I really hope someone can help me with these questions related to SPHINCS.
I am trying to understand the concept of “stateless” vs. “stateful” signature scheme which is the basis of SPHINCS. I went to read Chapter 6.4 of Goldreich’s Foundations of Cryptography, Volume 2. In page 546, image attached with this question, he gives a small example of one parent node x authenticating two children nodes, x_0 and x_1.
Here are my questions:
Q1: node x generates (S_x, V_x) pair where S_x is the Secret Key and V_x is the Public Key. Then node x generates two additional pairs (S_x0,V_x0 ) and (S_x1,V_x1) for nodes x_0 and x_1, respectively. Is there is any relationship between S_x and S_x0,S_x1 other than the fact that they use the same G(1^n) ?
Q2: the public key is only V_x?
Q3: signing only occurs with children nodes. Say I sign α with S_x0, thus the signature is going to be σ=f_Sx0(α),V_x0,auth_x0 where auth_x0= f_Sx(V_x0) ?
Q4: In SPHINCS, can we assume that all private keys are also generated with the same G(1^n) ? and thus we are always going to generate the same private keys given the same seed? Thus the “stateless” came from?
Thank you very much