On secret sharing from extended norm-trace curves

On secret sharing from extended norm-trace curv es Ola v Geil [0000-0002-9666-3399] Departmen t of Mathematical Sciences, Aalborg Univ ersity , Denmark olav@math.aau.dk Abstract. In [4] Camps-Moreno et al. treated (relative) generalized Hamming w eights of codes from extended norm-trace curves and they ga ve examples of resulting go o d asymmetric quan tum error-correcting co des emplo ying information on the relative distances. In the present pap er we study ramp secret sharing sc hemes whic h are ob jects that re- quire an analysis of higher relativ e weigh ts and we sho w that not only do sc hemes defined from one-p oint algebraic geometric co des from extended norm-trace curves hav e go o d parameters, they also p osses a second la yer of security along the lines of [11]. It is left undecided in [4, page 2889] if the “fo otprint-lik e approach” as employ ed by Camps-Moreno herein is strictly better for co des related to extended norm-trace co des than the general approach for treating one-point algebraic geometric codes and their likes as presented in [12]. W e demonstrate that the method used in [4] to estimate (relativ e) generalized Hamming weigh ts of co des from extended norm-trace curv es can b e viewed as a clever application of the enhanced Goppa b ound in [12] rather than a competing approach. Keyw ords: Extended norm-trace curves · One-p oint algebraic geomet- ric co des · Ramp secret sharing · Relative generalized Hamming weigh ts. 1 In tro duction Secret sharing is a branch of cryptography in whic h a central dealer shares a secret among a set of participants by giving them each a share. This is done in suc h a wa y that if few of them p o ol their shares then they obtain no information on the secret, but if many do then they can recov er it in full. One often requires the schemes to be linear meaning that (partial) recov ery , whenever p ossible, can b e done b y simple and fast linear algebra algorithms. In most schemes studied in the literature the shares and the secrets are elements from the same set, of- ten a finite field F q . This is in contrast to linear ramp secret sharing where the secret b elongs to F ℓ q , and the shares to F q , with ℓ ≥ 1 . Ramp sc hemes were orig- inally introduced by Blakley and Meadows in [2] and b y Y amamoto in [20], and ha ve gained a lot of in terest due to their ric h structures and their application in connection with for instance storage of bulk data and secure multipart y com- putation [7, 8]. A linear ramp scheme is synonymous to a pair of nested linear 2 O. Geil co des and worst case (partial) information leak age as well as worst case (partial) information recov ery can b e describ ed by relative parameters of the pair of co des and their duals. F or large num b er of participants compared to the field size knowledge on the w orst case scenarios, how ever, b y far giv es the complete picture, but a deep er analysis is typically very difficult. In [11] the author coined the concept of maxi- m um non- i -qualifying sets to address this difficult y . These are sets of maximum size b eing not able to recov er i log 2 ( q ) bits of information. It w as demonstrated that t wo sp ecific classes of monomial-Cartesian codes p ossesses families of max- im um non- i -qualifying sets with a very ric h structure and based on this infor- mation one can then imp ose a second lay er of security . By elab orating on results in [4] in the present pap er we demonstrate that also ramp secret sharing schemes constructed from nested one-p oint algebraic geometric co des from the extended norm-trace curves p osses families of w ell-structured maxim um non- i -qualifying sets giving again rise to a second lay er of security . The exp osition in [4] uses the language of Gröbner basis theory , but as we demonstrate their strategy for deriving estimates on co de parameters related to the extended norm-trace curve is not a comp eting metho d to the enhanced Goppa kind of b ounds in [12, Eq. 13, Eq. 17] for general algebraic function fields of transcendence degree 1 , but can b e view ed as a clever application thereof. By pro ving this w e answer an open question raised b y Camps-Moreno et al. in [4, page 2889]. The pap er is organized as follows. In Section 2 w e give the necessary bac k- ground on linear ramp secret sharing. W e next treat general theory of relativ e generalized Hamming weigh ts of nested one-p oin t co des and their relatives in Section 3. Then in Section 4 we elab orate on material in [4] regarding (relative) generalized Hamming weigh ts of nested decreasing norm-trace co des and their duals, and demonstrate the relationship b et ween the metho d used in [4] and that of [12]. Finally , in Section 5 w e apply suc h results to pro duce the alluded linear ramp sc hemes with desirable structure and consequently a second la yer of securit y . Section 6 con tains concluding remarks. 