text: ∈ N ∈ N It can be checked that (v, u) (u, v) and (u, v) / (u, v). Furthermore, in the special cases σ σ above we have the following: →− N − { } [P1] (u, v) = σ (u, v) . σ →− ∈ × N × ∩ [P2] If (u, v) A B then (u, v) = (B A ) σ . i i σ i i We can now formulate the sampling algorithm (see Algorithm 3). { } ⊆ ∈ Algorithm 3 Sampling from A(f , σ ) for M = e , . . . , e σ Ω − M k 1 k k 1: for i = k, k 1, . . . , 1 do { } 2: choose orientation (u, v) of −e−d−g−e−e−i −=−−→u, v (cid:48) (cid:48) ∈ N − { } 3: select (u , v ) σi(u, v) (cid:48) e1(cid:48), . . . , ei−1 uniformly at random 4: set σi−1 = Swapσi((u, v), (u , v )) 5: end for 6: return σ0 Let us verify the correctness of this algorithm. Using the deﬁnitions of ψˆ and Swap , the σ following fact can be easily checked. { } ∈ ∈ (cid:48) (cid:48) ∈ N (cid:48) (cid:48) Lemma{3{1. (a})} Suppose that u, v σi Ω, (u , v ) σi(u, v) and σi−1 = Swapσi((u, v), (u , v )). Then ψˆ( u, v , σ − ) = σ . i 1 i {{ }} { } ∈ ∈ (b) Conversely, suppose that ψˆ( u, v , σ − ) = σ for e = u, v E and σ − Ω. Then there (cid:48) (cid:48) ∈ N i 1 i i (cid:48) (cid:48) i 1 e{xi(cid:48)sts(cid:48)}un∈ique (−u ,{v )} σi(u, v) such that σi−1 = Swapσi((u, v), (u , v )). Furthermore, it satisﬁes u , v / σ − e . i 1 i Using this lemma, we can now show establish correctness of the sampling procedure. We say that two executions of Algorithm 3 are distinct if they made diﬀerent choices in line 3 for some ∈ i [k]. ∈ Proposition 32. Algorithm 3 is well-deﬁned, i.e. in line 3 we have e σ . It can generate object ∈ i i σ Ω if and only if ψˆ(M, σ ) = σ . Finally, distinct executions produce distinct outputs. 0 0 k Proof. The proof will have two parts corresponding to two directions. (a) Let σ , σ − , . . . be the sequence of objects produced by the algorithm. We will show using k k 1 { } ⊆ induction on i = k, . . . , 1, 0 that e , . . . , e σ (and therefore line 3 for index i is well-deﬁned) { } 1 i i ∈ and ψˆ( e , . . . , e , σ ) = σ . The base case i = k is trivial. Suppose the claim holds for i [k], i+1 k− i k { } ⊆ let us show it for i 1. We have e , . . . , e σ by the induction hypothesis; inspecting the rule (cid:48) (cid:48) 1 { i i } ⊆ for choosing (u , v ), we conclude that e , . . . , e − σ − . For the second claim we can write 1 i 1 i 1 { } { } { } { } ψˆ( e , . . . , e , σ − ) = ψˆ( e , . . . , e , ψˆ( e , σ − )) = ψˆ( e , . . . , e , σ ) = σ i k i 1 i+1 k i i 1 i+1 k i k where the ﬁrst equality is by the deﬁnition of ψˆ, the second is by Lemma 31(a) and third is by the induction hypothesis. This concludes the argument. ∈ { } (b) Suppose that ψˆ(M, σ ) = σ for σ Ω and M = e , . . . , e . Deﬁne objects σ , . . . , σ 0 k 0 1 k 1 k as in (13). We claim that Algorithm 3 can replicate this sequence (in the reverse order). Indeed, (cid:48) (cid:48) ∈ N { (cid:48) (cid:48)} ∈ − { } by Lemma 31(b) it suﬃces to show−−t−h−a−t−−fo−r−→any (u , v ) (u, v) with u ,−v−−−/−−σ−−−−→ e (cid:48) (cid:48) ∈ N − { } σi (cid:48) (cid:48) ∈ { i 1 } i we also have (u , v ) (u, v) e , . . . , e − . Suppose not, then (u , v ) e , . . . , e − . By σi{ 1 } ⊆ i 1 { (cid:48) (cid:48)} ∈ { (cid:48) 1 (cid:48)} i 1 Proposition 30(a) we have e , . . . , e − σ − , and so u , v σ−−−−.−−T−h−u−s−,→u , v = e . But e 1 i 1 i 1 i 1 i i { } (cid:48) (cid:48) ∈ { } does not appear in e , . . . , e − , and so we cannot have (u , v ) e , . . . , e − - a contradiction. 1 i 1 1 i 1 22 ##LTLine## ##LTLine## ##LTLine##

text: N

text: N

text: It

text: can

text: be

text: checked

text: that

text: (v,

text: u)

text: (u,

text: v)

text: and

text: (u,

text: v)

text: (u,

text: v).

