The Ladder

Full Ladder Table

The complete ladder exposes each recomputed row and the final modular result. There is no giant decimal intermediate to trust.

highlighted = computed this step

Read the whole ladder

The full table shows prior result, square, optional multiply, and row result for each bit.

one row per bit\text{one row per bit}
Full ladder tableThe ladder rows are recomputed from pinned base, exponent, and modulus.Full ladder table - 7^13 mod 33stepbitpriorsquaremultiplyresult01117711716131320134skip4314161313

First row starts from one

The first row starts from 1, squares to 1, multiplies by base, and ends at 7.

1→71\to7
Full ladder tableThe ladder rows are recomputed from pinned base, exponent, and modulus.Full ladder table - 7^13 mod 33stepbitpriorsquaremultiplyresult01117711716131320134skip4314161313

Next one bit multiplies again

The next row carries prior result 7, squares to 16, multiplies, and ends at 13.

16⋅7 mod 33=1316\cdot7\bmod{}33=13
Full ladder tableThe ladder rows are recomputed from pinned base, exponent, and modulus.Full ladder table - 7^13 mod 33stepbitpriorsquaremultiplyresult01117711716131320134skip4314161313

The zero bit keeps the square

The zero-bit row squares prior result 13 to 4 and skips multiply, so the result stays 4.

skip multiply⇒4\text{skip multiply}\Rightarrow4
Full ladder tableThe ladder rows are recomputed from pinned base, exponent, and modulus.Full ladder table - 7^13 mod 33stepbitpriorsquaremultiplyresult01117711716131320134skip4314161313

Last one bit finishes

The final one-bit row squares 4 to 16, multiplies by base, and returns 13.

16⋅7 mod 33=1316\cdot7\bmod{}33=13
Full ladder tableThe ladder rows are recomputed from pinned base, exponent, and modulus.Full ladder table - 7^13 mod 33stepbitpriorsquaremultiplyresult01117711716131320134skip4314161313

Count the rows

The exponent has 4 bits, so the table has 4 rows.

rows=4\text{rows}=4
Full ladder tableThe ladder rows are recomputed from pinned base, exponent, and modulus.Full ladder table - 7^13 mod 33stepbitpriorsquaremultiplyresult01117711716131320134skip4314161313

Final result

The last row result is 13.

713 mod 33=137^{13}\bmod{}33=13
Full ladder tableThe ladder rows are recomputed from pinned base, exponent, and modulus.Full ladder table - 7^13 mod 33stepbitpriorsquaremultiplyresult01117711716131320134skip4314161313

Summary

The displayed result is recomputed from the pinned inputs and every row of the ladder.

ladder result verified\text{ladder result verified}
Full ladder tableThe ladder rows are recomputed from pinned base, exponent, and modulus.Full ladder table - 7^13 mod 33stepbitpriorsquaremultiplyresult01117711716131320134skip4314161313