2 Linear ramp secret sharing In linear ramp secret sharing the shares giv en to the individual participants b elong to a finite field F q , but the secret is a tupple in F ℓ q , where ℓ is al- lo wed to b e strictly larger than 1 . The linearity means that if c (1) 1 , . . . , c (1) n are shares for a secret s (1) and c (2) 1 , . . . , c (2) n are shares for a secret s (2) then also ac (1) 1 + bc (2) 1 , . . . , ac (1) n + bc (2) n w ork as shares for the secret a s (1) + b s (2) for an y a, b ∈ F q . Suc h schemes by [7, Sec. 4.2] can be put into the form of a nested co de construction as follo ws. Let C 2 ⊆ C 1 ⊆ F n q , dim C 2 = k 2 < dim C 1 = k 1 and ℓ = k 1 − k 2 . Consider bases { b 1 , . . . , b k 1 } and { b 1 , . . . , b k 1 , b k 1 +1 , . . . , b k 2 } , resp ectiv ely , for C 2 and C 1 , resp ectiv ely , as vector spaces o v er F q . A secret On secret sharing from extended norm-trace curves 3 s = ( s 1 , . . . , s ℓ ) ∈ F ℓ q is enco ded to ( c 1 , . . . , c n ) = r 1 b 1 + · · · + r k 2 b k 2 + s 1 b k 2 +1 + · · · + s ℓ b k 1 (1) and participant i , then receives c i as a share i = 1 , . . . , n . The crucial parameters of a linear ramp secret sharing sc heme are the n umber of participants, which is n , the dimension of space of secrets, which is ℓ , the priv acy num b ers t 1 , . . . , t ℓ , and the reconstruction num b ers r 1 , . . . , r ℓ . Here, for i = 1 , . . . , ℓ , t i is largest p ossible and r i is smallest p ossible such that – no set of t i participan ts is able to recov er i times log 2 ( q ) bits of information ab out s – an y set of r i participan ts can recov er i times log 2 ( q ) bits of information. Of sp ecial in terest are full priv acy and full recov ery corresp onding to t = t 1 and r = r ℓ . As is well-kno wn [16, 12] t m = M m ( C ⊥ 2 , C ⊥ 1 ) − 1 (2) r m = n − M ℓ − m +1 ( C 1 , C 2 ) + 1 (3) where M t ( C 1 , C 2 ) = min { # Supp ( D ) | D ⊆ C 1 , D ∩ C 2 = { 0 } , dim D = t } is called the t th relative generalized Hamming weigh t. As is alw ays the case in co ding theory , one cannot choose parameters freely . If, say , q , n, ℓ and 1 ≤ r 1 < · · · < r ℓ ≤ n are fixed num bers then among the set of all pairs of nested co des C 2 ⊆ C 1 ⊆ F n q with such parameters there are limitations as to how close t i can b e to r i , i = 1 , . . . , ℓ , see [6]. How ever, by definition for any i there exist groups of size r i − 1 who cannot recov er i times log 2 ( q ) bits of information and when we we hav e knowledge ab out the structure of suc h groups we ma y employ it as a second la yer of security as demonstrated in the recent paper [11]. The imp ortance of the mentioned groups justify that w e giv e them a particular name [11, Def. 8]. Definition 1. Given a line ar r amp se cr et sharing scheme define d fr om C 2 ⊆ C 1 ⊆ F n q , a set A ⊆ { 1 , . . . , n } is c al le d maximum non- i -qualifying if # A = n − M ℓ − i +1 ( C 1 , C 2 ) , but fr om the c orr esp onding shar es one c annot r e c over i log 2 ( q ) bits of information. T o establish information on the access-structure (who can recov er ho w m uch) and in particular information on the maxim um non- i -qualifying sets w e may emplo y the follo wing result [11, Thm. 3] whic h is a sligh t reform ulation of [7, Thm. 10]. Theorem 1. L et A = { i 1 < · · · < i m } ⊆ { 1 , . . . , n } . Assume c ′ i 1 , . . . , c ′ i m ar e si- multane ously r e alizable shar es in those p ositions, se e (1). The amount of p ossible se cr ets s c orr esp onding to such shar es e quals q s with s = max { dim D | D ⊆ C 1 , D ∩ C 2 = { 0 } , Supp ( D ) ⊆ ¯ A } wher e ¯ A = { 1 , . . . , n }\ A . 4 O. Geil In [11] Theorem 1 was employ ed to show that ramp schemes coming from the particular nested monomial-Cartesian co des as describ ed in [9, Sec. 4] do es not only hav e go o d parameters n, ℓ, r 1 , . . . , r ℓ , r 1 , . . . , t ℓ , but also support a second la yer of securit y by p ossessing maximum non- i -qualifying sets of desirable w ell- structured form. This insight w as then used in [11, Sec. 4] to construct monomial- Cartesian code based sc hemes with larger ℓ maintaining desirable second la yer of security by p ossessing w ell-structured maximum non- i -qualifying sets whilst pa ying less interest in the w orst case security t 1 , . . . , t ℓ . The aim of the present con tribution is to in vestigate what can be said regarding a second lay er of securit y for ramp sc hemes based on decreasing norm-trace co des. 3 Relativ e parameters of one-p oint algebraic geometric co des and their relatives Before treating co des from extended norm-trace