Distinct Circular Subsum Sequences (DCSS)

• DCSS are modular sequences defined as compositions of n with unique circular subsums modulo n, enabling noncolliding shift patterns for error localization.
• They underpin the ERPA module in digital watermarking, spreading bit errors across a 64-bit patch to achieve near-perfect signature recovery under distortion.
• Alternative formulations of DCSS include permutation-based methods and sequenceability in cyclic groups, each imposing distinct combinatorial constraints.

Distinct Circular Subsum Sequences (DCSS) denote, in the most technically developed recent usage, modular sequences whose circular subsums are unique modulo a fixed length $n$, so that their cumulative offsets can be used as noncolliding shift patterns for error localization and correction. In README, DCSS form the combinatorial core of the ERror PAinting (ERPA) module, which spreads the bit-error pattern of one 64-bit watermark patch into another patch and permits subsequent recovery even when the painted pattern is itself corrupted under distortion (Choi et al., 6 Jul 2025). In the literature summarized here, the same expression also appears in a separate permutation-based sense tied to two distinct local sums on a circle, while adjacent work on sequenceability studies orderings in cyclic groups with distinct partial sums (Lau, 2020, Costa et al., 2022).

1. Terminological scope

In the sources considered here, “DCSS” is not a single standardized object. The term is used for at least two non-equivalent constructions, and a nearby line of work studies distinct partial sums in cyclic groups without using the README definition.

Source	Underlying object	Core distinctness requirement
README (Choi et al., 6 Jul 2025)	A sequence $S=(s_1,\dots,s_L)$ of positive integers summing to $n$	Every nonempty circular subsum is unique modulo $n$
Cyclic-group sequenceability (Costa et al., 2022)	An ordering of a subset of $G\setminus\{0\}$	Partial sums are distinct, with possible $y_0=y_k=0$ in the rotational case
Two-sum permutation usage (Lau, 2020)	A permutation of $\{1,2,\dots,n\}$ on a circle	Three-term local sums at positions $1 \bmod 4$ and $3 \bmod 4$ are constant but unequal

A common misconception is to treat these as interchangeable formulations. The available sources do not support that interpretation. The README object is an offset-design device for robust watermark decoding; the cyclic-group literature concerns sequenceability and distinct partial sums in finite abelian groups; and the permutation-based usage concerns circular arrangements with two prescribed local-sum classes.

2. Formal definition in README

Let $n$ be the length of a bit-string; in README, $S=(s_1,\dots,s_L)$0. A DCSS is a sequence

$S=(s_1,\dots,s_L)$1

together with cumulative offsets

$S=(s_1,\dots,s_L)$2

considered modulo $S=(s_1,\dots,s_L)$3. The defining property is that for every pair of index ranges $S=(s_1,\dots,s_L)$4 and $S=(s_1,\dots,s_L)$5 with

$S=(s_1,\dots,s_L)$6

one has

$S=(s_1,\dots,s_L)$7

Equivalently,

$S=(s_1,\dots,s_L)$8

In words, every nonempty circular subsum of $S=(s_1,\dots,s_L)$9 is unique modulo $n$0 (Choi et al., 6 Jul 2025).

In practical use, README does not paint with the raw $n$1 but with the prefix sums. The offset set is

$n$2

and an error at position $n$3 is shifted to

$n$4

This formulation is operational rather than purely structural: uniqueness of circular subsums is converted directly into a family of distinguishable modular shift patterns.

3. Error painting and inversion in ERPA

In README, a watermark decoder extracts a 64-bit message $n$5 from patch $n$6, and the bit-error pattern relative to the ground truth $n$7 is

$n$8

The stated motivation is that naively re-embedding the 64-bit vector $n$9 into a second patch $n$0 would leave further flips in $n$1 able to scramble the information needed to identify which bits in $n$2 were wrong. DCSS address this by spreading each single-bit error across a fixed subset of positions chosen so that no two different subsets can collide when viewed modulo 64 (Choi et al., 6 Jul 2025).

