From 160309eff757f045e4a03ce6212e4275b3da7410 Mon Sep 17 00:00:00 2001 From: Marius Drechsler Date: Sun, 11 Aug 2024 23:29:22 +0200 Subject: [PATCH] Work --- content/BACH.typ | 19 ++++++++++++++++++- glossary.typ | 3 ++- main.pdf | Bin 1197886 -> 1201215 bytes pseudocode/find_quantizers.typ | 10 ---------- 4 files changed, 20 insertions(+), 12 deletions(-) delete mode 100644 pseudocode/find_quantizers.typ diff --git a/content/BACH.typ b/content/BACH.typ index a0fa947..7e2cd1f 100644 --- a/content/BACH.typ +++ b/content/BACH.typ @@ -16,4 +16,21 @@ Before we take a look at the higher order quantization cases, we will start with ) If we overlay the PDF of a zero-mean Gaussian distributed variable $X$ with a sign-based quantizer function as shown in @fig:1-bit_normal, we can see that the expected value of the Gaussian distribution overlaps with the decision threshold of the sign-based quantizer. -Considering that the margin of error of the value $x$ is comparable with the one shown in @fig:tmhd_example_enroll, we can conclude that values of $X$ +Considering that the margin of error of the value $x$ is comparable with the one shown in @fig:tmhd_example_enroll, we can conclude that values of $X$ that reside near $0$ are to be considered more unreliable than values that are further away from the x-value 0. +This means that the quantizer used here is very unreliable without generated helper-data. + +Now, to increase the reliability of this quantizer, we can try to move our input values further away from the value $x = 0$. +To do so, we can define a new input value $x^"lin"$ as a linear combination of two realizations of $X$, $x_1$ and $x_2$ with a set of weights $h_1$ and $h_2$: +$ +x^"lin" = h_1 dot x_1 + h_2 dot x_2 . +$ +We can define the vector of all possible linear combinations $bold(x^"lin")$ as the vector-matrix multiplication of the two input values $x_i$ and the matrix of all weight combinations: +$ +bold(x^"lin") &= vec(x_1, x_2) dot mat(delim: "[", h_1, -h_1, h_1, -h_1; h_2, h_2, -h_2, -h_2)\ +&= vec(x_1, x_2) dot mat(delim: "[", +1, -1, +1, -1; +1, +1, -1, -1) +$ +We will choose the optimal weights based on the highest absolute value of $bold(x^"lin")$, as that value will be the furthest away from $0$. +We may encounter two entries in $bold(x^"lin")$ that both have the same highest absolute value. + +Lets take a look at the resulting random distribution of this process: + diff --git a/glossary.typ b/glossary.typ index 9b125d6..383ce4f 100644 --- a/glossary.typ +++ b/glossary.typ @@ -7,5 +7,6 @@ (key: "hda", short: "HDA", plural: "HDAs", long: "helper data algorithm", longplural: "helper data algorithms"), (key: "ecdf", short: "eCDF", plural: "eCDFs", long: "empirical Cumulative Distribution Function", longplural: "empirical Cumulative Distribution Functions"), (key: "ber", short: "BER", plural: "BERs", long: "bit error rate", longplural: "bit error rates"), - (key: "smhdt", short: "SMHD", plural: "SMHDs", long: "S-Metric Helper Data Method") + (key: "smhdt", short: "SMHD", plural: "SMHDs", long: "S-Metric Helper Data Method"), + (key: "puf", short: "PUF", plural: "PUFs", long: "physical unclonale function", longplural: "physical unclonale functions") )) diff --git a/main.pdf b/main.pdf index a4fb7258703174123ae16e44402eb686ac4f6880..2ec3774b3138ab1c8a55b4debd22ea3263321f95 100644 GIT binary patch delta 330498 zcmdnj=ed7{r+{jJi<_>Yo;jClLqkIY7uR%wR(8?Jr=tX?|4w8Up7=+3VvZ)0vBBht zQ6j8H3I+;6lh-k;PCg%{KJia4qv7PnXvfW`89y*iUd6&UJ-(WqZSr~MlN`n%gAVXy)--=z$VF6MFGB_?pVelV*RfAlW*6Noo*2kHX(`we zj?Hx`O%BrqTiHe69+`Z9srY37rJ778Ag>6snLzwfwi;xg`sDjd8yO8ZCoWTBo!q$w z?h%>En^zwZf_a83H7_MUDT|BCWO|}Cv*_k|YYNyIji)~}V;0@4vSpsv^rWera0^9P zJv_Za{3iYP0Q*mP`XM$(;ps0}86_s4kCK=?|35#Q4_L+IE&pp7&9)jb@-dm1C>WWW znWMhZqC!pzii`z&V0qm10f z#tNnu#wH3TmX_0FSQ&MAEG$eFOf4+T6wJ)br?;^(CQVQJz%DZRJhKecEz{-L7?;As zc)Eljqdc>bndRn%YZ#ffzvE!^V3sqpQ?PNdFgA5@HZro~vN3RUu|$ZNPB-9Tl;Ckf 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