curves we revisit the general theory of relative generalized Hamming weigh ts of any pair of nested one-p oin t algebraic geometric co des and their relatives [12]. Consider an arbitrary algebraic function field of transcendence degree 1 defined ov er some finite field F Q . Let P 1 , . . . , P n , Q b e pairwise differen t rational places. Let H ( Q ) b e the W eierstrass semigroup of Q , i.e. H ( Q ) = − ν Q ( R = ∪ ∞ m =0 L ( mQ )) where ν Q is the discrete v aluation corresponding to Q . Define H ∗ ( Q ) = { λ ∈ H ( Q ) | C L ( G = P 1 + · · · + P n , λQ )  = C L ( G, ( λ − 1) Q ) } . Clearly , # H ∗ ( Q ) = n . W e now recall material from [12] on ho w to estimate (rel- ativ e) generalized Hamming weigh ts. Let D ⊆ F n Q b e any subspace, of dimension sa y m . There exist functions f 1 , . . . , f m ∈ R such that { ( f i ( P 1 ) , . . . , f i ( P n )) | i = 1 , . . . , m } is a basis for D and suc h that − ν Q ( f 1 ) < · · · < − ν Q ( f m ) (4) holds, where without loss of generality w e ma y assume that all num b ers in (4) b elong to H ∗ ( Q ) . These v alues do not dep end on the c hoice of the f i ’s, but are in v ariants of D and we ma y therfore define ρ ( D ) = {− ν Q ( f 1 ) , . . . , − ν Q ( f m ) } ⊆ H ∗ ( Q ) . F rom [12, Prop. 17] we hav e the b ound # Supp ( D ) ≥ #  H ∗ ( Q ) ∩  ∪ m s =1 ( γ i + H ( Q ))  (5) where ρ ( D ) = { γ 1 , . . . , γ m } . With prop er care one can apply (5) to any set of nested co des defined from R , but for nested one-p oin t co des the situation is immediate [12, Thm. 19]. Let λ 2 < On secret sharing from extended norm-trace curves 5 λ 1 b e elements in H ∗ ( Q ) and consider m ≤ dim C L ( G, λ 1 Q ) − dim C L ( G, λ 2 Q ) . W e hav e M m ( C L ( G, λ 1 Q ) , C L ( G, λ 2 Q )) ≥ min  #  H ∗ ( Q ) ∩  ∪ m s =1 ( γ s + H ( Q ))   | γ s ∈ H ∗ ( Q ) for s = 1 , . . . , m, λ 2 < γ 1 < · · · < γ m ≤ λ 1  . (6) That this b ound (and in larger generality (5)) can b e seen as an enhancemen t of the Goppa b ound for a one-p oin t Goppa co de i.e. the bound d ( C L ( G, λQ )) ≥ n − λ follo ws from [14, Lem. 5.15] which sa ys that for any n umerical semigroup Λ and any element λ herein w e ha ve λ = #( Λ \ ( λ + Λ )) . In fact (6) can b e a strict improv ement ev en when applied to estimate the minimum distance of a one-p oin t co de. This happ ens when some of the elements in Λ \ ( λ + Λ ) are not con tained in H ∗ ( Q ) . As noted in [12] a translation of [12, Thm 14] into the particular case of one-p oin t algebraic geometric co des and their relatives pro duces a similar result as (5), but for dual spaces. W e no w fill in the missing details b y stating such result. Consider a subspace D ⊆ F n Q of dimension m and let η 1 , . . . , η m b e the unique n umbers such that for i = 1 , . . . , m there exists some c ∈ D satisfying c ∈ C ⊥ L ( G, ( η i − 1) , Q ) , but c / ∈ C ⊥ L ( G, η i Q ) . W e shall write κ ( D ) = { η 1 , . . . , η m } ⊆ H ∗ ( Q ) . The coun terpart to (5) is # Supp ( D ) ≥ #  H ( Q ) ∩  ∪ m s =1 ( η i − H ( Q ))  (7) F rom this one immediately obtains [12, Thm. 20] which w e no w state M m ( C ⊥ L ( G, λ 2 Q ) , C ⊥ L ( G, λ 1 Q )) ≥ min  #  H ( Q ) ∩  ∪ m s =1 ( γ s − H ( Q ))   | γ s ∈ H ∗ ( Q ) for s = 1 , . . . , m, λ 2 < γ 1 < · · · < γ m ≤ λ 1  . (8) (Note that in the ab o ve formula γ 1 , . . . , γ m pla y the role of η 1 , . . . , η m ). 4 Bounds on relativ e parameters of co des from the extended norm-trace curv e The extended norm-trace curve is the curve o ver F q s giv en b y the equation x u = y q s − 1 + y q s − 2 + · · · + y , (9) 6 O. Geil where u is a positive divisor of q s − 1 q − 1 . Recall, that the righ t hand side of (9) corresp onds to the trace map from F q s to F q , and similarly that x q s − 1 q − 1 corre- sp onds to the norm map. Related codes are therefore generalizations of norm- trace co des [10, 17] which again are generalizations of Hermitian co des [19, 21, 1]. Co des from the extended norm-trace curv e ha v e b een extensively studied in [3, 15, 5, 4] and one of the crucial observ ations employ ed here is the systematic structure of the affine v ariety , i.e. the set of affine ro ots of (9). As is easily seen from the prop erty of the norm-map and the trace-map the affine p oint set is the disjoin t union of the following sets [5, Lem. 3.1], A 0 = { (0 , b ) | b q s − 1 + · · · + b = 0 } = Γ (1) 0 × Γ (2) 0 and for i = 1 , . . . , q − 1 A i = { ( a, b ) | a u = α i , b q s − 1 + · · · + b = α i } = Γ (1) i × Γ (2) i , where α is a primitive elemen t of F q . Clearly , # A 0 = # Γ (2) 0 = q s − 1 and for i = 1 , . . . , q − 1 we hav e # A i = uq s − 1 where # Γ (1) i = u and Γ (2) i = q s − 1 , [5, Lem. 3.1]. The algebraic function field of transcendence degree 1 o ver F q s defined from (9) has exactly 1 + P q − 1 i =0 # A i rational places [5, Sec. 4.2], eac h of them, but one, b eing related to an affine p oint, and the last b eing the unique place at infinit y . F ollowing [5, Sec. 4.2] and [18] the W eierstrass semigroup corresp onding to the unique place Q at infinity equals H ( Q ) = ⟨ u, q s − 1 ⟩ and we hav e R = ∪ ∞ m =0 L ( mQ ) = F q s [ X, Y ] /I where I = ⟨ X u − Y q s − 1 − · · · − Y ⟩ . W riting x i y j = X i Y j + ⟨ X u − Y q s − 1 − · · · − Y ⟩ and following [5, Sec. 4.2] one sees that { x i y j | 0 ≤ i, 0 ≤ j < q s − 1 } is a basis for R as a v ector space ov er F q s and clearly − ν Q ( x i y j ) = iq s − 1 + j u , where ν Q indicates the discrete v aluation corresp onding to Q . No tw o elements of this basis ha ve the same ν Q -v alue, i.e. there is a one-to-one corresp ondence b et ween the basis and H ( Q ) . W e shall denote b y P 1 , . . . , P n the rational places differen t from Q , where of course n = ( u ( q − 1) + 1) q s − 1 . W riting G = P 1 + · · · + P n and as b efore H ∗ ( Q ) = { λ ∈ H ( Q ) | C L ( G, λQ )  = C L ( G, ( λ − 1) Q ) } , it is not difficult to see that H ∗ ( Q ) = { iq s − 1 + j u | 0 ≤ i < u ( q − 1) + 1 , 0 ≤ j < q s − 1 } On secret sharing from extended norm-trace curves 7 and that { ( f ( P 1 ) , . . . , f ( P n )) | f = x i y j , where 0 ≤ i < u ( q − 1) + 1 , 0 ≤ j < q s − 1 } is a basis for F n q s as a v ector space ov er F q s , and in particular that { ( f ( P 1 ) , . . . , f ( P n )) | f = x i y j , where 0 ≤ i < u ( q − 1) + 1 , 0 ≤ j < q s − 1 , iq s − 1 + j u ≤ λ } is a basis for C L ( G, λQ ) . In [5] similar results w ere prov ed using Gröbner basis theoretical arguments. W e next turn our attention to the problem of estimating relative generalized Hamming w eights. Let D ⊆ F n q s b e an y subspace, of dimension say m and con- sider ρ ( D ) = { γ 1 , . . . , γ m } ⊆ H ∗ ( Q ) as in Section 3. W e introduce the function ι : H ∗ ( Q ) → { ( i, j ) | 0 ≤ i < u ( q − 1) + 1 and 0 ≤ j ≤ q s − 1 − 1 } given b y ι ( λ = iq s − 1 + j u ) = ( i, j ) . In [4] # Supp ( D ) was estimated for the considered curv e using a Gröbner basis approach. W e explain their result and demonstrate the relationship with (5). Let ( a, b ) ∈ ι ( { γ 1 , . . . , γ m } ) = { ( i 1 , j 1 ) , . . . , ( i m , j m ) } b e chosen such that a = min { i 1 , . . . , i m } , then the b ound in [4, Eq. (3)+ Eq. (4)] reads # Supp ( D ) ≥ # { ( i, j ) | 0 ≤ i < u ( q − 1) + 1 , 0 ≤ j < q s − 1 , ( i s , j s ) ≤ p ( i, j ) for some s ∈ { 1 , . . . , m } or ( a + u, 0) ≤ p ( i, j ) } . (10) Here, we used the partial ordering ≤ p giv en by ( α, β ) ≤ p ( ϵ, δ ) if and only if α ≤ ϵ and β ≤ δ . Note, that the condition ( a + u, 0) ≤ p ( i, j ) of course only comes into action if a + u < u ( q − 1) + 1 . T o see that (10) is a consequence of (5) w e only need to show that ( a + u ) q s − 1 + 0 u b elongs to ( aq s − 1 , bu ) + H ( Q ) whenev er 0 < b . But, ( aq s − 1 + bu ) + (0 q s − 1 + ( q s − 1 − b ) u ) = ( a + u ) q s − 1 + 0 u and we are through. Com bining (10) and (6) one obtains M m ( C L ( G, λ 1 Q ) , C L ( G, λ 2 Q )) ≥ min { # { ( i, j ) | 0 ≤ i < u ( q − 1) + 1 , 0 ≤ j < q s − 1 , ι ( γ s ) ≤ p ( i, j ) for some s ∈ { 1 , . . . , m } or ( a + u, 0) ≤ p ( i, j ) } , for s = 1 , . . . , m, λ 2 < γ 1 < · · · < γ m ≤ λ 1 } . (11) W e con tinue the study of relativ e generalized Hamming weigh ts but now turn to dual co des. Given a subspace D ⊆ F n q s of dimension m let κ ( D ) = { η 1 , . . . , η m } b e as in Section 3. Cho ose ( a, b ) ∈ ι ( { η 1 , . . . , η m } ) = { ( i 1 , j 1 ) , . . . , ( i m , j m ) } with a maximal. W e hav e # Supp ( D ) ≥ # { ( i, j ) | 0 ≤ i < u ( q − 1) + 1 , 0 ≤ j < q s − 1 , ( i, j ) ≤ p ( i s , j s ) for some s ∈ { 1 , . . . , m } or ( i, j ) ≤ p ( a − u, q s − 1 − 1) } . (12) 8 O. Geil W e prov e this by applying (7) and by using similar arguments as ab ov e. W e only need to demonstrate that ( a − u ) q s − 1 + ( q s − 1) u ∈ ( aq s − 1 + bu ) − H ( Q ) for a ≥ u . W e ha ve ( aq s − 1 + bu ) − ( b + 1) u = ( a − u ) q s − 1 + ( q s − 1 − 1) u and w e are through. Com bining (12) and (8) one obtains M m ( C ⊥ L ( G, λ 2 Q ) , C ⊥ L ( G, λ 1 Q )) ≥ min { # { ( i, j ) | 0 ≤ i < u ( q − 1) + 1 , 0 ≤ j < q s − 1 , ( i, j ) ≤ p ι ( γ s ) for some s ∈ { 1 , . . . , m } or ( i, j ) ≤ p ( a − u, q s − 1) } , for s = 1 , . . . , m, λ 2 < γ 1 < · · · < γ m ≤ λ 1 } . (13) One of the important insights from [4] is that for so-called decreasing norm- trace co des C 2 ⊆ C 1 the estimate on M m ( C 1 , C 2 ) inferred from (10) b ecomes sharp. A co de of dimension k is called a decreasing norm-trace co de if it equals the span of functions g i , i = 1 , . . . , k ev aluated at P 1 , . . . , P n where w e hav e g i = x α i y β i with 0 ≤ α i < ( q − 