The ERPA painting map is defined by

$n$3

and the painted sequence is

$n$4

where $n$5 denotes the binary mask with ones at positions in $n$6, and $n$7 is bitwise OR. This 64-bit $n$8 is embedded into patch $n$9. After re-decoding $G\setminus\{0\}$0 under distortion, a noisy $G\setminus\{0\}$1 is obtained and passed to a small neural decoder

$G\setminus\{0\}$2

implemented as a single $G\setminus\{0\}$3 linear layer plus sigmoid, to produce $G\setminus\{0\}$4. The corrected main message is then

$G\setminus\{0\}$5

The significance of the circular-subsum condition is localized invertibility under corruption. README states that the distinct circular subsum property guarantees that even if some of the painted bits are corrupted, a noise-robust decoder can still invert the painting map and recover the original indices of the flipped bits.

4. Construction for $G\setminus\{0\}$6 and empirical behavior

For $G\setminus\{0\}$7, README uses a concrete DCSS of length $G\setminus\{0\}$8: $G\setminus\{0\}$9 Its cumulative offsets are

$y_0=y_k=0$0

all taken modulo 64. The summary describes a simple greedy search: starting from $y_0=y_k=0$1, one scans $y_0=y_k=0$2 to $y_0=y_k=0$3, appending $y_0=y_k=0$4 whenever doing so keeps $y_0=y_k=0$5 and preserves distinctness of all nonempty circular subsums modulo $y_0=y_k=0$6; in practice, one also prunes early and backtracks, and the reported outcome is the maximal-length DCSS above (Choi et al., 6 Jul 2025).

With this choice, any single-bit error at position $y_0=y_k=0$7 is spread to exactly seven positions. For the illustrative example $y_0=y_k=0$8,

$y_0=y_k=0$9

README states that if JPEG noise flips at most two of those seven bits, the neural decoder still recognizes the pattern as coming from $\{1,2,\dots,n\}$0, and so recovers $\{1,2,\dots,n\}$1. More generally, the neural decoder is reported to tolerate roughly $\{1,2,\dots,n\}$2–$\{1,2,\dots,n\}$3 random bit flips in $\{1,2,\dots,n\}$4 with over $\{1,2,\dots,n\}$5 accuracy. Tables 3–5 are described as reporting BER and Z.B.I.R. as functions of painting density $\{1,2,\dots,n\}$6, noise probability $\{1,2,\dots,n\}$7, permutation strategy, and decoder training distribution. One specific operating point is $\{1,2,\dots,n\}$8 with Bernoulli($\{1,2,\dots,n\}$9) training, for which the average BER is $1 \bmod 4$0 and Z.B.I.R. is $1 \bmod 4$1 under JPEG(50), versus $1 \bmod 4$2 Z.B.I.R. without ERPA.

At the image level, README embeds a 2048-bit RSA/ECDSA signature across 32 “message” patches $1 \bmod 4$3 and 32 “error-painting” patches $1 \bmod 4$4 in a $1 \bmod 4$5 image. Under JPEG-50, the reported numbers are Z.B.I.R.$1 \bmod 4$6, BER$1 \bmod 4$7 without ERPA, and Z.B.I.R.$1 \bmod 4$8, BER$1 \bmod 4$9 with ERPA using DCSS and Bernoulli($3 \bmod 4$0) training; under near-ideal conditions with no external distortion, ERPA pushes Z.B.I.R. to $3 \bmod 4$1. The same summary describes this as yielding near-perfect full 2048-bit signature recovery and as making README the first framework to embed 2048-bit cryptographic signatures in images with high assurance of zero-bit-error extraction.

