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

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?

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 $$d \bmod (p-1)$$.