1) u + 1 , 0 ≤ β i < q s − 1 , and where for any 0 ≤ α ≤ α i and 0 ≤ β ≤ β i there exists a j ∈ { 1 , . . . , k } such that g j = x α y β . Hence, the one-p oin t algebraic geometric co des related to the extended norm-trace curve are of this type. As a consequence of the sharpness in case of decreasing codes w e can conclude that in the case of the extended norm-trace curve the right hand side of (5) and the right hand side (10) are identical (this could alternatively ha ve b een prov ed directly by using [14, Lem. 5.15]). In [4, Thm. 5.3] the dual of decreasing norm-trace co de is sho wn to b e equiv- alen t to another decreasing norm-trace co de. W e leav e it for the reader to insp ect that the estimate one obtains by applying suc h corresp ondence to the relativ e generalized Hamming weigh ts of a pair of duals of nested decreasing norm-trace co des is in fact the same as one gets b y applying (12) directly . F rom the ab ov e w e conclude that b oth (11) and (13) are sharp. 5 Sc hemes with a second lay er of security As mentioned in the previous section it is prov ed in [4] that all their b ounds are sharp in that for any γ 1 < · · · < γ m in H ∗ ( Q ) there exists a corresp onding space D of dimension m satisfying ρ ( D ) = { γ 1 , . . . , γ m } and with equality in (10). Moreo ver, among such spaces there exist some which can be written D = Span F q s { ( f 1 ( P 1 , . . . , f 1 ( P n )) , . . . , ( f m ( P 1 ) , . . . , f m ( P n )) } with − ν Q ( f i ) = γ i , i = 1 , . . . , m and each elemen t f i b eing a product of linear factors. Our treatmen t of ramp secret sharing shall rely heavily on suc h obser- v ations. With the aim of establishing ramp secret sharing schemes with maximum non- i -qualifying sets allo wing for a second la yer of security we now revisit some of the results in [4]. The following theorem is a combination of [4, Lem. 3.1] and Case 1.1 in the pro of of [4, Thm. 3.2] adapted to our language. W ell-structured On secret sharing from extended norm-trace curves 9 maxim um non- i -qualifying sets deriv ed from the theorem are discussed in the subsequen t remark, corollary and examples. Theorem 2. Consider p airwise differ ent γ 1 , . . . , γ w ∈ H ∗ ( Q ) and write ι ( { γ 1 , . . . , γ w } ) = { ( a 1 , b 1 ) , . . . , ( a w , b w ) } . Assume a 1 < · · · < a w and b w < · · · < b 1 and that a w − a 1 < u . Assume a 1 ≤ u ( q − 2) + 1 Then the right hand side of (10) r e ads n −  a 1 q s − 1 + b r u + r − 1 X i =1 ( a i +1 − a i )( b i − b r )  . (14) The fol lowing functions f 1 , . . . , f w ∈ R satisfy − ν Q ( f j ) = a j q s − 1 + b j u , j = 1 , . . . , m and for D = Sp an F q s { ( f 1 ( P 1 ) , . . . , f 1 ( P n )) , . . . , ( f w ( P 1 ) , . . . , f w ( P n )) } e quality holds in (10) me aning that # Supp ( D ) is e qual to the right hand side of (14). L et i ′ ∈ { 1 , . . . , q − 1 } and cho ose α 1 , . . . , α a 1 ∈ Γ (1) 0 × Γ (1) 1 × · · · × Γ (1) i ′ − 1 × Γ (1) i ′ +1 × · · · × Γ (1) q − 1 Enumer ate Γ (1) i ′ = { α ′ 1 , . . . , α ′ u } and Γ (2) i ′ = { β 1 , . . . , β q s − 1 } . Final ly for j = 1 , . . . , r define f j = a 1 Y i =1 ( x − α i ) a j − a 1 Y i =1 ( x − α ′ i ) b j Y i =1 ( y − β i ) . (15) Pr o of. See [4, Lem. 3.1] and the first part of the pro of of [4, Thm. 3.2]. T o make Theorem 2 op erational in connection with deriving second la yer of securit y w e shall need the following lemma whic h is new. Lemma 1. Consider γ 1 < · · · < γ w in H ∗ ( Q ) . Assume γ w − γ 1 < min { u, q s − 1 } and write { ι ( γ 1 ) , . . . , ι ( γ w ) } = { ( a 1 , b 1 ) , . . . , ( a w , b w ) } . (16) The enumer ation on the right hand side of (16) c an b e done in such a way that a 1 < · · · < a w and b w < · · · < b 1 . (17) It holds that a w − a 1 < u . Under the additional c ondition γ 1 ≤ ( u ( q − 2) + w ) q s − 1 it holds that a 1 ≤ u ( q − 2) + 1 . Pr o of. The assumption γ j − γ i < min { u, q s − 1 } for 1 ≤ i < j ≤ w implies (17). Aiming for a con tradiction assume a w − a 1 ≥ u , and recall that by the very definition of the function ι we hav e b 1 − b w ≤ q s − 1 − 1 . W e obtain ( a w q s − 1 + b w u ) − ( a 1 q s − 1 + b 1 u ) ≥ uq s − 1 − ( q s − 1 − 1) u = u 10 O. Geil whic h b y assumption is imp ossible. Finally , the additional condition in com- bination with γ w − γ 1 < min { u, q s − 1 } ensures that ι ( γ i )  > p ( u ( q − 2) + w , 0) , i = 1 , . . . , w . But, the a i s constitute a strictly increasing sequence and we are through (to a void confusion be aw are that ι ( γ 1 ) needs not b e equal to ( a 1 , b 1 ) ). Corollary 1. Consider λ 2 < λ 1 with λ 2 + 1 , λ 1 ∈ H ∗ ( Q ) and λ 1 − ( λ 2 + 1) < min { u, q s − 1 } and λ 2 < ( u ( q − 2) + w ) q s − 1 and c onsider the neste d c o des C 2 = C L ( G, λ 2 Q ) ⊆ C L ( G, λ 1 Q ) = C 1 