text: Furthermore,

text: in

text: the

text: special

text: cases

text: σ

text: σ

text: above

text: we

text: have

text: the

text: following:

label: list

text: N

text: (u,

text: v)

text: σ

text: (u,

text: v)

text: σ

text: N

text: If

text: (u,

text: v)

text: A

text: B

text: then

text: (u,

text: v)

text: (B

text: A

text: σ

text: i

text: i

text: σ

text: i

text: i

text: We

text: can

text: now

text: formulate

text: the

text: sampling

text: algorithm

text: (see

text: Algorithm

text: Algorithm

text: Sampling

text: from

text: A(f

text: σ

text: for

text: M

text: e

text: e

text: σ

text: Ω

text: M

text: k

text: k

text: k

text: for

text: i

text: k,

text: k

text: do

text: choose

text: orientation

text: (u,

text: v)

text: of

text: −e−d−g−e−e−i

text: −=−−→u,

text: v

text: N

text: select

text: (u

text: v

text: σi(u,

text: v)

text: uniformly

text: at

text: random

text: set

text: σi−1

text: Swapσi((u,

text: v),

text: (u

text: v

text: end

text: for

text: return

text: σ0

text: Let

text: us

text: verify

text: the

text: correctness

text: of

text: this

text: algorithm.

text: Using

text: the

text: deﬁnitions

text: of

text: ψˆ

text: and

text: Swap

text: the

text: σ

text: following

text: fact

text: can

text: be

text: easily

text: checked.

text: N

text: (a})}

text: Suppose

text: that

text: u,

text: v

text: σi

text: Ω,

text: (u

text: v

text: σi(u,

text: v)

text: and

text: σi−1

text: Swapσi((u,

text: v),

text: (u

text: v

text: Then

text: ψˆ(

text: u,

text: v

text: σ

text: σ

text: i

text: i

text: (b)

text: Conversely,

text: suppose

text: that

text: ψˆ(

text: u,

text: v

text: σ

text: σ

text: for

text: e

text: u,

text: v

text: E

text: and

text: σ

text: Ω.

text: Then

text: there

text: N

text: i

text: i

text: i

text: i

text: (−u

text: ,{v

text: σi(u,

text: v)

text: such

text: that

text: σi−1

text: Swapσi((u,

text: v),

text: (u

text: v

text: Furthermore,

text: it

text: satisﬁes

text: u

text: v

text: σ

text: e

text: i

text: i

text: Using

text: this

text: lemma,

text: we

text: can

text: now

text: show

text: establish

text: correctness

text: of

text: the

text: sampling

text: procedure.

text: We

text: say

text: that

text: two

text: executions

text: of

text: Algorithm

text: are

text: distinct

text: if

text: they

text: made

text: diﬀerent

text: choices

text: in

text: line

text: for

text: some

text: i

text: [k].

text: Proposition

text: Algorithm

text: is

text: well-deﬁned,

text: i.e.

text: in

text: line

text: we

text: have

text: e

text: σ

text: It

text: can

text: generate

text: object

text: i

text: i

text: σ

text: Ω

text: if

text: and

text: only

text: if

text: ψˆ(M,

text: σ

text: σ

text: Finally,

text: distinct

text: executions

text: produce

text: distinct

text: outputs.

text: k

text: Proof.

text: The

text: proof

text: will

text: have

text: two

text: parts

text: corresponding

text: to

text: two

text: directions.

text: (a)

text: Let

text: σ

text: σ

text: be

text: the

text: sequence

text: of

text: objects

text: produced

text: by

text: the

text: algorithm.

text: We

text: will

text: show

text: using

text: k

text: k

text: induction

text: on

text: i

text: k,

text: that

text: e

text: e

text: σ

text: (and

text: therefore

text: line

text: for

text: index

text: i

text: is

text: well-deﬁned)

text: i

text: i

text: and

text: ψˆ(

text: e

text: e

text: σ

text: σ

text: The

text: base

text: case

text: i

text: k

text: is

text: trivial.

text: Suppose

text: the

text: claim

text: holds

text: for

text: i

text: [k],

text: k−

text: i

text: k

text: let

text: us

text: show

text: it

text: for

text: i

text: We

text: have

text: e

text: e

text: σ

text: by

text: the

text: induction

text: hypothesis;

text: inspecting

text: the

text: rule

text: i

text: i

text: for

text: choosing

text: (u

text: v

text: we

text: conclude

text: that

text: e

text: e

text: σ

text: For

text: the

text: second

text: claim

text: we

text: can

text: write

text: i

text: i

text: ψˆ(

text: e

text: e

text: σ

text: ψˆ(

text: e

text: e

text: ψˆ(

text: e

text: σ

text: ψˆ(

text: e

text: e

text: σ

text: σ

text: i

text: k

text: i

text: k

text: i

text: i

text: k

text: i

text: k

text: where

text: the

text: ﬁrst

text: equality

text: is

text: by

text: the

text: deﬁnition

text: of

text: ψˆ,

text: the

text: second

text: is

text: by

text: Lemma

text: and

text: third

text: is

text: by

text: the

text: induction

text: hypothesis.

text: This

text: concludes

text: the

text: argument.

text: (b)

text: Suppose

text: that

text: ψˆ(M,

text: σ

text: σ

text: for

text: σ

text: Ω

text: and

text: M

text: e

text: e

text: Deﬁne

text: objects

text: σ

text: σ

text: k

text: k

text: k

text: as

text: in

text: We

text: claim

text: that

text: Algorithm

text: can

text: replicate

text: this

text: sequence

text: (in

text: the

text: reverse

text: order).

text: Indeed,

text: N

text: by

text: Lemma

text: it

text: suﬃces

text: to

text: show−−t−h−a−t−−fo−r−→any

text: (u

text: v

text: (u,

text: v)

text: with

text: u

text: ,−v−−−/−−σ−−−−→

text: e

text: N

text: σi

text: i

text: i

text: we

text: also

text: have

text: (u

text: v

text: (u,

text: v)

text: e

text: e

text: Suppose

text: not,

text: then

text: (u

text: v

text: e

text: e

text: By

text: σi{

text: i

text: i

text: Proposition

text: we

text: have

text: e

text: e

text: σ

text: and

text: so

text: u

text: v

text: σ−−−−.−−T−h−u−s−,→u

text: v

text: e

text: But

text: e

text: i

text: i

text: i

text: i

text: i

text: does

text: not

text: appear

text: in

text: e

text: e

text: and

text: so

text: we

text: cannot

text: have

text: (u

text: v

text: e

text: e

text: a

text: contradiction.

