How deeply does Hacking Cryptography cover modulo arithmetic?

Explains recurrence, inverses, negative remainders, and recovery equations, then stops short of broader modular-arithmetic theory.

Modulo arithmetic

As explained in Hacking Cryptography

Arithmetic in which values are identified according to their remainders modulo a specified integer.

Modulo arithmetic identifies values by their remainders after division by a specified modulus. Clock arithmetic gives the intuition: 10 plus 3 hours is 1 modulo 12. section 2.2.2

In a linear congruential generator, each state is updated as Xₙ₊₁ = (aXₙ + c) mod m. This reduction keeps outputs within the modulus and supports parameter-recovery attacks. section 2.2.2

Inverses and implementation hazards

The multiplicative inverse of 7 modulo 12 is 7 because 7 × 7 = 49 ≡ 1 mod 12. In Go, -16 % 7 returns -2 rather than 5, so code must normalize negative results. section 2.2.2

Equation 2.3

\[X_{n+1} = \left( a X_n + c \right)\bmod m \label{equation:lcg}\]

Use the recurrence equation to connect modular reduction with the state transition of a linear congruential generator. section 2.2.2

How Hacking Cryptography covers it

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Also explained in

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Real-World CryptographyModular arithmetic is arithmetic in which values are identified according to their remainders after division by a fixed modulus.
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Grokking Data StructuresModular arithmetic is arithmetic in which values are considered equivalent when they have the same remainder under a specified modulus.

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This explainer was generated by Manning's knowledge graph from the text of Hacking Cryptography. The section links point to the parts of the book it draws on.