3-round DES: what bit do we find? (Regarding matsui's Paper on Linear Cryptanalysis of DES)

Question

In Matsui's paper, Section 5 ‘Linear Approximation of DES Cipher’, Matsui gives two similar equations: Eq. (6) for the first round and Eq. (7) for the third round. The two equations give an approximation of bit $K_1[22]$ and bit $K_3[22]$. Then he derives in Eq. (8) from Eqs. (6) and (7) an approximation of $K_1[22] \oplus K_3[22].$

From what I understand, the 3-round DES linear approximation is supposed to give an approximation of only one bit of the key, but here I see two bits from the partial key.

Which bit of the actual key does this approximation refer to?

Answer 1

The equation in (8) is obtained by summing (6) and (7), and is an approximation for the bit $$ K_1[22]\oplus K_3[22]. $$ The sum of two key bits is still a bit.