text: i

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: • | | ≥ − ∈ s 2, S = I2 Γ(I1) and (I2, . . . , Is) Stab. ∅ ∞ It can be seen that Stab(R, , t, ) = Stab(R, t) for an independent set R. ≥ ≥ We say that a tuple (R, S, t, (cid:96)) is valid if (R, S) is an independent pair and t 0, (cid:96) 1 are (cid:88) integers. We will prove the following for any valid tuple (R, S, t, (cid:96)): ≤ · λ µ(R)µ(S) θt (19) ϕ ∈ ϕ Stab(R,S,t,(cid:96)) (cid:118) Let us introduce a partial order on tuples (R, S, t, (cid:96)) as the lexicographical order on vectors | | ((cid:96), R ) (the ﬁrst component is more signiﬁcant). We use induction on this partial order. The ∅ ≤ | | base case is given by a tuple (R, S, t, 1). We can assume that S = and t R (otherwise (cid:88) (cid:89) (cid:89) Stab(R, S, t, 1) is empty). In this case Stab(R, S, t, 1) contains a single sequence ϕ = (R), and so ≤ · · | | ≤ · λ = λ (µ θ) = µ(R) θ R µ(R) θt ϕ f f ∈ ∅ ∈ ∈ ϕ Stab(R, ,t,1) f R f R ≤ · ∅ where we used inequality λ µ θ that follows from condition (2b) with S = . f f ≥ Now consider a valid tuple (R, S, t, (cid:96)) with (cid:96) 2, and assume that the claim holds for lower tuples. Two cases are possible. • ∅ ∅ ∅ − R = . We ha∅ve a natura{(cid:88)l i∅somorphism b| etwee(cid:88)n sets ∈Stab( , S,∅t, (cid:96)) a−nd S}tab(S, , t, (cid:96) 1), namely Stab( , S, t, (cid:96)) = ( , I1, . . . , Ir) (I1, . . . , Ir) Stab(S, , t, (cid:96) 1) . This gives ≤ · λ = λ µ(S) θt ϕ ϕ ∈ ∅ ∈ ∅ − ϕ Stab( ,S,t,(cid:96)) ϕ Stab(S, ,t,(cid:96) 1) where in the last inequality we used the induction hypothesis. • (cid:54) ∅ ∈ − − { } ∈ R = . Pick f R, and denote R = R f . Let Γ˜(f) be the set of ﬂaws g Γ(f) that ∈ (cid:48) (i) can occur in the second set of a sequence ϕ Stab(R, S, t, (cid:96)), and (ii) do belong to Γ(f ) (cid:48) ∈ − { ∈ | (cid:28) (cid:48) (cid:48) ∈ ∪ } for any f R . Formally, Γ˜(f) = g Γ(f) g f for all f R S . We can write (cid:88) (cid:88) (cid:88) · λϕ = (cid:88) λf λϕ ∈ ∈ ∈ − ∪ − ϕ Stab(R,S,t,(cid:96)) T Ind(Γ˜(f)) ϕ Stab(R ,S T,t 1,(cid:96)) (≤a) − ∪ · − λf µ(R )µ(S(cid:88) T ) θt 1 ∈ T Ind(Γ˜(f)) − · λ (=b) µ(R)µ(S)θt 1 f µ(T ) µ f ∈ T Ind(Γ˜(f)) (≤c) − · µ(R)µ(S)θt 1 θ − ∪ where (a) is by the induction hypothesis, (b) is true since µ(R ) = µ(R)/µ and µ(S T ) = f µ(S)µ(T ), and (c) follows from condition (2b). 26 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: s

text: S

text: and

text: Is)

text: Stab.

label: paragraph

text: It

text: can

text: be

text: seen

text: that

text: Stab(R,

text: t,

text: Stab(R,

text: t)

text: for

text: an

text: independent

text: set

text: R.

text: We

text: say

text: that

text: a

text: tuple

text: (R,

text: S,

text: t,

text: is

text: valid

text: if

text: (R,

text: S)

text: is

text: an

text: independent

text: pair

text: and

text: t

text: are

label: equation

text: integers.

text: We

text: will

text: prove

text: the

text: following

text: for

text: any

text: valid

text: tuple

text: (R,

text: S,

text: t,

text: λ

text: µ(R)µ(S)

text: θt

text: ϕ

text: ϕ

text: Let

text: us

text: introduce

text: a

text: partial

text: order

text: on

text: tuples

text: (R,

text: S,

text: t,

text: as

text: the

text: lexicographical

text: order

text: on

text: vectors

text: R

text: (the

text: ﬁrst

text: component

text: is

text: more

text: signiﬁcant).

text: We

text: use

text: induction

text: on

text: this

text: partial

text: order.

text: The

text: base

text: case

text: is

text: given

text: by

text: a

text: tuple

text: (R,

text: S,

text: t,

text: We

text: can

text: assume

text: that

text: S

text: and

text: t

text: R

text: (otherwise

text: Stab(R,

text: S,

text: t,

text: is

text: empty).

