponent x such that base**x fits within the precision of the given types field

exponentiation.exponent.secure_pow(base, exponents, trunc_to_domain=False, lower=None, upper=None, bits_buffer=0, bit_precision=None)

Secure exponentiation.

Computes approximate secret-sharing of base**x for all given secret-shared x in [lower, upper]. May additionally enforce that x is truncated to fit in [lower, upper].

The exponentiation is approximate since it splits the computation into an exact integral computation and a approximate non-integral computation. The approximated part leverages Taylor series.

If exponents are truncated to the feasible domain, for a exponent \(x\), the function returns \(base^{upper}\) if \(x > upper\), \(base^x\) if \(x\in [lower, upper]\), and \(0\) if \(x < lower\).

Parameters:

• base (float) – Base of the exponentiation.

• exponents (Union[SecureInteger, SecureFixedPoint, TypeVar(SecureObjectsContainerTV, List[SecureInteger], List[SecureFixedPoint])]) – Exponents of the base.

• lower (Optional[float]) – Lower bound of exponents range, maximized for type(exponents[0]) if None.

• upper (Optional[float]) – Upper bound of exponents range, maximized for type(exponents[0]) if None.

• bits_buffer (int) – Number of bits buffer between the actual and maximal values of upper and lower. Can be used to prevent overflows in subsequent computations without the need of computing lower and upper manually.

• trunc_to_domain (bool) – Truncates exponents so that they fall in the given range at cost of additional resources. Additionally ensures that base**[lower] evaluates to [0].

• bit_precision (Optional[int]) – Bound on relative error of approximation, set to half of fractional bits if None.

Return type:

Union[SecureInteger, SecureFixedPoint, TypeVar(SecureObjectsContainerTV, List[SecureInteger], List[SecureFixedPoint])]

Returns:

Secret-shared value of the exponentiation for every exponent provided.

Raises:

• AssertionError – Combination of parameters results in empty feasible domain.

• NotImplementedError – Size of domain is greater than twice the size of the maximum upper bound.

• TypeError – Secure type of exponents cannot deal with provided non-integer base.

• ValueError – Base equals zero.