VerifyTestVectors in CKKS

[Maokami]

Hello, Lattigo Team!

May I ask what is the purpose of the VerifyTestVectors in CKKS test?

lattigo/schemes/ckks/precision.go

Lines 70 to 93 in 1f00f3c

	func VerifyTestVectors(params Parameters, encoder *Encoder, decryptor *rlwe.Decryptor, valuesWant, valuesHave interface{}, log2MinPrec int, logprec float64, printPrecisionStats bool, t *testing.T) {
	precStats := GetPrecisionStats(params, encoder, decryptor, valuesWant, valuesHave, logprec, false)
	if printPrecisionStats {
	t.Log(precStats.String())
	}
	rf64, _ := precStats.MeanPrecision.Real.Float64()
	if64, _ := precStats.MeanPrecision.Imag.Float64()
	switch params.RingType() {
	case ring.Standard:
	log2MinPrec -= params.LogN() + 2 // Z[X]/(X^{N} + 1)
	case ring.ConjugateInvariant:
	log2MinPrec -= params.LogN() + 3 // Z[X + X^1]/(X^{2N} + 1)
	}
	if log2MinPrec < 0 {
	log2MinPrec = 0
	}
	require.GreaterOrEqual(t, rf64, float64(log2MinPrec))
	require.GreaterOrEqual(t, if64, float64(log2MinPrec))
	}

Specifically, from the following codes, it seems that the CKKS test codes want to verify whether mean(|(m1+m2) - Dec(Add(Enc(m1+m2)))|) > LogDefaultScale - LogN - 2.

lattigo/schemes/ckks/precision.go

Line 83 in 1f00f3c

log2MinPrec -= params.LogN() + 2 // Z[X]/(X^{N} + 1)

lattigo/schemes/ckks/ckks_test.go

Lines 426 to 441 in 1f00f3c

	func testEvaluatorAdd(tc *testContext, t *testing.T) {
	t.Run(GetTestName(tc.params, "Evaluator/AddNew/Ct"), func(t *testing.T) {
	values1, _, ciphertext1 := newTestVectors(tc, tc.encryptorSk, -1-1i, 1+1i, t)
	values2, _, ciphertext2 := newTestVectors(tc, tc.encryptorSk, -1-1i, 1+1i, t)
	for i := range values1 {
	values1[i].Add(values1[i], values2[i])
	}
	ciphertext3, err := tc.evaluator.AddNew(ciphertext1, ciphertext2)
	require.NoError(t, err)
	VerifyTestVectors(tc.params, tc.encoder, tc.decryptor, values1, ciphertext3, tc.params.LogDefaultScale(), 0, *printPrecisionStats, t)
	})

I wonder why this property must be checked, and is there any theoretical explanation of this?
Thanks!

[Pro7ech]

The CKKS scheme is approximate, we must therefore define some lower bound on the accepted precision when checking the correctness of the operations. This defines heuristic lower bound that is a function of the ring degree and scaling factor and is achieved with overwhelming probability.

[Maokami]

Thank you, @Pro7ech!

Is the heuristic lower bound calculated based on CKKS papers, such as the original CKKS paper and its reduced error version?