Change the starting point of logup elements

[weikengchen]

This PR changes how the alpha powers are being applied to the values. In addition, the number of lookup elements needed for the Poseidon example is indeed only N_STATE rather than N_STATE * 2 as previously defined.

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This change is 

[shaharsamocha7]

Reviewed 2 of 2 files at r1, all commit messages.
Reviewable status: all files reviewed, 1 unresolved discussion (waiting on @weikengchen)

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crates/prover/src/constraint_framework/logup.rs line 264 at r1 (raw file):

            Fraction::new(SecureField::one(), SecureField::one()),
        );
    }

Can you add this test here?

Suggestion:

    }
    
    #[test]
    fn test_lookup_elements_combine() {
        let mut channel = Blake2sChannel::default();
        let lookup_elements = LookupElements::<3>::draw(&mut channel);
        let values = [
            BaseField::from_u32_unchecked(123),
            BaseField::from_u32_unchecked(456),
            BaseField::from_u32_unchecked(789),
        ];

        assert_eq!(
            lookup_elements.combine::<BaseField, SecureField>(&values),
            BaseField::from_u32_unchecked(123)
                + BaseField::from_u32_unchecked(456) * lookup_elements.alpha
                + BaseField::from_u32_unchecked(789) * lookup_elements.alpha.pow(2)
                - lookup_elements.z
        );
    }

[weikengchen]

@shaharsamocha7 The test has been added.

[shaharsamocha7]

Reviewed 1 of 1 files at r2, all commit messages.
Reviewable status: complete! all files reviewed, all discussions resolved (waiting on @weikengchen)