the c o-dimension ℓ of which e quals #  H ∗ ( Q ) ∩ { λ 2 + 1 , . . . , λ 1 }  . F or w ∈ { 1 , . . . , ℓ } let { γ 1 , . . . , γ w } ⊆ H ∗ ( Q ) with λ 2 < γ 1 < · · · < γ w ≤ λ 1 b e such that the minimum value is attaine d in (10) among al l p ossible choic es of { γ 1 , . . . γ w } of this form. I.e. the right hand side of (10) for the given γ 1 , . . . , γ w achieves the value of M w ( C 1 , C 2 ) . The set of p ositions c orr esp ondning to c ommon r o ots of r elate d functions f 1 , . . . , f w as in The or em 2 c onstitutes a maximum non- ( ℓ − w + 1) -qualifying set. In ot her wor ds, by le aving out al l p articip ants c orr esp onging to non-c ommon r o ots of these functions the r emaining p articip ants c annot dete ct ( ℓ − w + 1) log 2 ( q ) bits of in- formation. Corollary 1 in com bination with the particular structure of { P 1 , . . . , P n } as w ell as the particular structure of each of f 1 , . . . , f w imp oses a second la yer of securit y in a large family of ramp secret sharing schemes defined from nested one-p oin t algebraic geometric codes ov er the extended norm-trace curves. Such schemes ha ve families of well-structured sets of participants who cannot all b e left out if ( ℓ − w + 1) log 2 ( q ) bits of information are to b e retrieved. This is discussed in the following remark. R emark 1. Recall, that the affine v ariet y of the extended norm-trace curve equals the disjoint union ∪ q − 1 v =0 A v where A i = Γ (1) i × Γ (2) i , and where # Γ (2) i = q s − 1 for i = 0 , . . . , q − 1 , where # Γ (1) 0 = 1 and where for i = 1 , . . . , q − 1 # Γ (1) i = u . T o illustrate the second lay er of securit y assume in the follo wing that w e hav e an organization with q − 1 large departments each ha ving uq s − 1 mem b ers. Sa y , A i corresp onds to a large departmen t i , i = 1 , . . . , q − 1 . Assume further that w e hav e a single small department of size q s − 1 . This department corresp onds to A 0 . If a 1 < u ( q − 2) + 1 then one may c ho ose α 1 , . . . , α a 1 in Theorem 2 in such a w ay that the common roots of the f 1 , . . . , f w in (15) corresp ond to the disjoin t union of the following sets: ⌊ a 1 u ⌋ entire large departmen ts A i 1 ∪ · · · ∪ A i ⌊ a 1 /u ⌋ , and for some i ′′  = i ′ b oth b elonging to { 1 , . . . , q − 1 }\{ i 1 , . . . , i ⌊ a 1 /u ⌋ } a subset S × Γ (2) i ′′ ⊆ A i ′′ , where S ⊆ Γ (1) i ′′ , # S = a 1 − ⌊ a 1 u ⌋ and finally a subset of A i ′ of size equal to b r u + P r − 1 i =1 ( a i +1 − a i )( b i − b r ) with the form of some (p ossibly irregular) staircase. W e call the latter subset “the set locally induced by γ 1 , . . . , γ w ” (or “the lo cally induced set” for short). The subset S × Γ (2) i ′′ as w ell a the lo cally induced set can b e given their own meaning. This is done b y dividing A i ′′ and A i ′ , resp ectively , into u horizon tal levels each con taining q s − 1 elemen ts. Here, w e enumerate the levels according to Γ (1) i ′′ and Γ i ′ , resp ectively . Similarly , we can in an obvious wa y divide A i ′′ and A ′ , resp ectively , in to q s − 1 v ertical lev els eac h con taining u members. Hence, for instance S × Γ (2) i ′′ consist of a 1 − ⌊ a 1 u ⌋ On secret sharing from extended norm-trace curves 11 horizon tal levels. Note, that for fixed A i ′′ there are  u a 1 −⌊ a 1 /u ⌋  p ossibilities for that. If a 1 = u ( q − 2) + 1 then there is in the set of common ro ots no set S × Γ i ′′ and w e need to include the entire set A 0 . Returning to the case a 1 < u ( q − 2) + 1 w e may of course also include A 0 in the zero-set, whic h then causes a minor c hange in ho w w e may include the other departmen ts. The man y different wa ys one can define f 1 , . . . , f w giv en fixed γ 1 , . . . , γ w pro vides us with man y differen t v ery systematic patterns of common ro ots, and similarly , of course, of the same n umber of different v ery systematic patterns of non-common ro ots, the latter b eing those participants if all left out, the remaining participan ts c annot detect ℓ − w + 1 log 2 ( q ) bits of information. This concludes the remark. W e illustrate the idea with a couple of examples where the first describ es a situation where the set of common ro ots (and therefore also the set of non- common ro ots) are as simple as can p ossibly be. Example 1. Let q ≥ 3 b e a prime-pow er, s ≥ 2 , and u ≥ 2 . Consider co des C 2 = C L ( G, λ 2 Q ) ⊆ C L ( G, λ 1 Q ) = C 1 ⊆ F n q s of co-dimension ℓ = 1 defined by ι ( λ 1 ) = ( τ u, 0) with τ ∈ { 1 , . . . , q − 2 } . The conditions of Corollary 1 are clearly satisfied and b y applying (2), (3), (14) and (13) we obtain t = M 1 ( C ⊥ 2 , C ⊥ 1 ) − 1 = ( τ − 1) uq s − 1 + q s − 1 + u − 1 r = n − M 1 ( C 1 , C 2 ) + 1 = τ uq s − 1 + 1 r − t = ( q s − 1 − 1)( u − 1) + 1 In the follo