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conflicting definitions for dP / dQ and exponent1 / exponent2 in PKCS 1?

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neubert

In Section 2 dP and dQ are defined thusly:

      dP             p's CRT exponent, a positive integer such that

                       e * dP == 1 (mod (p-1))

      dQ             q's CRT exponent, a positive integer such that

                       e * dQ == 1 (mod (q-1))

In Appendix A.1.2 we have this:

   o  exponent1 is d mod (p - 1).

   o  exponent2 is d mod (q - 1).

I believe exponent1 = dP and exponent2 = dQ but they're using different formulas. If the formulas are equivalent it is not immediately obvious to me how.

The first formula leads to $ d_p = e^{-1} \mathrm{mod}~(p−1)$ but idk how to get the second formula from that, even when taking the $ e \cdot d \equiv 1 \pmod{\lambda(n)} $ identity into consideration.

Maybe I'm mistaken in my belief that exponent1 = dP and exponent2 = dQ? Maybe the RFC is in error? Maybe the formulas are the equivalentand I'm just not seeing it?

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rsa
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Gilles 'SO- stop being evil' avatar Crypto
cn flag
Gilles 'SO- stop being evil'

The formulas are equivalent.

From §3.2: $e \cdot d \equiv 1 \pmod{\lambda(n)}$, i.e. $e \cdot d - 1 \equiv 0 \pmod{\lambda(n)}$, i.e. $\lambda(n)$ divides $e \cdot d - 1$.

From §3.1: $\lambda(n) = \mathrm{lcm}(p-1, q-1)$, so $p-1$ divides $\lambda(n)$. Therefore $p-1$ divides $e \cdot d - 1$, i.e. $e \cdot d - 1 \equiv 0 \pmod{p-1}$, i.e. $e \cdot d \equiv 1 \pmod{p-1}$. Thus $d \bmod (p-1)$ is the inverse of $e$ modulo $p-1$.

Conversely, suppose $e \cdot x \equiv 1 \pmod{p-1}$ and $0 \le x \lt p-1$. Then $x$ is the inverse of $e$ in $\mathbb{Z}_{p-1}$, which is unique, so $x \equiv d \pmod{p-1}$. Since I chose $x$ in the range $[0, p-1]$, it is $d \bmod (p-1)$.

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