GCM Create M tables and R

Question

I am trying to optimize GCM multiplication. This PDF explains GCM:

The Galois/Counter Mode of Operation (GCM)

The algorithms are in section 4.1

In algorithm 3 I have to multiply table M [128] with the element P that represents the polynomial α. Formulas (3) and (4) appear on page 9. From what I can understand, this would be:

**if V127 =0 then V ← rightshift(V )

else V ← rightshift(V ) ⊕ R**

The polynomial f would be 0xe1, no?

But in algorithm 3 I do not understand this: M[i] ← M[2i] · P. What would be the value of the element P if it corresponds to the polynomial α?

Hopefully, someone knows some code of the implementation of the tables that I can read. I have not found any code that can help me understand all of this.

Does this answer your question? GCM cipher M0 tables : semantic questions on how to implement GCM — kelalaka, Feb 05 '21 at 16:22
Please indicate if the given answer or previous Q/A's answered your question. — Maarten Bodewes, Feb 18 '21 at 23:38
The document cited in the question had at least one update, formerly hosted by NIST, archived there. Some of the changes seem to be significant. It is still cited by the current spec. — fgrieu, Feb 27 '23 at 10:38

Efimster · Answer 1 · 2021-02-06T21:28:49.837

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The polynomial f would be 0xe1, no?

it is $e1 \mathbin \|0^{120}$ i.e. 0xe1000000000000000000000000000000

What would be the value of the element P, if it corresponds to the polynomial α?

it is 2 reversed i.e. 0x40000000000000000000000000000000

edited Feb 06 '21 at 21:28

answered Feb 05 '21 at 16:00

Efimster

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Could you turn the belief into reality? – kelalaka Feb 05 '21 at 16:42
@kelalaka updated – Efimster Feb 06 '21 at 21:32
and references? – kelalaka Feb 06 '21 at 21:33
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@kelalaka The response based on my own understanding of the paper mentioned in the question. There is no other references. – Efimster Feb 08 '21 at 22:29
if 0xE1 ($11100001_b$) represents $x^7+x^2+x+1$ then 0x40 ($01000000_b$) represents $x$ – Ruggero Oct 25 '23 at 07:05

GCM Create M tables and R

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