wing we use the language of Remark 1. No w for all 1 ≤ i 1 < · · · < i τ ≤ q − 1 the set A i 1 ∪ · · · ∪ A i τ constitutes a maxim um non- 1 -qualifying set. Hence, by leaving out all members of ( q − 1) − τ large departments and the small departmen t one do es not obtain any information. In particular if τ = q − 2 one cannot lea ve out an en tire large department in combination with the small departmen t if one wan ts to generate information. Next let ι ( λ 1 ) = ( τ u + 1 , 0) with τ ∈ { 1 , . . . , q − 2 } . W e obtain r = ( τ u + 1) q s − 1 + 1 with the same v alue of r − t as before. By lea ving out all members of an y set of ( q − 1) − τ large departmen ts one do es not obtain an y information. And b y lea ving out an y ( q − 1) − τ − 1 large departmen ts, the small departmen t as well as S × Γ (2) i where A i i ∈ { 1 , . . . , q − 1 } is not one of the already left out large departmen t and where S ⊆ Γ (1) i is of size equal to u − 1 one neither obtains an y information. The complementary sets are maxim um non- 1 -qualifying. In particular if τ = q − 2 one cannot leav e out an en tire large department and obtain an y information. Similarly , one cannot leav e out the small department and S × Γ (2) i , i ∈ { 1 , . . . , q − 1 } where S is given as ab ov e. Example 2. Consider the Hermitian curv e x q +1 − y q − y ov er F q 2 . W e hav e u = q + 1 , s = 2 , and q s − 1 = q . Let λ 2 + 1 = a ℓ q where q − 1 ≤ a ℓ < u ( q − 2) + 1 = 12 O. Geil ( q + 1)( q − 2) + 1 . Let ℓ = q and λ 1 = λ 2 + ℓ . Then λ 2 + 1 , . . . , λ 2 + ℓ all b elong to H ∗ ( Q ) . Moreov er, we hav e ι ( λ 2 + 1 + i ) = ( a ℓ − i, i ) for i = 0 , . . . , ℓ − 1 . Hence, with C 2 ⊆ C 1 as in Corollary 1 all conditions therein are satisfied. Consider the related ramp secret sharing scheme. W e hav e t 1 = M 1 ( C ⊥ 2 , C ⊥ 1 ) − 1 = ( a ℓ + 1) + ( q − 1)( a ℓ − q ) − 1 t ℓ = M ℓ ( C ⊥ 2 , C ⊥ 1 ) − 1 = a 1 q + X i =0 ℓ − 1 i ) − 1 = ( a ℓ − q + 1) q + ( q − 1) q 2 − 1 r 1 = n − M ℓ ( C 1 , C 2 ) + 1 = a ℓ q − ( ℓ X i =1 ( i − 1)) + 1 = a ℓ q − ( q − 1) q 2 + 1 r ℓ = n − M 1 ( C 1 , C 2 ) + 1 = n − ( n − λ 1 ) + 1 = ( a ℓ + 1) q . Using the approac h describ ed in Remark 1 we hav e maximum non- 1 -qualifying sets of the follo wing form. Namely , the disjoint union of ⌊ a ℓ − ( q − 1) q +1 large de- partmen ts, the follo wing subset of a large departmen t S × Γ i ′′ ⊆ A i ′′ , with # S = a ℓ − ( q − 1) − ⌊ ( a ℓ − ( q − 1)) / ( q + 1) ⌋ and finally a lo cally induced set (subset of A i ′ ) which has the form of a staircase eac h step ha ving height equal to 1 . W e obtain maximum non- ℓ -qualifying sets as ab ov e, but with the latter lo cally induced set b eing replaced by q − 1 vertical levels each containing q + 1 elements. One, of course can also in vestigate non- i -qualifying sets with 1 < i < ℓ = q , but w e shall refrain from that in the present exposition. Example 3. In this example w e consider q = 4 , s = 3 and u = ( q s − 1) ( q − 1)3 = 7 . W e obtain co des ov er F 64 of length n = ( u ( q − 1) + 1) q s − 1 = 352 . W e hav e 88 , 89 / ∈ H ∗ ( Q ) ⊆ ⟨ 7 , 16 ⟩ , but 87 , 90 , 91 , 92 ∈ H ∗ ( Q ) . W e shall consider a pair of nested co des of co-dimension ℓ = 3 , namely , C 2 ⊆ C 1 where C 2 = C L ( G, 87 Q ) and C 1 = C L ( G, 92 Q ) for which we note that all conditions of Corollary 1 are satisfied. W e hav e ι (90) = (3 , 6) , ι (91) = (0 , 13) , and ι (92) = (4 , 4) Applying (14) w e calculate M 1 ( C 1 , C 2 ) = min { 262 , 261 , 260 } = 260 (18) M 2 ( C 1 , C 2 ) = min { 289 , 276 , 274 } = 274 (19) M 3 ( C 1 , C 2 ) = 295 , (20) where the three v alues on the righ t hand side of (18) each corresp onds to a calculation concerning one of the v alues 90 , 91 , 92 the minim um being attained for 92 . The three v alues on the righ t hand sides of (19) eac h corresp onds to a calculation concerning a pair of such v alues, the minimum b eing attained for { 90 , 92 } . Finally , (20) corresp onds to a calculation concerning all three n umbers 90 , 91 , 92 at the same time. T o calculate the relative weigh ts of the dual codes w e apply (13) directly . W e obtain M 1 ( C ⊥ 2 , C ⊥ 1 ) = min { 14 , 28 , 25 } = 14 (21) M 2 ( C ⊥ 2 , C ⊥ 1 ) = min { 35 , 34 , 33 } = 33 (22) M 3 ( C ⊥ 2 , C ⊥ 1 ) = 40 . (23) On secret sharing from extended norm-trace curves 13 F rom the ab ov e in combination with (2) and (3) w e obtain t 1 = 259 , t 2 = 273 , t 3 = 294 , r 1 = 313 , r 2 = 320 , and r 3 = 339 . T o detect maximum non- 1 - qualifying sets we should apply Remark 1 to { ι (90) ι (91) , ι (92) } . Here, a 1 = 0 and therefore all the sets w e obtain consist of a lo cally induced set and nothing else. So leaving out the entire set of mem b ers from all departments but one large departmen t, there are limits as to which patterns