Adaptive Secure Non-zero Inner Product Encryption Scheme with Small-scale Public Parameters
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摘要: 内积加密是一种支持内积形式的函数加密，已有内积加密方案的公开参数规模较大，为解决该问题，该文基于素数阶熵扩张引理，利用双对偶向量空间(DPVS)技术，提出一个公开参数规模较小的具有适应安全性的内积加密方案。在方案的私钥生成算法中，将用户的属性向量的分量与主私钥向量结合，生成一个可与熵扩张引理中密钥分量结合的向量；在方案的加密算法中，将内积向量的每一分量与熵扩张引理中的部分密文分量结合。在素数阶熵扩张引理和
${\rm{MDDH}}_{k, k + 1}^n$ 困难假设成立条件下，证明了方案具有适应安全性。该文方案公开参数仅有10个群元素，与现有内积加密方案相比，公开参数规模最小。-
关键词:
- 内积加密 /
- 素数阶熵扩张引理 /
- $ {\rm{MDDH}}_{k,k+1}^n$困难假设 /
- 适应安全
Abstract: Inner product encryption is a kind of function encryption which supports inner product form. The public parameter scale of the existing inner product encryption schemes are large. In order to solve this problem, based on prime-order bilinear entropy expansion lemma and Double Pairing Vector Space (DPVS), an inner product encryption scheme is proposed in this paper, which has fewer public parameters and adaptive security. In the private key generation algorithm of the scheme, the components of the user’s attribute with the main private key are combined to generate a vector that can be combined with the key components in the entropy expansion lemma, and in encryption algorithm of the scheme, each component of the inner product vector is combined with ciphertext component in the entropy expansion lemma. Finally, under the condition of prime order bilinear entropy extension lemma and$\textstyle{{\rm{MDDH}}_{k, k + 1}^n}$ difficult assumption, the adaptive secure of the scheme is proved. The proposed scheme has only 10 group elements as public parameters, which is the smallest compared with the existing inner product encryption schemes. -
表 1
${\rm{Game}}$ 序列Game ct sk $\kappa < i$ $\kappa = i$ $\kappa > i$ 0 标准 标准 0’ 熵扩张 熵扩张 $i$ 熵扩张 半功能 熵扩张 熵扩张 $i,1$ – – 伪标准 – $i,2$ – – 伪半功能 – $i,3$ – – 半功能 – Final 随机消息 半功能 
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表 2 与现有内积加密方案的数据长度比较
方案 公开参数长度 私钥长度 密文长度 安全性假设 安全性 文献[5] $(4{n^2} + 3)|{G_1}|$ $(2n + 1)|{G_1}|$ $(2n + 1)|{G_1}|$ 2 variants of GSD 选择安全 文献[7] $(4{n^2} + 2n)|{G_1}|$ $(2n + 3)|{G_1}|$ $(2n + 3)|{G_1}|{\rm{ + }}|{G_T}|$ n-eDDH 适应安全 文献[8] $(4{n^2} + 3)|{G_1}|$ $(3n + 2)|{G_1}|$ $(3n + 2)|{G_1}| + |{G_T}|$ DLIN 适应安全 文献[9](type1) $105|{G_1}|$ $(3n + 2)|{G_1}|$ $(3n + 2)|{G_1}| + |{G_T}|$ DLIN 适应安全 文献[10] $28|{G_1}|$ $7n|{G_2}|{\rm{ + }}\alpha $ $7n|{G_1}|$ SXDH 适应安全 本方案 $9|{G_1}| + {G_T}$ $8n|{G_2}|$ $(5n + 3)|{G_1}|{\rm{ + |}}{G_T}{\rm{|}}$ ${\rm{MDDH}}_{k,k + 1}^n$ 适应安全 注：其中n表示系统属性的个数，$|{G_1}|,|{G_2}|,|{G_T}|$分别表示${G_1},{G_2},{G_T}$中群元素的长度。
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