Understanding CKKS Moduli

FHE Questions
1

I print my fresh ciphertext moduli that has depth 5, and it shows:

Element 0 params:
  q0 = 2147352577
  q1 = 1071513601
  q2 = 1070727169
  q3 = 1077477377
  q4 = 1073479681
  q5 = 1073872897
  q6 = 786433
Element 1 params:
  q0 = 2147352577
  q1 = 1071513601
  q2 = 1070727169
  q3 = 1077477377
  q4 = 1073479681
  q5 = 1073872897
  q6 = 786433

My question:

  • why is q6 much smaller than others?
  • I remember a few months ago, someone who has a better understanding of FHE than me told me that the CKKS moduli should start small and then increase ( I am not sure what his reference is). But the moduli here is decreasing instead of increasing. I am unsure now what he was talking about or if there is any different consideration in OpenFHE. Can someone please help explain this?

Thank you

2

This should answer your question.

The moduli q_1 to q_5 alternate between values above and below the value 2^{scaleModSize}, with one modulus larger and one smaller.

3

Thank you! for my understanding of the referenced discussion:

  • q_6 is the special auxiliary modulus, which is why it has smaller bit
  • with depth=5 CKKS moduli is q_0 \dots q_5