5. Relation to distinct partial sums and sequenceability

A nearby combinatorial framework is the theory of sequenceability in finite abelian groups. For a finite abelian group $3 \bmod 4$2 and a subset $3 \bmod 4$3 of size $3 \bmod 4$4, an ordering $3 \bmod 4$5 has partial sums

$3 \bmod 4$6

It is a linear sequencing if $3 \bmod 4$7 are all distinct, and a rotational sequencing if the only repetition is $3 \bmod 4$8. A subset is sequenceable if it admits either a linear sequencing or a rotational sequencing, and a group is strongly sequenceable if every nonzero subset is sequenceable (Costa et al., 2022).

The 2022 results establish sequenceability for many subsets of $3 \bmod 4$9 when $n$0. In particular, if $n$1 with $n$2 prime, every subset $n$3 of size $n$4 is sequenceable when $n$5 and $n$6, when $n$7 and $n$8, and in several partial cases for $n$9. An asymptotic extension states that the same conclusions hold for $S=(s_1,\dots,s_L)$00 when every prime divisor of $S=(s_1,\dots,s_L)$01 exceeds $S=(s_1,\dots,s_L)$02. The proofs proceed by selecting a quotient sequencing in $S=(s_1,\dots,s_L)$03, constructing a polynomial $S=(s_1,\dots,s_L)$04, and applying Alon’s Combinatorial Nullstellensatz.

This literature does not define DCSS in the README sense, but it studies a related noncollision phenomenon: modular cumulative sums are required not to repeat. A plausible implication is that README’s circular-offset design belongs to a broader combinatorial tradition in which the uniqueness of cumulative modular structure is the crucial resource. The distinction remains important: sequenceability concerns orderings of group elements, whereas README DCSS concern integer compositions of $S=(s_1,\dots,s_L)$05 whose circular subsums produce a usable offset set.

6. Alternative permutation-based usage and open directions

A separate source uses “Distinct Circular Subsum Sequence” for a permutation-based object derived from work on “two-sum” sequences. In that formulation, for $S=(s_1,\dots,s_L)$06, a DCSS of length $S=(s_1,\dots,s_L)$07 is a permutation

$S=(s_1,\dots,s_L)$08

of $S=(s_1,\dots,s_L)$09 such that, with indices taken modulo $S=(s_1,\dots,s_L)$10 and

$S=(s_1,\dots,s_L)$11

there exist two integers $S=(s_1,\dots,s_L)$12 for which $S=(s_1,\dots,s_L)$13 whenever $S=(s_1,\dots,s_L)$14 and $S=(s_1,\dots,s_L)$15 whenever $S=(s_1,\dots,s_L)$16. The stated existence theorem is that for every integer $S=(s_1,\dots,s_L)$17, $S=(s_1,\dots,s_L)$18, there exists such a DCSS, with explicit constructions organized by congruence classes modulo $S=(s_1,\dots,s_L)$19 and with $S=(s_1,\dots,s_L)$20 as the sole exception (Lau, 2020).

The same summary gives concrete examples. For $S=(s_1,\dots,s_L)$21,

$S=(s_1,\dots,s_L)$22

with $S=(s_1,\dots,s_L)$23 and $S=(s_1,\dots,s_L)$24. For $S=(s_1,\dots,s_L)$25,

$S=(s_1,\dots,s_L)$26

with $S=(s_1,\dots,s_L)$27 and $S=(s_1,\dots,s_L)$28. It also states that these two-sum sequences arise from a local antimagic total labeling of a path, and that similar phenomena can be studied on cycles, wheels, and other graph families. Lau is reported to conjecture and in part prove that one can force more than two distinct sums under more general mod-$S=(s_1,\dots,s_L)$29 congruence conditions, leaving open the systematic treatment of $S=(s_1,\dots,s_L)$30-sum sequences for $S=(s_1,\dots,s_L)$31.

Taken together, the sources support a careful, non-unified view of the term. In README, DCSS are a modular offset design for robust digital-signature watermarking; in the sequenceability literature, the closest analogue is the study of distinct partial sums in cyclic groups; and in the two-sum literature, DCSS refer to a different circular permutation problem. The shared theme is distinctness of modular or circularly indexed sum structures, but the objects, constraints, and applications differ substantially.