text: In

text: this

text: case

text: Stab(R,

text: S,

text: t,

text: contains

text: a

text: single

text: sequence

text: ϕ

text: (R),

text: and

text: so

text: λ

text: λ

text: (µ

text: θ)

text: µ(R)

text: θ

text: R

text: µ(R)

text: θt

text: ϕ

text: f

text: f

text: ϕ

text: Stab(R,

text: f

text: R

text: f

text: R

text: where

text: we

text: used

text: inequality

text: λ

text: µ

text: θ

text: that

text: follows

text: from

text: condition

text: with

text: S

text: f

text: f

text: Now

text: consider

text: a

text: valid

text: tuple

text: (R,

text: S,

text: t,

text: with

text: and

text: assume

text: that

text: the

text: claim

text: holds

text: for

text: lower

text: tuples.

text: Two

text: cases

text: are

text: possible.

text: R

text: We

text: ha∅ve

text: a

text: i∅somorphism

text: b|

text: sets

text: ∈Stab(

text: S,∅t,

text: a−nd

text: S}tab(S,

text: t,

text: namely

text: Stab(

text: S,

text: t,

text: Ir)

text: Ir)

text: Stab(S,

text: t,

text: This

text: gives

text: λ

text: λ

text: µ(S)

text: θt

text: ϕ

text: ϕ

text: ϕ

text: Stab(

text: ϕ

text: Stab(S,

text: where

text: in

text: the

text: last

text: inequality

text: we

text: used

text: the

text: induction

text: hypothesis.

text: R

text: Pick

text: f

text: R,

text: and

text: denote

text: R

text: R

text: f

text: Let

text: Γ˜(f)

text: be

text: the

text: set

text: of

text: ﬂaws

text: g

text: Γ(f)

text: that

text: (i)

text: can

text: occur

text: in

text: the

text: second

text: set

text: of

text: a

text: sequence

text: ϕ

text: Stab(R,

text: S,

text: t,

text: and

text: (ii)

text: do

text: belong

text: to

text: Γ(f

text: for

text: any

text: f

text: R

text: Formally,

text: Γ˜(f)

text: g

text: Γ(f)

text: g

text: f

text: for

text: all

text: f

text: R

text: S

text: We

text: can

text: write

text: λϕ

text: λf

text: λϕ

text: ϕ

text: T

text: Ind(Γ˜(f))

text: ϕ

text: Stab(R

text: ,S

text: T,t

text: (≤a)

text: λf

text: µ(R

text: T

text: θt

text: T

text: Ind(Γ˜(f))

text: λ

text: (=b)

text: µ(R)µ(S)θt

text: f

text: µ(T

text: µ

text: f

text: T

text: Ind(Γ˜(f))

text: (≤c)

text: µ(R)µ(S)θt

text: θ

text: where

text: (a)

text: is

text: by

text: the

text: induction

text: hypothesis,

text: (b)

text: is

text: true

text: since

text: µ(R

text: µ(R)/µ

text: and

text: µ(S

text: T

text: f

text: µ(S)µ(T

text: and

text: (c)

text: follows

text: from

text: condition

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [15] Austin Mohr. Applications of the lopsided Lov´asz local lemma regarding hypergraphs. PhD thesis, University of South Carolina, 2013. [16] Robin A. Moser and G´abor Tardos. A constructive proof of the general Lov´asz local lemma. J. ACM, 57(2), 2010. [17] Sokol Ndreca, Aldo Procacci, and Benedetto Scoppola. Improved bounds on coloring of graphs. European Journal of Combinatorics, 33(4):592–609, 2012. [18] Wesley Pegden. An extension of the Moser-Tardos algorithmic local lemma. SIAM Journal on Discrete Mathematics, 28(2):911–917, 2014. [19] James B. Shearer. On a problem of Spencer. Combinatorica, 5(3):241–245, 1985. [20] Mario Szegedy. The Lov´asz local lemma - a survey. In Andrei A. Bulatov and Arseny M. Shur, editors, Computer Science – Theory and Applications, volume 7913, pages 1–11. Springer, Lecture Notes in Computer Science, 2013. 28

label: reference

text: Austin

text: Mohr.

text: Applications

text: of

text: the

text: lopsided

text: Lov´asz

text: local

text: lemma

text: regarding

text: hypergraphs.

text: PhD

text: thesis,

text: University

text: of

text: South

text: Carolina,

text: Robin

text: A.

text: Moser

text: and

text: G´abor

text: Tardos.

text: A

text: constructive

text: proof

text: of

text: the

text: general

text: Lov´asz

text: local

text: lemma.

text: J.

text: ACM,

text: Sokol

text: Ndreca,

text: Aldo

text: Procacci,

text: and

text: Benedetto

text: Scoppola.

text: Improved

text: bounds

text: on

text: coloring

text: of

text: graphs.

text: European

text: Journal

text: of

text: Combinatorics,

text: Wesley

text: Pegden.

text: An

text: extension

text: of

text: the

text: Moser-Tardos

text: algorithmic

text: local

text: lemma.

text: SIAM

text: Journal

text: on

text: Discrete

text: Mathematics,

text: James

text: B.

text: Shearer.

text: On

text: a

text: problem

text: of

text: Spencer.

text: Combinatorica,

text: Mario

text: Szegedy.

text: The

text: Lov´asz

text: local

text: lemma

text: a

text: survey.

text: In

text: Andrei

text: A.