of mem b ers from the remaining departmen t that could b e left out. T o detect maxim um non- 2 -qualifying sets we should consider { ι (90) , ι (92) } . Here, a 1 = 3 . Hence, we obtain sets consisting of a Cartesian pro duct S × Γ (2) i ′′ , # S = 3 in combination with a lo cally induced set. Finally , to detect maximum non- 1 -qualifying sets we should consider { ι (92) } . Here, a 1 = 4 , and we obtain sets that are the union of a Cartesian pro duct S × Γ (2) i ′′ , # S = 4 and a lo cally induced set. R emark 2. Revisiting Theorem 2 and the conditions therein keep the assumption γ 1 < · · · < γ w in H ∗ ( Q ) again with ι ( { γ 1 , . . . , γ w } ) = { ( a 1 , b 1 ) , . . . , ( a w , b w ) } and with the condition that a 1 < · · · < a w and b w < · · · < b 1 . But instead of requiring a 1 ≤ u ( q − 2) + 1 assume no w u ( q − 2) + 1 < a 1 . Insp ecting (10) we see that the last option do es not come into action as u ( q − 1) + 1 < a 1 + u whic h do es not correspond to the first coordinate of any ι ( δ ) where δ ∈ H ∗ ( Q ) . Hence, the righ t hand side of (10) b ecomes # { ( i, j ) | 0 ≤ i < u ( q − 1) + 1 , 0 ≤ j ≤ q s − 1 − 1 , ( a s , b s ) < p ( i, j ) for some s ∈ { 1 , . . . , w }} . But then the situation is similar to that of monomial-Cartesian co des regard- ing parameters of primary co des (see [11, Eq. (5)]) and the maxim um non- i - qualifying sets hav e a similar structure, how ev er with a little less freedom of c hoice (see [11, Thm. 6]). The adv antage of using co des from extended norm- trace curv es for such high v alue of a 1 do es not lie in the reconstruction num b ers, nor the structure of maxim um non- i -qualifying sets. Rather, it is the priv acy n umbers that are impro ved. W e illustrate Remark 2 with a final example. Example 4. W e first consider nested one-p oin t algebraic geometric co des ov er F 16 using the Hermitian curve X 5 − Y 4 − Y . W e hav e ι (66) = (14 , 2) and ι (67) = (13 , 3) b oth 66 and 67 b elonging to H ∗ ( Q ) as ι ( H ∗ ( Q )) = { ( i, j ) | 0 ≤ i < 16 , 0 ≤ j < 4 } . Hence, the co-dimension of C 2 = C L ( G, 65 Q ) ⊆ C L ( G, 67 Q ) = C 1 is ℓ = 2 . W e ha ve M 2 ( C 1 , C 2 ) = # { (14 , 2) , (13 , 3) , (14 , 3) , (15 , 2) , (15 , 3) } = 5 M 1 ( C 1 , C 2 ) = min { # { ((13 , 3) , (14 , 3) , (15 , 3) } , # { (14 , 2) , 14 , 3) , (15 , 2) , (15 , 3) }} = 3 14 O. Geil W e hav e maxim um non- 1 -qualifying sets of the form: all 64 points in the entire affine v ariety except { ( α ′ 1 , β 1 ) , ( α ′ 2 , β 1 ) , ( α ′ 1 , β 2 ) , ( α ′ 2 , β 2 ) , ( α ′ 3 , β 2 ) } where w e use the notation from Theorem 2 with the arbitrary enumeration from there. W e ha ve maximum non- 2 -qualifying sets of the form: all 64 p oints except { ( α ′ 1 , β 1 ) , ( α ′ 2 , β 1 ) , ( α ′ 3 , β 1 ) } . T urning to comparable monomial-Cartesian co des we consider as p oint set (affine v ariet y) now F 16 × S 2 = { P ′ 1 , . . . , P ′ 64 } i.e. # S 2 = 4 . W e write F 16 = { δ 1 , . . . , δ 16 } and S 2 = { ϵ 1 , . . . , ϵ 4 } and consider an ev aluation map ev : F 16 [ X, Y ] → F 64 16 giv en b y ev ( F ) = ( F ( P ′ 1 ) , . . . , F ( P ′ 64 )) . Define C ′ 2 = Span F 16 { ev ( X i Y j ) | 0 ≤ i < 16 , 0 ≤ j < 4 , 4 i + 5 j ≤ 65 } ⊆ C ′ 1 = Span F 16 { ev ( X i Y j ) | 0 ≤ i < 16 , 0 ≤ j < 4 , 4 i + 5 j ≤ 67 } . The relative parameters are the same as before (see [11, Eq. (5)]), but now we ha ve that the entire affine v ariety (still of size 64 ) min us { ( δ 1 , ϵ 1 ) , ( δ 2 , ϵ 1 ) , ( δ 1 , ϵ 2 ) , ( δ 2 , ϵ 2 ) , ( δ 3 , ϵ 2 ) } is a maximum non- 1 -qualifying set (see [11, Thm. 6]). Similarly , the entire affine v ariet y minus { ( δ 1 , ϵ 1 ) , ( δ 2 , ϵ 1 ) , ( δ 3 , ϵ 1 ) } is a maxim um non- 2 -qualifying set. Due to the man y wa ys one can enumerate the elements of F 16 and similarly enumerate the elements of S 2 the second lay er of security b ecomes m uch stronger for the secret sharing scheme defined from the latter pair of co des. In this pap er we only considered the case of not too large co-dimension of t wo one-point algebraic geometric codes C 2 ⊆ C 1 imp osing a situation where a 1 < · · · < a w and b w < · · · < b 1 . It is also p ossible to consider higher co- dimension for whic h we refer the reader to employ the last part of [4, Prf. of Thm. 3.2]. 6 Concluding remarks The second lay er of security resulting from families of maximum non- i -qualifying sets of systematic form w as first considered in [11] in the case of monomial- Cartesian co des. In the present work it was analyzed for schemes defined from nested one-p oint algebraic geometric codes defined o ver extended norm-tr ace curv es. There should b e other algebraic co des for whic h the resulting schemes ha ve a second lay er of security . 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