MODULAR ARITHMETIC / SQUARE AND MULTIPLY
Calculate a nonnegative integer power residue with bounded exact square-and-multiply arithmetic.
METHOD / WORKED EXAMPLE
Calculate a nonnegative integer power residue with bounded exact square-and-multiply arithmetic. The page exposes the recurrence, factorization, or residue evidence used to obtain the bounded exact result.
WORKED DEFAULT
For 7^128 modulo 13, repeated squaring and reduction return the exact residue 3 without constructing 7^128.
READ THE RESULT
The result is the least nonnegative representative of the power's residue class, not the unreduced ordinary power.
ASSUMPTIONS AND LIMITS
The modulus cap keeps intermediate products exactly representable; this page is not a constant-time security implementation.
COMMON QUESTIONS
Normalize the base, scan the nonnegative exponent in binary, square the factor, multiply selected powers, and reduce modulo m after every operation. Exponent zero returns one modulo m, including residue zero only for modulus one, which this page excludes. Inputs must be whole numbers inside the displayed safe bound. The calculator does not round decimals into the domain or silently substitute a different convention, so verify each entered integer before interpreting the exact result.
The result is the least nonnegative representative of the power's residue class, not the unreduced ordinary power. A direct check with a small exponent can multiply and reduce manually after every step. Use the displayed factors, recurrence step, or modular residue as an independent check. Exact integer output means the implemented arithmetic has no decimal approximation within its bound; it does not prove an unrelated theorem or an unstated combinatorial model.
The modulus cap keeps intermediate products exactly representable; this page is not a constant-time security implementation. Production cryptography additionally requires constant-time code, vetted libraries, approved parameters, and secure key handling. The bound keeps browser work predictable and every returned integer within JavaScript's exact safe range. Larger inputs need arbitrary-precision software, explicit resource controls, and independent verification rather than treating an overflowed floating-point value as exact.
RELATED TOOLS
Use boundary
Normalize the base, scan the nonnegative exponent in binary, square the factor, multiply selected powers, and reduce modulo m after every operation. The workspace preserves the bounded integer method and verification evidence beside the result.
r = a^e mod m by repeated squaring.