text: Bulatov

text: and

text: Arseny

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text: Shur,

text: editors,

text: Computer

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text: volume

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text: 1 On the Security of MTA-OTIBASs (Multiple-TA One-Time Identity-Based Aggregate Signatures) Lei Zhang, Member, IEEE, Qianhong Wu, Member, IEEE, Josep Domingo-Ferrer, Fellow, IEEE Bo Qin, Chuanyan Hu Abstract— In [3] the authors proposed a new aggregate signature scheme referred to as multiple-TA (trusted authority) one-time identity-based aggregate signature (MTA-OTIBAS). Further, they gave a concrete MTA-OTIBAS scheme. We recall here the deﬁnition of MTA-OTIBAS and the concrete proposed scheme. Then we prove that our MTA-OTIBAS concrete scheme is existentially unforgeable against adaptively chosen-message attacks in the random oracle model under the co-CDH problem assumption. Index Terms—Identity based cryptosystem, Signature, Aggregate signature (cid:70) 1 INTRODUCTION 2 DEFINITION OF MTA-OTIBAS In [3] we proposed a new aggregate signature scheme An MTA-OTIBAS scheme consists of six algorithms, referred to as multiple-TA (trusted authority) one-time i.e., Root.Setup, LowLevel.Setup, Extract, Sign, Aggregate, identity-based aggregate signature (MTA-OTIBAS). Further, and Verify. Root.Setup is run by the root TA to gener- we gave a concrete MTA-OTIBAS scheme. We ﬁrst recall the ate the global system parameters and system master key. notion of MTA-OTIBAS; we then recall its formal deﬁnition LowLevel.Setup is an interactive protocol run between a and the concrete scheme proposed in [3]. Then, we give the lower-level TA and the root TA. It generates the secret key, detailed security proof of MTA-OTIBAS (not given in [3]). public key and certiﬁcate of the lower-level TA. Extract An MTA-OTIBAS scheme has the following features. takes as input a lower-level TA’s secret key and a signer’s Firstly, each user’s public key is his identity, so no certiﬁcate identity, and outputs a private key for the signer. Sign takes is needed on the public key, which avoids the certiﬁcate as input a signer’s identity, his private key, the certiﬁcate of management overhead. Secondly, a signer’s private key the signer’s corresponding lower-level TA and any message, (corresponding to an identity and a lower-level TA) is re- and outputs a signature on the message. The signature is stricted to be used only once; after that, the signer’s private only valid under the signer’s identity and the certiﬁcate of key should be updated. Thirdly, the MTA-OTIBAS scheme his corresponding lower-level TA. A restriction here is that also allows signature aggregation and fast veriﬁcation, i.e., a private key corresponding to a speciﬁc identity issued n signatures can be aggregated into a single short signature by a lower-level TA can be used only once. However, the (even signatures generated by signers enrolled by different same identity can be enrolled by different lower-level TAs. lower-level TAs), which greatly saves storage space, and can This implies that the corruption of a lower-level TA does be veriﬁed simultaneously. not inﬂuence the signers enrolled by other lower-level TAs. We recall the formal deﬁnition of MTA-OTIBAS in Sec- Aggregate is used to aggregate n message-signature pairs tion . In Section we recall the concrete MTA-OTIBAS generated by the Sign procedure into a single signature, i.e., scheme. Then in Section we prove that our MTA-OTIBAS an aggregate signature. Verify is used to check the validity concrete scheme is existentially unforgeable against adap- of an aggregate signature. It takes as input n messages, the tively chosen-message attacks in the random oracle model corresponding aggregate signature, n identities enrolled by under the co-CDH problem assumption. l lower-level TAs, and outputs 1 or 0 to represent whether the aggregate signature is valid or not. • Lei Zhang and Chuanyan Hu are with the Shanghai Key Laboratory of 3 A CONCRETE MTA-OTIBAS SCHEME Trustworthy Computing, East China Normal University, and the State Key Laboratory of Integrated Services Networks, Xidian University; Qian- Our MTA-OTIBAS scheme is realized using bilinear hong Wu is with the School of Electronic and Information Engineering, maps which are widely employed in identity-based cryp- Beihang University, and the State Key Laboratory of Information Security G ×G → G tosystems. A map eˆ : is called a bilinear map (JIonsesptitDutoemoinfgIon-fFoerrmreartiiosnwEitnhgtihneeeDrinepga,rtCmheinntesoef ACocamdpeumtyeroEfnSgcinieenecreins)g; if eˆ(g , g ) (cid:54)= 1 and eˆ(gα1, gβ)2= eˆ(gT, g )αβ for all α, β ∈ Z∗, and Mathematics, Universitat Rovira i Virgili; Bo Qin is with the Key where1 G2 , G are two1cyc2lic group1s o2f prime order q, Gq 1 2 T Laboratory of Data Engineering and Knowledge Engineering, Ministry of is a multiplicative cyclic group of the same order, g is a Education, School of Information, Renmin University of China, the Key G G 1 Laboratory of Cryptologic Technology and Information Security, Min- generator of 1, and g2 is a generator of 2. By exploiting istry of Education, Shandong University, and the State Key Laboratory bilinear maps, we implement our MTA-OTIBAS scheme. of Cryptology (e-mail: leizhang@sei.ecnu.edu.cn, qhwu@xidian.edu.cn, josep.domingo@urv.cat, bo.qin@ruc.edu.cn, chuanyanhu@ecnu.edu.cn). Root.Setup: The root TA runs this algorithm to generate the system parameters as follows: ##LTLine## ##LTLine## ##LTLine##

font: PSBFWJ+URWPalladioL-Roma

text: On

label: title

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text: the

text: Security

text: of

text: MTA-OTIBASs

text: (Multiple-TA

text: One-Time

text: Identity-Based

text: Aggregate

text: Signatures)

text: Lei

label: author

text: Zhang,

text: Member,

font: JCKFXU+NimbusSanL-ReguItal

text: IEEE,

text: Qianhong

text: Wu,

text: Member,

text: IEEE,

text: Josep

text: Domingo-Ferrer,

text: Fellow,

text: IEEE

text: Bo

text: Qin,

text: Chuanyan

text: Hu

text: Abstract—

font: DYPPSW+NimbusSanL-Bold

text: In

label: abstract

text: the

text: authors

text: proposed

text: a

text: new

text: aggregate

text: signature

text: scheme

text: referred

text: to

text: as

text: multiple-TA

text: (trusted

text: authority)

text: one-time

text: identity-based

text: aggregate

text: signature

text: (MTA-OTIBAS).

text: Further,

text: they

text: gave

text: a

text: concrete

text: MTA-OTIBAS

text: scheme.

text: We

text: recall

text: here

text: the

text: deﬁnition

text: of

text: MTA-OTIBAS

text: and

text: the

text: concrete

text: proposed

text: scheme.

text: Then

text: we

text: prove

text: that

text: our

text: MTA-OTIBAS

text: concrete

text: scheme

text: is

text: existentially

text: unforgeable

text: against

text: adaptively

text: chosen-message

text: attacks

text: in

text: the

text: random

text: oracle

text: model

text: under

text: the

text: co-CDH

text: problem

text: assumption.

text: Index

text: Terms—Identity

text: based

text: cryptosystem,

text: Signature,

text: Aggregate

text: signature

text: INTRODUCTION

text: DEFINITION

text: OF

text: MTA-OTIBAS

text: In

text: we

text: proposed

text: a

text: new

text: aggregate

text: signature

text: scheme

text: An

text: MTA-OTIBAS

text: scheme

text: consists

text: of

text: six

text: algorithms,

text: referred

text: to

text: as

text: multiple-TA

text: (trusted

text: authority)

text: one-time

text: i.e.,

text: Root.Setup,

text: LowLevel.Setup,

text: Extract,

text: Sign,

text: Aggregate,

text: identity-based

text: aggregate

text: signature

text: (MTA-OTIBAS).

text: Further,

text: and

text: Verify.

text: Root.Setup

text: is

text: run

text: by

text: the

text: root

text: TA

text: to

text: gener-

text: we

text: gave

text: a

text: concrete

text: MTA-OTIBAS

text: scheme.

text: We

text: ﬁrst

text: recall

text: the

text: ate

text: the

text: global

text: system

text: parameters

text: and

text: system

text: master

text: key.

text: notion

text: of

text: MTA-OTIBAS;

text: we

text: then

text: recall

text: its

text: formal

text: deﬁnition

text: LowLevel.Setup

text: is

text: an

text: interactive

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text: root

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text: detailed

text: security

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text: MTA-OTIBAS

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text: certiﬁcate

text: of

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text: lower-level

text: TA.

text: Extract

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text: MTA-OTIBAS

text: scheme

text: has

text: the

text: following

text: features.

text: takes

text: as

text: input

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text: lower-level

text: TA’s

text: secret

text: key

text: and

text: a

text: signer’s

text: Firstly,

text: each

text: user’s

text: public

text: key

text: is

text: his

text: identity,

text: so

text: no

text: certiﬁcate

text: identity,

text: and

text: outputs

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text: private

text: key

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text: the

text: signer.

text: Sign

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text: is

text: needed

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text: management

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text: Secondly,

text: a

text: signer’s

text: private

text: key

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text: corresponding

text: lower-level

text: TA

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text: any

text: message,

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text: TA)

text: is

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text: and

text: outputs

text: a

text: signature

text: on

text: the

text: message.

text: The

text: signature

text: is

text: stricted

text: to

text: be

text: used

text: only

text: once;

text: after

text: that,

text: the

text: signer’s

text: private

text: only

text: valid

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text: the

text: signer’s

text: identity

text: and

text: the

text: certiﬁcate

text: of

text: key

text: should

text: be

text: updated.

text: Thirdly,

text: the

text: MTA-OTIBAS

text: scheme

text: his

text: corresponding

text: lower-level

text: TA.

text: A

text: restriction

text: here

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text: that

text: also

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text: signature

text: aggregation

text: and

text: fast

text: veriﬁcation,

text: i.e.,

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text: private

text: key

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text: identity

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text: signatures

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text: short

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text: a

text: lower-level

text: TA

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text: be

text: used

text: only

text: once.

text: However,

text: the

text: (even

text: signatures

text: generated

text: by

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text: enrolled

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text: same

text: identity

text: can

text: be

text: enrolled

text: by

text: different

text: lower-level

text: TAs.

text: lower-level

text: TAs),

text: which

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text: space,

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text: implies

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text: corruption

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text: lower-level

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text: simultaneously.

text: not

text: inﬂuence

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text: signers

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text: other

text: lower-level

text: TAs.

text: We

text: recall

text: the

text: formal

text: deﬁnition

text: of

text: MTA-OTIBAS

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text: Sec-

text: Aggregate

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text: pairs

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text: signature,

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text: scheme.

text: Then

text: in

text: Section

text: we

text: prove

text: that

text: our

text: MTA-OTIBAS

text: an

text: aggregate

text: signature.

text: Verify

text: is

text: used

text: to

text: check

text: the

text: validity

text: concrete

text: scheme

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text: existentially

text: unforgeable

text: against

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text: co-CDH

text: problem

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text: l

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text: and

text: outputs

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text: whether

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text: aggregate

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text: valid

text: or

text: not.

text: Lei

text: Zhang

text: and

text: Chuanyan

text: Hu

text: are

text: with

text: the

text: Shanghai

text: Key

text: Laboratory

text: of

text: A

text: CONCRETE

text: MTA-OTIBAS

text: SCHEME

text: Trustworthy

text: Computing,

text: East

text: China

text: Normal

text: University,

text: and

text: the

text: State

text: Key

text: Laboratory

text: of

text: Integrated

text: Services

text: Networks,

text: Xidian

text: University;

text: Qian-

text: Our

text: MTA-OTIBAS

text: scheme

text: is

text: realized

text: using

text: bilinear

text: hong

text: Wu

text: is

text: with

text: the

text: School

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text: Electronic

text: and

text: Information

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text: maps

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text: of

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text: (e-mail:

text: leizhang@sei.ecnu.edu.cn,

text: qhwu@xidian.edu.cn,

text: josep.domingo@urv.cat,

text: bo.qin@ruc.edu.cn,

text: chuanyanhu@ecnu.edu.cn).

text: Root.Setup:

text: The

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text: algorithm

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text: E. Keiha¨nen et al.: Impact of beam deconvolution on noise properties Our calculations are carried out in harmonic space, which is Table 1. Noise parameters used in simulations: Knee fre- the natural domain for beam deconvolution. We construct a har- quency (fknee), slope (β), and white noise rms (σ). From Planck monic noise covariance matrix that describes the noise proper- Collaboration II (2015). ties of the harmonic a coeﬃcients, which are the primary out- X(cid:96)m put of artDeco. Because the problem is huge, it is not possible fknee β σ to construct the covariance matrix for all multipoles. In practice, Horn M S M S M S the noise covariance matrix approach is limited to multipoles (cid:96) (cid:47) 100. 70 GHz We further produce a pixel noise covariance matrix, which LFI-18 14.82 17.78 -1.060 -1.180 4.553 4.146 describes the noise properties of deconvolved sky maps, con- LFI-19 11.72 13.71 -1.207 -1.111 5.144 4.926 LFI-20 7.96 5.67 -1.198 -1.298 5.212 5.507 structed from the a coeﬃcients through spherical harmonic Xlm LFI-21 37.89 13.27 -1.247 -1.205 4.003 4.971 transform. A noise covariance matrix is an essential ingredient in LFI-22 9.68 14.80 -1.424 -1.237 4.356 4.715 many CMB power spectrum estimation methods. A method for LFI-23 29.65 59.03 -1.073 -1.211 4.476 4.790 producing a noise covariance matrix for un-deconvolved maps 44 GHz has been presented in Keskitalo et al. (2010). Its application to LFI-24 26.78 88.30 -0.942 -0.908 3.159 2.734 Planck LFI data is presented in Planck Collaboration VI (2015). LFI-25 20.07 46.37 -0.845 -0.904 2.834 2.698 The paper is structured as follows. In Sect. we perform a LFI-26 64.42 68.19 -0.918 -0.758 3.295 2.887 series of full-resolution simulations, to gain a general view of 30GHz the impact of deconvolution on residual noise. In Sect. we fo- cus on white noise. We derive a white noise covariance matrix, LFI-27 174.53 108.79 -0.927 -0.907 1.605 1.729 LFI-28 130.08 43.08 -0.931 -0.900 1.812 1.633 and compare its predictions with Monte Carlo simulations. The main result of this paper is the derivation of a full low-resolution noise covariance matrix, presented in Sect. . In Sect. we per- form a thorough validation of the harmonic covariance matrix through Monte Carlo simulations. We show results from χ2 tests and from noise bias comparison. In Sect. we perform analysis in pixel space. We summarise our results in Sect. . disk in “4D map” format. The four dimensions in this context re- fer to the three angles θ, φ, ψ which deﬁne the detector pointing, and pointing period index as fourth dimension. Pointing period 2. Simulations refers to a period of typical duration of 40 min, where the Planck We start by performing a series of full-resolution noise sim- scanning pattern follows a ﬁxed ring on the sky. ulations to assess the impact of deconvolution. Our simulations Madam generates a three-dimensional grid by discretising the mimic the data from the LFI instrument of Planck experiment. detector pointing angles. The pixelization obeys HEALPix pix- We created a time-ordered data stream, which we ﬁlled with sim- elization in θ and φ, and the ψ angle is divided uniformly into ulated noise, and fed it as input to the artDeco deconvolver. The Npsi bins. Another parameter Nside controls the resolution in θ, φ. simulations were run on Sisu, the Cray XC40 supercomputer of The TOI is binned on the grid according to its pointing, and CSC, Finland. the grid is stored by pointing period. The data volume is com- 2.1. Generating noise pressed typically by a factor of 20 compared to the full TOI. Radiometer-speciﬁc parameters required in subsequent analysis We regenerated the LFI detector pointing with LevelS soft- steps are stored as keywords in the 4D object header. For the ware (Reinecke et al. 2006). The simulated mission covers 4 years of data. We used the internal noise generator of the Madam present study we used resolution Nside=1024, Npsi=4096. At this resolution the 4D maps take roughly 8 GB of disk space per de- map-making code to generate a noise time stream. The noise tector. properties mimic those of the real Planck data. We considered the three LFI channels: 30 GHz, 44 GHz, and 70 GHz. The In white noise simulations, we use Madam only to generate noise parameters used in simulations were taken from Planck the noise, to apply the ﬂags, and to store the data stream in 4D Collaboration II (2015). For convenience, the parameters are map format. No destriping is applied. listed in Table 1. The 4D map is an intermediate data object that is useful for We consider two types of noise. In the ﬁrst case we generated many purposes. In particular, it can further be compressed into pure white noise, with variances taken from Table 1. The vari- ance is assumed constant in time, but varies between radiome- the “3D map” format that serves as input to artDeco. This step is trivial, it consists of coadding all pointing periods on a sin- ters. In the second series of simulations we generated realistic gle grid. Further, we can construct a binned map from the same 1/f noise, again with parameters from Table 1. We destriped the 4D map input. The binning operation represents “normal” map- noise stream with Madam, and stored the cleaned data on disk. making without beam deconvolution. The combined operation The data was then fed as input to artDeco. of destriping, 4D map production, and map binning, is nearly The destriping procedure follows the one applied to LFI equivalent to the operation of producing a destriped map directly mapmaking (Planck Collaboration VI 2015). We applied the same destriping mask, and used the same HEALPix map resolu- by Madam. The only diﬀerence is the discretization of the ψ an- gle, which at the chosen resolution N =4096 has a negligible tion N =1024 (3.44’), and same baseline length (1s for 70 GHz psi side eﬀect on the ﬁnal map. and 0.25 s for 30 GHz) as in real Planck mapmaking. We refer to these simulations as realistic noise simulations, in contrast to the To reduce the scatter in results, we produced several noise white noise simulations. In both simulations we applied realis- realizations for each case studied. At 30 GHz, we generated 40 tic Planck ﬂags to the TOI stream. This excludes roughly 8% of realizations for each case. At 44 GHz and 70 GHz, where the the data. Madam has the capacity of storing the destriped TOI on simulations are more demanding, we produced 10 realizations. 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: E.

text: Keiha¨nen

text: et

text: al.:

text: Impact

text: of

text: beam

text: deconvolution

text: on

text: noise

text: properties

text: Our

text: calculations

text: are

text: carried

text: out

text: in

text: harmonic

text: space,

text: which

text: is

text: Table

text: Noise

label: caption

text: parameters

text: used

text: in

text: simulations:

text: Knee

text: fre-

text: the

text: natural

text: domain

text: for

text: beam

text: deconvolution.

text: We

text: construct

text: a

text: har-

text: quency

text: (fknee),

text: slope

text: (β),

text: and

text: white

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text: rms

text: (σ).

text: From

text: Planck

text: monic

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text: covariance

text: matrix

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text: ]2)Kµ([ C ‘ sesac fo rebmuN E. Keiha¨nen et al.: Impact of beam deconvolution on noise properties 0.08 0.06 0.03 TT 0.06 EE 0.04 BB 0.02 0.04 TE×10 TB×10 0.02 0.02 EB×10 0.01 0.00 0.00 0.00 0.02 0.02 30GHz 44GHz 70GHz 0.04 0.04 0.01 100 101 100 101 100 101 Multipole ‘ Fig. 9. Predicted noise bias for LFI frequencies: TT, EE, BB, TE, T B, EB. The cross spectra have be multiplied by 10 to bring out the structure more clearly. χp2df Wn Ideal Real 30 GHz 1.0004 1.0016 40 30 1.0196 30 10 44 GHz 0.9998 1.0037 20 30 1.0138 10 10 70 GHz 0.9992 1.0002 00 10 20 30 40 30 1.0037 Fig. 10. Structure of the TT block of the 30 GHz NCVM at low multipoles ((cid:96) = 0 – 6). Index on x and y axis is (cid:96)2+(cid:96)+m. Positive 10 correlation is shown in red colour, negative in blue. 1.00 1.05 1.00 1.05 Reduced χ2 The aim of the ideal simulation set is to verify that the NCVM is constructed correctly. The realistic simulation set tests Fig. 11. Reduced χ2 distribution for 30 GHz, 44 GHz, 70 GHz, how accurately the matrix produced models the actual LFI data for white noise (left), ideal, and realistic simulation (right), for products, given the approximations involved. 100 realizations. Also shown is the χ2 distribution corresponding to an ideal result. The numbers within the plot give the mean value of the distribution. 5.2. Chi2 tests We validate the noise covariance matrices through χ2 tests. We generated 100 realizations of of harmonic vectors aX(cid:96)m, as tion becomes√Gaussian, with expectation value 1 and standard described above. For each vector and corresponding matrix we deviation of 2/Ndof, which for Ndof=7795 is 0.016. compute the value We compared the white noise covariance matrix of Eq. (11) − against white noise simulations, and the full noise matrix of Eq. χ2 = aTC 1a . (40) (29) against ideal and realistic simulations. Results from tests at N the three Planck LFI frequencies, 30, 44, and 70 GHz are shown dof in Fig. 11. We show a histogram of the χ2 values for 100 real- Here C is the covariance matrix, and N is the number of izations, and quote their mean. The white noise matrix models dof degrees of freedom. Multipole (cid:96) contributes 2(cid:96) + 1 degrees of the white noise residual nearly perfectly, χ2 mean ranging from freedom, except for the monopole and dipole, which are identi- 0.9992 to 1.0004, Comparison between the full matrix and ideal − cally zero for E and B. This yields in total Ndof=3((cid:96)max+1)2 8 simulation gives slightly larger values, from 1.0002 at 70 GHz degrees of freedom, which for (cid:96)max=50 gives Ndof=7795. For a to 1.0037 at 44 GHz, while the expected one-sigma deviation for matrix that perfectly describes the noise properties, the results 100 realizations is 0.0016. 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